hcju/proofwikips

1{"_id": "0", "title": "Closed Form for Triangular Numbers", "text": "The closed-form expression for the $n$th triangular number is: :$\\displaystyle T_n = \\sum_{i \\mathop = 1}^n i = \\frac {n \\paren {n + 1} } 2$"}2{"_id": "1", "title": "Sum of Sequence of Squares", "text": ":$\\displaystyle \\forall n \\in \\N: \\sum_{i \\mathop = 1}^n i^2 = \\frac {n \\paren {n + 1} \\paren {2 n + 1} } 6$"}3{"_id": "2", "title": "Union is Associative", "text": "Set union is associative: :$A \\cup \\paren {B \\cup C} = \\paren {A \\cup B} \\cup C$"}4{"_id": "3", "title": "Pythagoras's Theorem", "text": "Let $\\triangle ABC$ be a right triangle with $c$ as the hypotenuse. Then: :$a^2 + b^2 = c^2$"}5{"_id": "4", "title": "Euclid's Theorem", "text": "For any finite set of prime numbers, there exists a prime number not in that set. {{:Euclid:Proposition/IX/20}}"}6{"_id": "5", "title": "Limsup Squeeze Theorem", "text": "Let $\\left \\langle {x_n} \\right \\rangle$ and $\\left \\langle {y_n} \\right \\rangle$ be sequences in $\\R$. Let: : $(1): \\quad \\forall n \\ge n_0: \\left|{x_n}\\right| \\le y_n$ : $(2): \\quad \\displaystyle \\limsup_{n \\mathop \\to \\infty} \\left({y_n}\\right) = 0$, where $\\limsup$ denotes the limit superior. Then: : $\\displaystyle \\lim_{n \\mathop \\to \\infty} x_n = 0$"}7{"_id": "6", "title": "Square Root of Prime is Irrational", "text": "The square root of any prime number is irrational."}8{"_id": "7", "title": "Derivative of Exponential Function", "text": "Let $\\exp$ be the exponential function. Then: :$\\map {\\dfrac \\d {\\d x} } {\\exp x} = \\exp x$"}9{"_id": "8", "title": "Derivative of Sine Function", "text": ":$\\map {\\dfrac \\d {\\d x} } {\\sin x} = \\cos x$"}10{"_id": "9", "title": "Cauchy-Bunyakovsky-Schwarz Inequality", "text": "=== Semi-Inner Product Spaces === {{:Cauchy-Bunyakovsky-Schwarz Inequality/Inner Product Spaces}}"}11{"_id": "10", "title": "0.999...=1", "text": ":$0.999 \\ldots = 1$"}12{"_id": "12", "title": "Sum of Sequence of Cubes", "text": ":$\\displaystyle \\sum_{i \\mathop = 1}^n i^3 = \\paren {\\sum_{i \\mathop = 1}^n i}^2 = \\frac {n^2 \\paren {n + 1}^2} 4$"}13{"_id": "13", "title": "Law of Cosines", "text": "Let $\\triangle ABC$ be a triangle whose sides $a, b, c$ are such that $a$ is opposite $A$, $b$ is opposite $B$ and $c$ is opposite $C$. Then: :$c^2 = a^2 + b^2 - 2 a b \\cos C$"}14{"_id": "14", "title": "Euler's Formula", "text": "Let $z \\in \\C$ be a complex number. Then: :$e^{i z} = \\cos z + i \\sin z$"}15{"_id": "16", "title": "Lagrange's Theorem (Group Theory)", "text": "Let $G$ be a finite group. Let $H$ be a subgroup of $G$. Then: : $\\order H$ divides $\\order G$ where $\\order G$ and $\\order H$ are the order of $G$ and $H$ respectively. In fact: :$\\index G H = \\dfrac {\\order G} {\\order H}$ where $\\index G H$ is the index of $H$ in $G$. When $G$ is an infinite group, we can still interpret this theorem sensibly: :A subgroup of finite index in an infinite group is itself an infinite group. :A finite subgroup of an infinite group has infinite index."}16{"_id": "17", "title": "ProofWiki:Sandbox", "text": "$\\forall n \\in \\Z_{>0}: \\displaystyle \\sum_{k \\mathop = 0}^{n} \\binom {2n} {2k} E_{2n - 2k } = 0$ where $E_k$ denotes the $k$th Euler number."}17{"_id": "18", "title": "Carathéodory's Theorem (Analysis)", "text": "Let $I \\subseteq \\R$. Let $c \\in I$ be an interior point of $I$. {{Disambiguate|Definition:Interior Point}} {{explain|In this case, there appears not to be a definition for \"interior point\" which appropriately captures the gist of this. It is clearly a point inside a real interval, but the concept has not yet been defined.}} Let $f : I \\to \\R$ be a real function. Then $f$ is differentiable at $c$ {{iff}}: :There exists a real function $\\varphi : I \\to \\R$ that is continuous at $c$ and satisfies: :$(1): \\quad \\forall x \\in I: f \\left({x}\\right) - f \\left({c}\\right) = \\varphi \\left({x}\\right) \\left({x - c}\\right)$ :$(2): \\quad \\varphi \\left({c}\\right) = f' \\left({c}\\right)$"}18{"_id": "19", "title": "One-Step Subgroup Test", "text": "Let $\\struct {G, \\circ}$ be a group. Let $H$ be a subset of $G$. Then $\\struct {H, \\circ}$ is a subgroup of $\\struct {G, \\circ}$ {{iff}}: : $(1): \\quad H \\ne \\O$, that is, $H$ is non-empty : $(2): \\quad \\forall a, b \\in H: a \\circ b^{-1} \\in H$."}19{"_id": "20", "title": "Two-Step Subgroup Test", "text": "Let $\\struct {G, \\circ}$ be a group. Let $H$ be a subset of $G$. Then $\\struct {H, \\circ}$ is a subgroup of $\\struct {G, \\circ}$ {{iff}}: :$(1): \\quad H \\ne \\O$, that is, $H$ is non-empty :$(2): \\quad a, b \\in H \\implies a \\circ b \\in H$ :$(3): \\quad a \\in H \\implies a^{-1} \\in H$. That is, $\\struct {H, \\circ}$ is a subgroup of $\\struct {G, \\circ}$ {{iff}} $\\struct {H, \\circ}$ is a $H$ be a nonempty subset of $G$ which is: :closed under its operation and: :closed under inversion."}20{"_id": "21", "title": "Fundamental Theorem of Arithmetic", "text": "For every integer $n$ such that $n > 1$, $n$ can be expressed as the product of one or more primes, uniquely up to the order in which they appear."}21{"_id": "22", "title": "Euclid's Lemma", "text": "Let $a, b, c \\in \\Z$. Let $a \\divides b c$, where $\\divides$ denotes divisibility. Let $a \\perp b$, where $\\perp$ denotes relative primeness. Then $a \\divides c$."}22{"_id": "23", "title": "Fermat's Little Theorem", "text": "Let $p$ be a prime number. Let $n \\in \\Z_{>0}$ be a positive integer such that $p$ is not a divisor of $n$. Then: :$n^{p - 1} \\equiv 1 \\pmod p$"}23{"_id": "24", "title": "De Morgan's Laws (Set Theory)", "text": "{{:De Morgan's Laws (Set Theory)/Set Difference}}"}24{"_id": "25", "title": "De Moivre's Formula", "text": "Let $z \\in \\C$ be a complex number expressed in complex form: :$z = r \\paren {\\cos x + i \\sin x}$ Then: :$\\forall \\omega \\in \\C: \\paren {r \\paren {\\cos x + i \\sin x} }^\\omega = r^\\omega \\paren {\\map \\cos {\\omega x} + i \\, \\map \\sin {\\omega x} }$"}25{"_id": "26", "title": "First Isomorphism Theorem", "text": "=== Groups === {{:First Isomorphism Theorem/Groups}} === Rings === {{:First Isomorphism Theorem/Rings}}"}26{"_id": "27", "title": "Power Rule for Derivatives", "text": "Let $n \\in \\R$. Let $f: \\R \\to \\R$ be the real function defined as $\\map f x = x^n$. Then: :$\\map {f'} x = n x^{n - 1}$ everywhere that $\\map f x = x^n$ is defined. When $x = 0$ and $n = 0$, $\\map {f'} x$ is undefined."}27{"_id": "28", "title": "Basel Problem", "text": ":$\\displaystyle \\map \\zeta 2 = \\sum_{n \\mathop = 1}^\\infty {\\frac 1 {n^2} } = \\frac {\\pi^2} 6$ where $\\zeta$ denotes the Riemann zeta function."}28{"_id": "30", "title": "1+2+...+n+(n-1)+...+1 = n^2", "text": ":$\\forall n \\in \\N: 1 + 2 + \\cdots + n + \\paren {n - 1} + \\cdots + 1 = n^2$"}29{"_id": "31", "title": "Necessary Conditions for Existence of Skolem Sequence", "text": "A Skolem sequence of order $n$ can only exist if $n \\equiv 0, 1 \\pmod 4$."}30{"_id": "32", "title": "Divisibility by 9", "text": "A number expressed in decimal notation is divisible by $9$ {{iff}} the sum of its digits is divisible by $9$. That is: :$N = \\sqbrk {a_0 a_1 a_2 \\ldots a_n}_{10} = a_0 + a_1 10 + a_2 10^2 + \\cdots + a_n 10^n$ is divisible by $9$  {{iff}}: :$a_0 + a_1 + \\ldots + a_n$ is divisible by $9$."}31{"_id": "33", "title": "Existence of Rational Powers of Irrational Numbers", "text": "There exist irrational numbers $a$ and $b$ such that $a^b$ is rational."}32{"_id": "34", "title": "Intersection is Associative", "text": "Set intersection is associative: :$A \\cap \\paren {B \\cap C} = \\paren {A \\cap B} \\cap C$"}33{"_id": "35", "title": "Center of Symmetric Group is Trivial", "text": "Let $n \\in \\N$ be a natural number. Let $S_n$ denote the symmetric group of order $n$. Let $n \\ge 3$. Then the center $\\map Z {S_n}$ of $S_n$ is trivial."}34{"_id": "36", "title": "Square Root of 2 is Irrational", "text": ":$\\sqrt 2$ is irrational."}35{"_id": "38", "title": "Empty Set is Unique", "text": "The empty set is unique."}36{"_id": "39", "title": "Principle of Non-Contradiction", "text": "The '''Principle of Non-Contradiction''' is a valid deduction sequent in propositional logic."}37{"_id": "40", "title": "Rule of Substitution", "text": "Let $S$ be a sequent that has been proved. Then a proof can be found for any substitution instance of $S$."}38{"_id": "41", "title": "Rule of Sequent Introduction", "text": "Let the statements $P_1, P_2, \\ldots, P_n$ be conclusions in a proof, on various assumptions. Let $P_1, P_2, \\ldots, P_n \\vdash Q$ be a substitution instance of a sequent for which we already have a proof. {{explain|Question the use of \"substitution instance\": can we not \"just\" allow for $P_1, \\ldots, P_n$ to be \"just\" statements?}} Then we may introduce, at any stage of a proof (citing '''SI'''), either: :The conclusion $Q$ of the sequent already proved or: :A substitution instance of such a conclusion, together with a reference to the sequent that is being cited. This conclusion depend upon the pool of assumptions upon which $P_1, P_2, \\ldots, P_n \\vdash Q$ rests."}39{"_id": "42", "title": "Law of Identity", "text": "Every proposition entails itself:"}40{"_id": "43", "title": "Rule of Idempotence", "text": "The '''rule of idempotence''' is two-fold:"}41{"_id": "46", "title": "Hypothetical Syllogism", "text": "The '''(rule of the) hypothetical syllogism''' is a valid deduction sequent in propositional logic: :If we can conclude that $p$ implies $q$, and if we can also conclude that $q$ implies $r$, then we may infer that $p$ implies $r$."}42{"_id": "49", "title": "Extended Rule of Implication", "text": "Any sequent can be expressed as a theorem. That is: :$P_1, P_2, P_3, \\ldots, P_n \\vdash Q$ means the same thing as: :$\\vdash P_1 \\implies \\paren {P_2 \\implies \\paren {P_3 \\implies \\paren {\\ldots \\implies \\paren {P_n \\implies Q} \\ldots} } }$ The latter expression is known as the corresponding conditional of the former. Thus every sequent containing the symbol $\\vdash$ can, if so desired, be expressed in the form of a theorem which has $\\implies$."}43{"_id": "51", "title": "Equivalences are Interderivable", "text": "If two propositional formulas are interderivable, they are equivalent: :$\\paren {p \\dashv \\vdash q} \\dashv \\vdash \\paren {p \\iff q}$"}44{"_id": "52", "title": "Union is Commutative", "text": "Set union is commutative: :$S \\cup T = T \\cup S$"}45{"_id": "53", "title": "Intersection is Commutative", "text": "Set intersection is commutative: :$S \\cap T = T \\cap S$"}46{"_id": "54", "title": "Russell's Paradox", "text": "The comprehension principle leads to a contradiction."}47{"_id": "57", "title": "De Morgan's Laws (Logic)", "text": "{{:De Morgan's Laws (Logic)/Disjunction of Negations}}"}48{"_id": "62", "title": "Solution to Quadratic Equation", "text": "The quadratic equation of the form $a x^2 + b x + c = 0$ has solutions: :$x = \\dfrac {-b \\pm \\sqrt {b^2 - 4 a c} } {2 a}$"}49{"_id": "63", "title": "Normal Subgroup Test", "text": "Let $G$ be a group and $H \\le G$. Then $H$ is a normal subgroup of $G$ {{iff}}: :$\\forall x \\in G: x H x^{-1} \\subseteq H$."}50{"_id": "64", "title": "Equality of Ordered Pairs", "text": "Two ordered pairs are equal {{iff}} corresponding coordinates are equal: :$\\tuple {a, b} = \\tuple {c, d} \\iff a = c \\land b = d$"}51{"_id": "65", "title": "Identity is Unique", "text": "Let $\\left({S, \\circ}\\right)$ be an algebraic structure that has an identity element $e \\in S$. Then $e$ is unique."}52{"_id": "66", "title": "Cancellation Laws", "text": "Let $G$ be a group. Let $a, b, c \\in G$. Then the following hold: ;Right cancellation law :$b a = c a \\implies b = c$ ;Left cancellation law :$a b = a c \\implies b = c$"}53{"_id": "67", "title": "Center of Group is Normal Subgroup", "text": "The center $\\map Z G$ of any group $G$ is a normal subgroup of $G$ which is abelian."}54{"_id": "68", "title": "Composition of Relations is Associative", "text": "The composition of relations is an associative binary operation: :$\\paren {\\RR_3 \\circ \\RR_2} \\circ \\RR_1 = \\RR_3 \\circ \\paren {\\RR_2 \\circ \\RR_1}$"}55{"_id": "69", "title": "Intersection of Subsemigroups", "text": "Let $\\left({S, \\circ}\\right)$ be a semigroup. Let $\\left({T_1, \\circ}\\right)$ and $\\left({T_2, \\circ}\\right)$ be subsemigroups of $\\left({S, \\circ}\\right)$. Then the intersection of $\\left({T_1, \\circ}\\right)$ and $\\left({T_2, \\circ}\\right)$ is itself a subsemigroup of that $\\left({S, \\circ}\\right)$. If $\\left({T, \\circ}\\right)$ is that intersection of $\\left({T_1, \\circ}\\right)$ and $\\left({T_2, \\circ}\\right)$, it follows that $\\left({T, \\circ}\\right)$ is also a subsemigroup of both $\\left({T_1, \\circ}\\right)$ and $\\left({T_2, \\circ}\\right)$."}56{"_id": "70", "title": "Cancellable Elements of Semigroup form Subsemigroup", "text": "Let $\\struct {S, \\circ}$ be a semigroup. Let $C$ be the set of cancellable elements of $\\struct {S, \\circ}$. Then $\\struct {C, \\circ}$ is a subsemigroup of $\\struct {S, \\circ}$."}57{"_id": "71", "title": "Subgroup of Cyclic Group is Cyclic", "text": "Let $G$ be a cyclic group. Let $H$ be a subgroup of $G$. Then $H$ is cyclic."}58{"_id": "72", "title": "Empty Set is Subset of All Sets", "text": "The empty set $\\O$ is a subset of every set (including itself). That is: :$\\forall S: \\O \\subseteq S$"}59{"_id": "73", "title": "Relation Reflexivity", "text": "Every relation has exactly one of these properties: it is either: :reflexive, :antireflexive or :non-reflexive."}60{"_id": "74", "title": "Union is Idempotent", "text": "Set union is idempotent: :$S \\cup S = S$"}61{"_id": "75", "title": "Intersection is Idempotent", "text": "Set intersection is idempotent: :$S \\cap S = S$"}62{"_id": "76", "title": "Set is Subset of Union", "text": "The union of two sets is a superset of each: :$S \\subseteq S \\cup T$ :$T \\subseteq S \\cup T$"}63{"_id": "77", "title": "Set is Subset of Itself", "text": "Every set is a subset of itself: :$\\forall S: S \\subseteq S$ Thus, by definition, the relation '''is a subset of''' is reflexive."}64{"_id": "78", "title": "Subset of Set with Propositional Function", "text": "Let $S$ be a set. Let $P: S \\to \\set {\\text{true}, \\text{false} }$ be a propositional function on $S$. Then: :$\\set {x \\in S: \\map P x} \\subseteq S$"}65{"_id": "80", "title": "Universal Generalisation", "text": "Let $\\mathbf a$ be any arbitrarily selected object in the universe of discourse. Then: :$\\map P {\\mathbf a} \\vdash \\forall x: \\map P x$ In natural language: :''Suppose $P$ is true of any arbitrarily selected $\\mathbf a$'' in the universe of discourse.'' :''Then $P$ is true of everything in the universe of discourse.''"}66{"_id": "82", "title": "Existential Instantiation", "text": ":$\\exists x: P \\left({x}\\right), P \\left({\\mathbf a}\\right) \\implies y \\vdash y$ Suppose we have the following: : From our universe of discourse, ''any'' arbitrarily selected object $\\mathbf a$ which has the property $P$ implies a conclusion $y$ : $\\mathbf a$ is not free in $y$ : It is known that there ''does'' actually exists an object that has $P$. Then we may infer $y$. This is called the '''Rule of Existential Instantiation''' and often appears in a proof with its abbreviation '''EI'''. When using this rule of existential instantiation: :$\\exists x: P \\left({x}\\right), P \\left({\\mathbf a}\\right) \\implies y \\vdash y$ the instance of $P \\left({\\mathbf a}\\right)$ is referred to as the '''typical disjunct'''."}67{"_id": "83", "title": "Singleton of Element is Subset", "text": "Let $S$ be a set. Let $\\set x$ be the singleton of $x$. Then: :$x \\in S \\iff \\set x \\subseteq S$"}68{"_id": "84", "title": "Subset Relation is Transitive", "text": "The relation \"is a subset of\" is transitive: :$\\paren {R \\subseteq S} \\land \\paren {S \\subseteq T} \\implies R \\subseteq T$"}69{"_id": "85", "title": "Set Inequality", "text": ":$S \\ne T \\iff \\left({S \\nsubseteq T}\\right) \\lor \\left({T \\nsubseteq S}\\right)$"}70{"_id": "86", "title": "Set Equals Itself", "text": "All sets are equal to themselves: :$\\forall S: S = S$"}71{"_id": "87", "title": "Union is Smallest Superset", "text": "Let $S_1$ and $S_2$ be sets. Then $S_1 \\cup S_2$ is the smallest set containing both $S_1$ and $S_2$. That is: :$\\paren {S_1 \\subseteq T} \\land \\paren {S_2 \\subseteq T} \\iff \\paren {S_1 \\cup S_2} \\subseteq T$"}72{"_id": "88", "title": "Union with Empty Set", "text": "The union of any set with the empty set is the set itself: :$S \\cup \\O = S$"}73{"_id": "89", "title": "Intersection is Subset", "text": "The intersection of two sets is a subset of each: :$S \\cap T \\subseteq S$ :$S \\cap T \\subseteq T$"}74{"_id": "90", "title": "Intersection with Empty Set", "text": "The intersection of any set with the empty set is itself the empty set: :$S \\cap \\O = \\O$"}75{"_id": "91", "title": "Intersection is Largest Subset", "text": "Let $T_1$ and $T_2$ be sets. Then $T_1 \\cap T_2$ is the largest set contained in both $T_1$ and $T_2$. That is: :$S \\subseteq T_1 \\land S \\subseteq T_2 \\iff S \\subseteq T_1 \\cap T_2$"}76{"_id": "92", "title": "Intersection is Subset of Union", "text": "The intersection of two sets is a subset of their union: :$S \\cap T \\subseteq S \\cup T$"}77{"_id": "93", "title": "Absorption Laws (Set Theory)/Union with Intersection", "text": ":$S \\cup \\paren {S \\cap T} = S$"}78{"_id": "94", "title": "Absorption Laws (Set Theory)/Intersection with Union", "text": ":$S \\cap \\paren {S \\cup T} = S$"}79{"_id": "95", "title": "Union Distributes over Intersection", "text": "Set union is distributive over set intersection: :$R \\cup \\paren {S \\cap T} = \\paren {R \\cup S} \\cap \\paren {R \\cup T}$"}80{"_id": "96", "title": "Intersection Distributes over Union", "text": "Set intersection is distributive over set union: :$R \\cap \\paren {S \\cup T} = \\paren {R \\cap S} \\cup \\paren {R \\cap T}$"}81{"_id": "97", "title": "Set Difference is Subset", "text": ":$S \\setminus T \\subseteq S$"}82{"_id": "98", "title": "Set Difference with Empty Set is Self", "text": "The set difference between a set and the empty set is the set itself: :$S \\setminus \\O = S$"}83{"_id": "99", "title": "Set Difference with Self is Empty Set", "text": "The set difference of a set with itself is the empty set: :$S \\setminus S = \\O$"}84{"_id": "100", "title": "Set Difference Equals First Set iff Empty Intersection", "text": ":$S \\setminus T = S \\iff S \\cap T = \\O$"}85{"_id": "101", "title": "Equal Set Differences iff Equal Intersections", "text": ":$R \\setminus S = R \\setminus T \\iff R \\cap S = R \\cap T$"}86{"_id": "102", "title": "Euler's Number is Irrational", "text": "Euler's number $e$ is irrational."}87{"_id": "104", "title": "Results Concerning Set Difference with Union", "text": "Let: * $S \\setminus T$ denote set difference * $S \\cup T$ denote set union * $S \\cap T$ denote set intersection."}88{"_id": "105", "title": "Set Difference with Intersection", "text": "Let $S$ and $T$ be sets."}89{"_id": "106", "title": "Results Concerning Set Difference with Intersection", "text": "Let: :$S \\setminus T$ denote set difference :$S \\cap T$ denote set intersection."}90{"_id": "107", "title": "Pascal's Rule", "text": "For positive integers $n, k$ with $1 \\le k \\le n$: :$\\dbinom n {k - 1} + \\dbinom n k = \\dbinom {n + 1} k$ This is also valid for the real number definition: :$\\forall r \\in \\R, k \\in \\Z: \\dbinom r {k - 1} + \\dbinom r k = \\dbinom {r + 1} k$"}91{"_id": "108", "title": "Set Difference Union Intersection", "text": ":$S = \\paren {S \\setminus T} \\cup \\paren {S \\cap T}$"}92{"_id": "109", "title": "Absorption Laws (Logic)", "text": "For any two propositions $p$ and $q$, we have:"}93{"_id": "110", "title": "Set Difference with Set Difference", "text": ":$S \\setminus \\paren {S \\setminus T} = S \\cap T = T \\setminus \\paren {T \\setminus S}$"}94{"_id": "112", "title": "Dirichlet's Test for Uniform Convergence", "text": "Let $D$ be a set. Let $\\struct {V, \\norm {\\,\\cdot\\,} }$ be a normed vector space. Let $a_i, b_i$ be mappings from $D \\to M$. Let the following conditions be satisfied: :$(1): \\quad$ The sequence of partial sums of $\\displaystyle \\sum_{n \\mathop = 1}^\\infty \\map {a_n} x$ be bounded on $D$ :$(2): \\quad \\sequence {\\map {b_n} x}$ be monotonic for each $x \\in D$ :$(3): \\quad \\map {b_n} x \\to 0$ converge uniformly on $D$. Then: :$\\displaystyle \\sum_{n \\mathop = 1}^\\infty \\map {a_n} x \\, \\map {b_n} x$ converges uniformly on $D$."}95{"_id": "114", "title": "Relative Complement of Empty Set", "text": "The relative complement of the empty set is the set itself: :$\\relcomp S \\O = S$"}96{"_id": "115", "title": "Relative Complement with Self is Empty Set", "text": "The relative complement of a set in itself is the empty set: :$\\relcomp S S = \\O$"}97{"_id": "116", "title": "Relative Complement of Relative Complement", "text": ":$\\relcomp S {\\relcomp S T} = T$"}98{"_id": "117", "title": "Intersection with Relative Complement is Empty", "text": "The intersection of a set and its relative complement is the empty set: :$T \\cap \\relcomp S T = \\O$"}99{"_id": "118", "title": "Union with Relative Complement", "text": "The union of a set $T$ and its relative complement in $S$ is the set $S$: :$\\relcomp S T \\cup T = S$"}100{"_id": "119", "title": "Set Difference as Intersection with Relative Complement", "text": "Let $A, B \\subseteq S$. Then the set difference between $A$ and $B$ can be expressed as the intersection with the relative complement with respect to $S$: :$A \\setminus B = A \\cap \\relcomp S B$"}101{"_id": "120", "title": "Equivalence of Axiom Schemata for Groups", "text": "In the definition of a group, the axioms for the existence of an identity element and for closure under taking inverses can be replaced by the following two axioms: : Given a group $G$, there exists at least one element $e \\in G$ such that $e$ is a '''left identity'''; : For any element $g$ in a group $G$, there exists at least one '''left inverse''' of $g$. Alternatively, we can also replace the aforementioned axioms with the following two: : Given a group $G$, there exists at least one element $e \\in G$ such that $e$ is a '''right identity'''; : For any element $g$ in a group $G$, there exists at least one '''right inverse''' of $g$. Thus we can formulate the group axioms as either of the following:"}102{"_id": "121", "title": "Group has Latin Square Property", "text": "Let $\\struct {G, \\circ}$ be a group. Then $G$ satisfies the Latin square property. That is, for all $a, b \\in G$, there exists a unique $g \\in G$ such that $a \\circ g = b$. Similarly, there exists a unique $h \\in G$ such that $h \\circ a = b$."}103{"_id": "122", "title": "Symmetric Difference is Commutative", "text": "Symmetric difference is commutative: :$S * T = T * S$"}104{"_id": "124", "title": "Symmetric Difference of Equal Sets", "text": "The symmetric difference of two equal sets is the empty set: :$S = T \\iff S * T = \\O$"}105{"_id": "125", "title": "Union is Empty iff Sets are Empty", "text": ":$S \\cup T = \\O \\iff S = \\O \\land T = \\O$"}106{"_id": "126", "title": "Symmetric Difference with Empty Set", "text": ":$S * \\O = S$ where $*$ denotes the symmetric difference."}107{"_id": "127", "title": "Intersection Distributes over Symmetric Difference", "text": "Intersection is distributive over symmetric difference: :$\\paren {R * S} \\cap T = \\paren {R \\cap T} * \\paren {S \\cap T}$ :$T \\cap \\paren {R * S} = \\paren {T \\cap R} * \\paren {T \\cap S}$"}108{"_id": "128", "title": "Symmetric Difference of Unions", "text": "Let $R$, $S$ and $T$ be sets. Then: :$\\paren {R \\cup T} * \\paren {S \\cup T} = \\paren {R * S} \\setminus T$ where: :$*$ denotes the symmetric difference :$\\setminus$ denotes set difference :$\\cup$ denotes set union"}109{"_id": "129", "title": "Intersection with Universe", "text": "The intersection of a set with the universe is the set itself: :$\\mathbb U \\cap S = S$"}110{"_id": "130", "title": "Union with Universe", "text": "The union of a set with the universe is the universe: :$\\mathbb U \\cup S = \\mathbb U$"}111{"_id": "131", "title": "Complement of Empty Set is Universe", "text": "The complement of the empty set is the universe: :$\\map \\complement \\O = \\mathbb U$"}112{"_id": "132", "title": "Complement of Universe is Empty Set", "text": "The complement of the universe is the empty set: :$\\map \\complement {\\mathbb U} = \\O$"}113{"_id": "133", "title": "Complement of Complement", "text": "The complement of the complement of a set is the set itself: :$\\map \\complement {\\map \\complement S} = S$"}114{"_id": "134", "title": "Intersection with Complement", "text": "The intersection of a set and its complement is the empty set: :$S \\cap \\map \\complement S = \\O$"}115{"_id": "135", "title": "Union with Complement", "text": "The union of a set and its complement is the universe: :$S \\cup \\map \\complement S = \\mathbb U$"}116{"_id": "137", "title": "Set Difference as Intersection with Complement", "text": "Set difference can be expressed as the intersection with the set complement: :$A \\setminus B = A \\cap \\map \\complement B$"}117{"_id": "138", "title": "Set Difference of Complements", "text": ":$\\complement \\left({S}\\right) \\setminus \\complement \\left({T}\\right) = T \\setminus S$"}118{"_id": "139", "title": "Symmetric Difference of Complements", "text": "The symmetric difference of two sets equals the symmetric difference of their complements: :$\\map \\complement S * \\map \\complement T = S * T$"}119{"_id": "141", "title": "Symmetric Difference with Self is Empty Set", "text": "The symmetric difference of a set with itself is the empty set: :$S * S = \\O$"}120{"_id": "142", "title": "Symmetric Difference with Complement", "text": "The symmetric difference of a set with its complement is the universe: :$S * \\relcomp {} S = \\mathbb U$"}121{"_id": "143", "title": "Symmetric Difference is Associative", "text": "Symmetric difference is associative: :$R * \\paren {S * T} = \\paren {R * S} * T$"}122{"_id": "144", "title": "Cartesian Product is Empty iff Factor is Empty", "text": ":$S \\times T = \\O \\iff S = \\O \\lor T = \\O$ Thus: :$S \\times \\O = \\O = \\O \\times T$"}123{"_id": "146", "title": "Cartesian Product of Subsets", "text": "Let $A, B, S, T$ be sets such that $A \\subseteq B$ and $S \\subseteq T$. Then: :$A \\times S \\subseteq B \\times T$ In addition, if $A, S \\ne \\O$, then: :$A \\times S \\subseteq B \\times T \\iff A \\subseteq B \\land S \\subseteq T$"}124{"_id": "147", "title": "Cartesian Product of Intersections", "text": ":$\\paren {S_1 \\cap S_2} \\times \\paren {T_1 \\cap T_2} = \\paren {S_1 \\times T_1} \\cap \\paren {S_2 \\times T_2}$ where $S_1, S_2, T_1, T_2$ are sets."}125{"_id": "148", "title": "Cartesian Product Distributes over Union", "text": "Cartesian product is distributive over union: :$A \\times \\paren {B \\cup C} = \\paren {A \\times B} \\cup \\paren {A \\times C}$ :$\\paren {B \\cup C} \\times A = \\paren {B \\times A} \\cup \\paren {C \\times A}$"}126{"_id": "149", "title": "Cartesian Product of Unions", "text": ":$\\paren {S_1 \\cup S_2} \\times \\paren {T_1 \\cup T_2} = \\paren {S_1 \\times T_1} \\cup \\paren {S_2 \\times T_2} \\cup \\paren {S_1 \\times T_2} \\cup \\paren {S_2 \\times T_1}$"}127{"_id": "150", "title": "Cartesian Product Distributes over Set Difference", "text": "Cartesian product is distributive over set difference: :$(1): \\quad S \\times \\paren {T_1 \\setminus T_2} = \\paren {S \\times T_1} \\setminus \\paren {S \\times T_2}$ :$(2): \\quad \\paren {T_1 \\setminus T_2} \\times S = \\paren {T_1 \\times S} \\setminus \\paren {T_2 \\times S}$"}128{"_id": "151", "title": "Set Difference of Cartesian Products", "text": ":$\\paren {S_1 \\times S_2} \\setminus \\paren {T_1 \\times T_2} = \\paren {S_1 \\times \\paren {S_2 \\setminus T_2} } \\cup \\paren {\\paren {S_1 \\setminus T_1} \\times S_2}$"}129{"_id": "152", "title": "Inverse of Inverse Relation", "text": "The inverse of an inverse relation is the relation itself: :$\\paren {\\RR^{-1} }^{-1} = \\RR$"}130{"_id": "153", "title": "Diagonal Relation is Equivalence", "text": "The diagonal relation $\\Delta_S$ on a set $S$ is always an equivalence in $S$."}131{"_id": "154", "title": "Trivial Relation is Equivalence", "text": "The trivial relation on $S$: :$\\RR = S \\times S$ is always an equivalence in $S$."}132{"_id": "155", "title": "Equality of Relations", "text": "Let $\\RR_1$ and $\\RR_2$ be relations on $S_1 \\times T_1$ and $S_2 \\times T_2$ respectively. Then $\\RR_1$ and $\\RR_2$ are '''equal''' {{iff}}: :$S_1 = S_2$ :$T_1 = T_2$ :$\\tuple {s, t} \\in \\RR_1 \\iff \\tuple {s, t} \\in \\RR_2$ It is worth labouring the point that for two relations to be '''equal''', not only must their domains be equal, but so must their codomains."}133{"_id": "156", "title": "Image of Singleton under Relation", "text": "Let $\\RR \\subseteq S \\times T$ be a relation. Then the image of an element of $S$ is equal to the image of a singleton containing that element, the singleton being a subset of $S$: :$\\forall s \\in S: \\map \\RR s = \\RR \\sqbrk {\\set s}$"}134{"_id": "157", "title": "Image of Subset under Relation is Subset of Image", "text": "Let $S$ and $T$ be sets. Let $\\RR \\subseteq S \\times T$ be a relation from $S$ to $T$. Let $A, B \\subseteq S$ such that $A \\subseteq B$. Then the image of $A$ is a subset of the image of $B$: :$A \\subseteq B \\implies \\RR \\sqbrk A \\subseteq \\RR \\sqbrk B$ In the notation of direct image mappings, this can be written: :$A \\subseteq B \\implies \\map {\\RR^\\to} A \\subseteq \\map {\\RR^\\to} B$"}135{"_id": "158", "title": "Image of Element is Subset", "text": "Let $S$ and $T$ be sets. Let $\\RR \\subseteq S \\times T$ be a relation. Let $A \\subseteq S$. Then: :$s \\in A \\implies \\map \\RR s \\subseteq \\RR \\sqbrk A$"}136{"_id": "159", "title": "Image is Subset of Codomain", "text": "Let $\\RR = S \\times T$ be a relation. For all subsets $A$ of the domain of $\\RR$, the image of $A$ is a subset of the codomain of $\\RR$: :$\\forall A \\subseteq \\Dom \\RR: \\RR \\sqbrk A \\subseteq T$ In the notation of direct image mappings, this can be written as: :$\\forall A \\in \\powerset S: \\map {\\RR^\\to} A \\in \\powerset T$"}137{"_id": "160", "title": "Image of Empty Set is Empty Set", "text": "Let $\\RR \\subseteq S \\times T$ be a relation. The image of the empty set is the empty set: :$\\RR \\sqbrk \\O = \\O$"}138{"_id": "161", "title": "Domain of Composite Relation", "text": "Let $\\RR_2 \\circ \\RR_1$ be a composite relation. Then the domain of $\\RR_2 \\circ \\RR_1$ is the domain of $\\RR_1$: :$\\Dom {\\RR_2 \\circ \\RR_1} = \\Dom {\\RR_1}$"}139{"_id": "162", "title": "Codomain of Composite Relation", "text": "Let $\\RR_2 \\circ \\RR_1$ be a composite relation. Then the codomain of $\\RR_2 \\circ \\RR_1$ is the codomain of $\\RR_2$: :$\\Cdm {\\RR_2 \\circ \\RR_1} = \\Cdm {\\RR_2}$"}140{"_id": "163", "title": "Preimage of Mapping equals Domain", "text": "The preimage of a mapping is the same set as its domain: :$\\Preimg f = \\Dom f$"}141{"_id": "164", "title": "Preimage of Relation is Subset of Domain", "text": "Let $\\mathcal R \\subseteq S \\times T$ be a relation. Then the preimage of $\\mathcal R$ is a subset of its domain: :$\\Preimg {\\mathcal R} \\subseteq S$"}142{"_id": "165", "title": "Preimage of Image under Left-Total Relation is Superset", "text": "Let $\\RR \\subseteq S \\times T$ be a left-total relation. Then: :$A \\subseteq S \\implies A \\subseteq \\paren {\\RR^{-1} \\circ \\RR} \\sqbrk A$ where: :$\\RR \\sqbrk A$ denotes the image of $A$ under $\\RR$ :$\\RR^{-1} \\sqbrk A$ denotes the preimage of $A$ under $\\RR$ :$\\RR^{-1} \\circ \\RR$ denotes composition of $\\RR^{-1}$ and $\\RR$. This can be expressed in the language and notation of direct image mappings and inverse image mappings as: :$\\forall A \\in \\powerset S: A \\subseteq \\map {\\paren {\\RR^\\gets \\circ \\RR^\\to} } A$"}143{"_id": "166", "title": "Inverse of Composite Relation", "text": "Let $\\RR_2 \\circ \\RR_1 \\subseteq S_1 \\times S_3$ be the composite of the two relations $\\RR_1 \\subseteq S_1 \\times S_2$ and $\\RR_2 \\subseteq S_2 \\times S_3$. Then: :$\\paren {\\RR_2 \\circ \\RR_1}^{-1} = \\RR_1^{-1} \\circ \\RR_2^{-1}$"}144{"_id": "167", "title": "Image of Union under Relation", "text": "Let $S$ and $T$ be sets. Let $\\RR \\subseteq S \\times T$ be a relation. Let $S_1$ and $S_2$ be subsets of $S$. Then: :$\\RR \\sqbrk {S_1 \\cup S_2} = \\RR \\sqbrk {S_1} \\cup \\RR \\sqbrk {S_2}$ That is, the image of the union of subsets of $S$ is equal to the union of their images."}145{"_id": "168", "title": "Image of Intersection under Relation", "text": "Let $S$ and $T$ be sets. Let $\\RR \\subseteq S \\times T$ be a relation. Let $S_1$ and $S_2$ be subsets of $S$. Then: :$\\RR \\sqbrk {S_1 \\cap S_2} \\subseteq \\RR \\sqbrk {S_1} \\cap \\RR \\sqbrk {S_2}$ That is, the image of the intersection of subsets of $S$ is a subset of the intersection of their images."}146{"_id": "169", "title": "Image of Set Difference under Relation", "text": "Let $\\RR \\subseteq S \\times T$ be a relation. Let $A$ and $B$ be subsets of $S$. Then: :$\\RR \\sqbrk A \\setminus \\RR \\sqbrk B \\subseteq \\RR \\sqbrk {A \\setminus B}$ where: :$\\setminus$ denotes set difference :$\\RR \\sqbrk A$ denotes image of $A$ under $\\RR$."}147{"_id": "170", "title": "Preimage of Union under Relation", "text": "Let $S$ and $T$ be sets. Let $\\RR \\subseteq S \\times T$ be a relation. Let $T_1$ and $T_2$ be subsets of $T$. Then: :$\\RR^{-1} \\sqbrk {T_1 \\cup T_2} = \\RR^{-1} \\sqbrk {T_1} \\cup \\RR^{-1} \\sqbrk {T_2}$"}148{"_id": "171", "title": "Preimage of Intersection under Relation", "text": "Let $S$ and $T$ be sets. Let $\\mathcal R \\subseteq S \\times T$ be a relation. Let $C$ and $D$ be subsets of $T$. Then: :$\\mathcal R^{-1} \\left[{C \\cap D}\\right] \\subseteq \\mathcal R^{-1} \\left[{C}\\right] \\cap \\mathcal R^{-1} \\left[{D}\\right]$"}149{"_id": "173", "title": "Restriction is Subset of Relation", "text": "Let $\\mathcal R \\subseteq S \\times T$ be a relation. Let $X \\subseteq S$. Then the restriction of $\\mathcal R$ to $X$ is a subset of $\\mathcal R$."}150{"_id": "174", "title": "Null Relation is Antireflexive, Symmetric and Transitive", "text": "Let $S$ be a set which is non-empty. Let $\\RR \\subseteq S \\times S$ be the null relation. Then $\\RR$ is antireflexive, symmetric and transitive. If $S = \\O$ then Relation on Empty Set is Equivalence applies."}151{"_id": "175", "title": "Symmetry of Relations is Symmetric", "text": "Let $\\mathcal R$ be a relation on $S$ which is symmetric. Then: :$\\left({x, y}\\right) \\in \\mathcal R \\iff \\left({y, x}\\right) \\in \\mathcal R$."}152{"_id": "176", "title": "Relation is Symmetric and Antisymmetric iff Coreflexive", "text": "Let $S$ be a set. Let $\\RR \\subseteq S \\times S$ be a relation in $S$. Then: :$\\RR$ is both symmetric and antisymmetric {{iff}}: :$\\RR$ is coreflexive."}153{"_id": "177", "title": "Relation both Symmetric and Asymmetric is Null", "text": "Let $\\RR$ be a relation in $S$ which is both symmetric and asymmetric. Then: :$\\RR = \\O$"}154{"_id": "178", "title": "Equivalent Characterizations of Abelian Group", "text": "Let $G$ be a group. {{TFAE}} {{begin-eqn}} {{eqn | n = 1       | o =       | c = $G$ is abelian }} {{eqn | n = 2       | o =       | c = $\\forall a, b \\in G: \\paren {a b}^{-1} = a^{-1} b^{-1}$ }} {{eqn | n = 3       | o =       | c = Cross cancellation property: $\\forall a, b, c \\in G: a b = c a \\implies b = c$ }} {{eqn | n = 4       | o =       | c = Middle cancellation property: $\\forall a, b, c, d, x \\in G: a x b = c x d \\implies a b = c d$  }} {{end-eqn}}"}155{"_id": "180", "title": "Asymmetric Relation is Antisymmetric", "text": "Let $\\RR$ be an asymmetric relation. Then $\\RR$ is also antisymmetric."}156{"_id": "181", "title": "Asymmetric Relation is Antireflexive", "text": "Let $S$ be a set. Let $\\RR \\subseteq S \\times S$ be a relation on $S$. Let $\\RR$ be asymmetric. Then $\\RR$ is also antireflexive."}157{"_id": "182", "title": "Antireflexive and Transitive Relation is Asymmetric", "text": "Let $\\mathcal R \\subseteq S \\times S$ be a relation which is not null. Let $\\mathcal R$ be antireflexive and transitive. Then $\\mathcal R$ is also asymmetric."}158{"_id": "183", "title": "Antitransitive Relation is Antireflexive", "text": "Let $S$ be a set. Let $\\mathcal R \\subseteq S \\times S$ be a relation on $S$. Let $\\mathcal R$ be antitransitive. Then $\\mathcal R$ is also antireflexive."}159{"_id": "184", "title": "Symmetric Transitive and Serial Relation is Reflexive", "text": "Let $\\RR$ be a relation which is: :symmetric :transitive :serial. Then $\\RR$ is reflexive. Thus such a relation is an equivalence."}160{"_id": "185", "title": "Inverse Relation Properties", "text": "Let $\\mathcal R$ be a relation on a set $S$. If $\\mathcal R$ has any of the properties: * Reflexive * Antireflexive * Non-reflexive * Symmetric * Asymmetric * Antisymmetric * Non-symmetric * Transitive * Antitransitive * Non-transitive ... then its inverse $\\mathcal R^{-1}$ has the same properties."}161{"_id": "186", "title": "Properties of Restriction of Relation", "text": "Let $S$ be a set. Let $\\mathcal R \\subseteq S \\times S$ be a relation on $S$. Let $T \\subseteq S$ be a subset of $S$. Let $\\mathcal R \\restriction_T \\ \\subseteq T \\times T$ be the restriction of $\\mathcal R$ to $T$. If $\\mathcal R$ on $S$ has any of the properties: * Reflexive * Antireflexive * Symmetric * Antisymmetric * Asymmetric * Transitive * Antitransitive * Connected ... then $\\mathcal R \\restriction_T$ on $T$ has the same properties."}162{"_id": "188", "title": "Relation is Antireflexive iff Disjoint from Diagonal Relation", "text": "Let $\\mathcal R \\subseteq S \\times S$ be a relation on a set $S$. Then: : $\\mathcal R$ is antireflexive iff : $\\Delta_S \\cap \\mathcal R = \\varnothing$ where $\\Delta_S$ is the diagonal relation."}163{"_id": "189", "title": "Inverse Relation Equal iff Subset", "text": "If a relation $\\RR$ is a subset or superset of its inverse, then it equals its inverse. That is, the following are equivalent: {{begin-axiom}} {{axiom | n = 1         | m = \\RR \\subseteq \\RR^{-1} }} {{axiom | n = 2         | m = \\RR^{-1} \\subseteq \\RR }} {{axiom | n = 3         | m = \\RR = \\RR^{-1} }} {{end-axiom}}"}164{"_id": "191", "title": "Relation is Symmetric iff Inverse is Symmetric", "text": "A relation $\\mathcal R$ is symmetric {{iff}} its inverse $\\mathcal R^{-1}$ is also symmetric."}165{"_id": "194", "title": "Inverse of Many-to-One Relation is One-to-Many", "text": "The inverse of a many-to-one relation is a one-to-many relation, and vice versa."}166{"_id": "195", "title": "Many-to-One Relation Composite with Inverse is Transitive", "text": "Let $\\mathcal R \\subseteq S \\times T$ be a relation which is many-to-one. Then the composites (both ways) of $\\mathcal R$ and its inverse $\\mathcal R^{-1}$, that is, both $\\mathcal R^{-1} \\circ \\mathcal R$ and $\\mathcal R \\circ \\mathcal R^{-1}$, are transitive."}167{"_id": "196", "title": "One-to-Many Relation Composite with Inverse is Coreflexive", "text": "Let $\\mathcal R \\subseteq S \\times S$ be a relation which is one-to-many. Then the composite of $\\mathcal R$ with its inverse is a coreflexive relation: :$\\mathcal R^{-1} \\circ \\mathcal R \\subseteq \\Delta_X$ {{refactor|This is a separate result.}} That is, by Relation is Symmetric and Antisymmetric iff Coreflexive, $\\mathcal R^{-1} \\circ \\mathcal R$ is both symmetric and antisymmetric."}168{"_id": "197", "title": "Element in its own Equivalence Class", "text": "Let $\\mathcal R$ be an equivalence relation on a set $S$. Then every element of $S$ is in its own $\\mathcal R$-class: :$\\forall x \\in S: x \\in \\eqclass x {\\mathcal R}$"}169{"_id": "198", "title": "Equivalence Class of Element is Subset", "text": "Let $\\mathcal R$ be an equivalence relation on a set $S$. The $\\mathcal R$-class of every element of $S$ is a subset of the set the element is in: :$\\forall x \\in S: \\eqclass x {\\mathcal R} \\subseteq S$"}170{"_id": "199", "title": "Equivalence Class is not Empty", "text": "Let $\\mathcal R$ be an equivalence relation on a set $S$. Then no $\\mathcal R$-class is empty."}171{"_id": "200", "title": "Handshake Lemma", "text": "Let $G$ be a $\\tuple {p, q}$-undirected graph, which may be a multigraph or a loop-graph, or both. Let $V = \\set {v_1, v_2, \\ldots, v_p}$ be the vertex set of $G$. Then: :$\\displaystyle \\sum_{i \\mathop = 1}^p \\map {\\deg_G} {v_i} = 2 q$ where $\\map {\\deg_G} {v_i}$ is the degree of vertex $v_i$. That is, the sum of all the degrees of all the vertices of an graph is equal to twice its size. This result is known as the '''Handshake Lemma''' or '''Handshaking Lemma'''."}172{"_id": "201", "title": "Equivalence Class holds Equivalent Elements", "text": "Let $\\RR$ be an equivalence relation on a set $S$. Then: :$\\tuple {x, y} \\in \\RR \\iff \\eqclass x \\RR = \\eqclass y \\RR$"}173{"_id": "202", "title": "Equivalence Classes are Disjoint", "text": "Let $\\RR$ be an equivalence relation on a set $S$. Then all $\\RR$-classes are pairwise disjoint: :$\\tuple {x, y} \\notin \\RR \\iff \\eqclass x \\RR \\cap \\eqclass y \\RR = \\O$"}174{"_id": "203", "title": "Fundamental Theorem on Equivalence Relations", "text": "Let $\\RR \\subseteq S \\times S$ be an equivalence on a set $S$. Then the quotient $S / \\RR$ of $S$ by $\\RR$ forms a partition of $S$."}175{"_id": "204", "title": "Equivalence Class is Unique", "text": "Let $\\mathcal R$ be an equivalence relation on $S$. For each $x \\in S$, the one and only one $\\mathcal R$-class to which $x$ belongs is $\\eqclass x {\\mathcal R}$."}176{"_id": "205", "title": "Equivalence Class Equivalent Statements", "text": "Let $\\mathcal R$ be an equivalence on $S$. {{refactor|Use TFAE template}} Then $\\forall x, y \\in S$, the following statements are all equivalent: :$(1): \\quad x$ and $y$ are in the same $\\mathcal R$-class :$(2): \\quad \\eqclass x {\\mathcal R} = \\eqclass y {\\mathcal R}$ :$(3): \\quad x \\mathrel {\\mathcal R} y$ :$(4): \\quad x \\in \\eqclass y {\\mathcal R}$ :$(5): \\quad y \\in \\eqclass x {\\mathcal R}$ :$(6): \\quad \\eqclass x {\\mathcal R} \\cap \\eqclass y {\\mathcal R} \\ne \\O$"}177{"_id": "206", "title": "Relation Induced by Partition is Equivalence", "text": "Let $\\mathbb S$ be a partition of a set $S$. Let $\\RR$ be the relation induced by $\\mathbb S$. Then: :$(1): \\quad \\RR$ is unique :$(2): \\quad \\RR$ is an equivalence relation on $S$. Hence $\\mathbb S$ is the quotient set of $S$ by $\\RR$, that is: :$\\mathbb S = S / \\RR$"}178{"_id": "207", "title": "Relation Partitions Set iff Equivalence", "text": "Let $\\RR$ be a relation on a set $S$. Then $S$ can be partitioned into subsets by $\\RR$ {{iff}} $\\RR$ is an equivalence relation on $S$. The partition of $S$ defined by $\\RR$ is the quotient set $S / \\RR$."}179{"_id": "208", "title": "Intersection of Equivalences", "text": "The intersection of two equivalence relations is itself an equivalence relation."}180{"_id": "210", "title": "Equivalence of Definitions of Equivalence Relation", "text": "Let $\\RR$ be a relation on a set $S$. {{TFAE|def = Equivalence Relation}}"}181{"_id": "211", "title": "Equivalence iff Diagonal and Inverse Composite", "text": "Let $\\RR$ be a relation on $S$. Then $\\RR$ is an equivalence relation on $S$ {{iff}}: :$\\Delta_S \\subseteq \\RR$ and: :$\\RR = \\RR \\circ \\RR^{-1}$"}182{"_id": "212", "title": "Equality of Elements in Range of Mapping", "text": "Let $f: S \\to T$ be a mapping. Then: :$\\exists y \\in \\Rng f: \\tuple {x_1, y} \\in f \\land \\tuple {x_2, y} \\in f \\iff \\map f {x_1} = \\map f {x_2}$"}183{"_id": "213", "title": "Inverse of Mapping is One-to-Many Relation", "text": "Let $f$ be a mapping. Then its inverse $f^{-1}$ is a one-to-many relation."}184{"_id": "214", "title": "Equality of Mappings", "text": "Two mappings $f_1: S_1 \\to T_1, f_2: S_2 \\to T_2$ are equal {{iff}}: :$(1): \\quad S_1 = S_2$ :$(2): \\quad T_1 = T_2$ :$(3): \\quad \\forall x \\in S_1: \\map {f_1} x = \\map {f_2} x$"}185{"_id": "215", "title": "Mapping is Constant iff Image is Singleton", "text": "A mapping is a constant mapping {{iff}} its image is a singleton."}186{"_id": "216", "title": "Diagonal Relation is Right Identity", "text": "Let $\\RR \\subseteq S \\times T$ be a relation on $S \\times T$. Then: :$\\RR \\circ \\Delta_S = \\RR$ where $\\Delta_S$ is the diagonal relation on $S$, and $\\circ$ signifies composition of relations."}187{"_id": "217", "title": "Identity Mapping is Left Identity", "text": "Let $S$ and $T$ be sets. Let $f: S \\to T$ be a mapping. Then: :$I_T \\circ f = f$ where $I_T$ is the identity mapping on $T$, and $\\circ$ signifies composition of mappings."}188{"_id": "218", "title": "Image of Element under Inverse Mapping", "text": "Let $S$ and $T$ be sets. Let $f: S \\to T$ be a mapping such that its inverse $f^{-1}: T \\to S$ is also a mapping. Then: :$\\forall x \\in S, y \\in T: \\map f x = y \\iff \\map {f^{-1} } y = x$"}189{"_id": "219", "title": "Injection iff Left Cancellable", "text": "A mapping $f$ is an injection {{iff}} $f$ is left cancellable."}190{"_id": "221", "title": "Identity Mapping is Injection", "text": "On any set $S$, the identity mapping $I_S: S \\to S$ is an injection."}191{"_id": "222", "title": "Composite of Injections is Injection", "text": "A composite of injections is an injection. That is: :If $f$ and $g$ are injections, then so is $f \\circ g$."}192{"_id": "223", "title": "Injection if Composite is Injection", "text": "Let $f$ and $g$ be mappings such that their composite $g \\circ f$ is an injection. Then $f$ is an injection."}193{"_id": "224", "title": "Injection iff Left Inverse", "text": "A mapping $f: S \\to T, S \\ne \\O$ is an injection {{iff}}: :$\\exists g: T \\to S: g \\circ f = I_S$ where $g$ is a mapping. That is, {{iff}} $f$ has a left inverse."}194{"_id": "225", "title": "Inclusion Mapping is Injection", "text": "Let $S, T$ be sets such that $S$ is a subset of $T$. Then the inclusion mapping $i_S: S \\to T$ defined as: :$\\forall x \\in S: \\map {i_S} x = x$ is an injection. For this reason the inclusion mapping can be known as the '''canonical injection of $S$ to $T$'''."}195{"_id": "226", "title": "Surjection iff Right Cancellable", "text": "Let $f$ be a mapping. Then $f$ is a surjection {{iff}} $f$ is right cancellable."}196{"_id": "227", "title": "Identity Mapping is Surjection", "text": "On any set $S$, the identity mapping $I_S: S \\to S$ is a surjection."}197{"_id": "228", "title": "Restriction of Mapping to Image is Surjection", "text": "Let $f: S \\to T$ be a mapping. Let $g: S \\to \\Img f$ be the restriction of $f$ to $S \\times \\Img f$. Then $g$ is a surjective restriction of $f$."}198{"_id": "229", "title": "Composite of Surjections is Surjection", "text": "A composite of surjections is a surjection. That is: :If $g$ and $f$ are surjections, then so is $g \\circ f$."}199{"_id": "230", "title": "Surjection if Composite is Surjection", "text": "Let $f: S_1 \\to S_2$ and $g: S_2 \\to S_3$ be mappings such that $g \\circ f$ is a surjection. Then $g$ is a surjection."}200{"_id": "231", "title": "Surjection iff Right Inverse", "text": "A mapping $f: S \\to T, S \\ne \\O$ is a surjection {{iff}}: :$\\exists g: T \\to S: f \\circ g = I_T$ where: :$g$ is a mapping :$I_T$ is the identity mapping on $T$. That is, {{iff}} $f$ has a right inverse."}201{"_id": "232", "title": "Identity Mapping is Bijection", "text": "The identity mapping $I_S: S \\to S$ on the set $S$ is a bijection."}202{"_id": "233", "title": "Identity Mapping is Permutation", "text": "The identity mapping $I_S: S \\to S$ on the set $S$ is a permutation."}203{"_id": "234", "title": "Bijection iff Left and Right Inverse", "text": "Let $f: S \\to T$ be a mapping. $f$ is a bijection {{iff}}: :$(1): \\quad \\exists g_1: T \\to S: g_1 \\circ f = I_S$ :$(2): \\quad \\exists g_2: T \\to S: f \\circ g_2 = I_T$ where both $g_1$ and $g_2$ are mappings. It also follows that it is necessarily the case that $g_1 = g_2$ for such to be possible."}204{"_id": "235", "title": "Inverse of Bijection is Bijection", "text": "Let $f: S \\to T$ be a bijection in the sense that: :$(1): \\quad f$ is an injection :$(2): \\quad f$ is a surjection. Then the inverse $f^{-1}$ of $f$ is itself a bijection by the same definition."}205{"_id": "236", "title": "Inverse Element of Bijection", "text": "Let $S$ and $T$ be sets. Let $f: S \\to T$ be a bijection. Then: :$\\map {f^{-1} } y = x \\iff \\map f x = y$ where $f^{-1}$ is the inverse mapping of $f$."}206{"_id": "237", "title": "Composite of Bijection with Inverse is Identity Mapping", "text": "Let $f: S \\to T$ be a bijection. Then: : $f^{-1} \\circ f = I_S$ : $f \\circ f^{-1} = I_T$ where $I_S$ and $I_T$ are the identity mappings on $S$ and $T$ respectively."}207{"_id": "238", "title": "Inverse of Inverse of Bijection", "text": "Let $f: S \\to T$ be a bijection. Then: :$\\paren {f^{-1} }^{-1} = f$ where $f^{-1}$ is the inverse of $f$."}208{"_id": "239", "title": "Composite of Bijections is Bijection", "text": "Let $f$ and $g$ be mappings such that $\\Dom f = \\Cdm g$. Then: :If $f$ and $g$ are both bijections, then so is $f \\circ g$ where $f \\circ g$ is the composite mapping of $f$ with $g$."}209{"_id": "240", "title": "Inverse of Composite Bijection", "text": "Let $f$ and $g$ be bijections such that $\\Dom g = \\Cdm f$. Then: :$\\paren {g \\circ f}^{-1} = f^{-1} \\circ g^{-1}$ and $f^{-1} \\circ g^{-1}$ is itself a bijection."}210{"_id": "241", "title": "Composite of Permutations is Permutation", "text": "Let $f, g$ are permutations of a set $S$. Then their composite $g \\circ f$ is also a permutation of $S$."}211{"_id": "242", "title": "Inverse of Permutation is Permutation", "text": "If $f$ is a permutation of $S$, then so is its inverse $f^{-1}$."}212{"_id": "243", "title": "Left Inverse Mapping is Surjection", "text": "Let $f: S \\to T$ be an injection. Let $g: T \\to S$ be a left inverse of $f$. Then $g$ is a surjection."}213{"_id": "244", "title": "Right Inverse Mapping is Injection", "text": "Let $f: S \\to T$ be a mapping. Let $g: T \\to S$ be a right inverse of $f$. Then $g$ is an injection."}214{"_id": "245", "title": "Set Equivalence is Equivalence Relation", "text": "Set equivalence is an equivalence relation."}215{"_id": "246", "title": "Set Equivalence Less One Element", "text": "Let $S$ and $T$ be sets which are equivalent: :$S \\sim T$ Let $a \\in S$ and $b \\in T$. Then: : $S \\setminus \\left\\{{a}\\right\\} \\sim T \\setminus \\left\\{{b}\\right\\}$ where $\\setminus$ denotes set difference."}216{"_id": "247", "title": "Image of Union under Mapping", "text": "Let $S$ and $T$ be sets. Let $f: S \\to T$ be a mapping. Let $A$ and $B$ be subsets of $S$. Then: :$f \\sqbrk {A \\cup B} = f \\sqbrk A \\cup f \\sqbrk B$ This can be expressed in the language and notation of direct image mappings as: :$\\forall A, B \\in \\powerset S: \\map {f^\\to} {A \\cup B} = \\map {f^\\to} A \\cup \\map {f^\\to} B$"}217{"_id": "248", "title": "Image of Intersection under Mapping", "text": "Let $S$ and $T$ be sets. Let $f: S \\to T$ be a mapping. Let $S_1$ and $S_2$ be subsets of $S$. Then: :$f \\sqbrk {S_1 \\cap S_2} \\subseteq f \\sqbrk {S_1} \\cap f \\sqbrk {S_2}$ This can be expressed in the language and notation of direct image mappings as: :$\\forall S_1, S_2 \\in \\powerset S: \\map {f^\\to} {S_1 \\cap S_2} \\subseteq \\map {f^\\to} {S_1} \\cap \\map {f^\\to} {S_2}$"}218{"_id": "249", "title": "Image of Intersection under Injection", "text": "Let $S$ and $T$ be sets. Let $f: S \\to T$ be a mapping. Then: :$\\forall A, B \\subseteq S: f \\sqbrk {A \\cap B} = f \\sqbrk A \\cap f \\sqbrk B$ {{iff}} $f$ is an injection."}219{"_id": "250", "title": "Image of Intersection under One-to-Many Relation", "text": "Let $S$ and $T$ be sets. Let $\\RR \\subseteq S \\times T$ be a relation. Then: :$\\forall S_1, S_2 \\subseteq S: \\RR \\sqbrk {S_1 \\cap S_2} = \\RR \\sqbrk {S_1} \\cap \\RR \\sqbrk {S_2}$  {{iff}} $\\RR$ is one-to-many."}220{"_id": "251", "title": "One-to-Many Image of Set Difference", "text": "Let $\\mathcal R \\subseteq S \\times T$ be a relation. Let $A$ and $B$ be subsets of $S$. Then: :$(1): \\quad \\mathcal R \\left[{A}\\right] \\setminus \\mathcal R \\left[{B}\\right] = \\mathcal R \\left[{A \\setminus B}\\right]$ {{iff}} $\\mathcal R$ is one-to-many."}221{"_id": "252", "title": "Preimage of Union under Mapping", "text": "Let $S$ and $T$ be sets. Let $f: S \\to T$ be a mapping. Let $T_1$ and $T_2$ be subsets of $T$. Then: :$f^{-1} \\sqbrk {T_1 \\cup T_2} = f^{-1} \\sqbrk {T_1} \\cup f^{-1} \\sqbrk {T_2}$ This can be expressed in the language and notation of inverse image mappings as: :$\\forall T_1, T_2 \\in \\powerset T: \\map {f^\\gets} {T_1 \\cup T_2} = \\map {f^\\gets} {T_1} \\cup \\map {f^\\gets} {T_2}$"}222{"_id": "253", "title": "Preimage of Intersection under Mapping", "text": "Let $f: S \\to T$ be a mapping. Let $T_1$ and $T_2$ be subsets of $T$. Then: :$f^{-1} \\sqbrk {T_1 \\cap T_2} = f^{-1} \\sqbrk {T_1} \\cap f^{-1} \\sqbrk {T_2}$ This can be expressed in the language and notation of inverse image mappings as: :$\\forall T_1, T_2 \\in \\powerset T: \\map {f^\\gets} {T_1 \\cap T_2} = \\map {f^\\gets} {T_1} \\cap \\map {f^\\gets} {T_2}$"}223{"_id": "254", "title": "Preimage of Set Difference under Mapping", "text": "Let $f: S \\to T$ be a mapping. Let $T_1$ and $T_2$ be subsets of $T$. Then: :$f^{-1} \\sqbrk {T_1 \\setminus T_2} = f^{-1} \\sqbrk {T_1} \\setminus f^{-1} \\sqbrk {T_2}$ where: :$\\setminus$ denotes set difference :$f^{-1} \\sqbrk {T_1}$ denotes preimage."}224{"_id": "255", "title": "Projection is Surjection", "text": "Let $S$ and $T$ be non-empty sets. Let $S \\times T$ be the Cartesian product of $S$ and $T$. Let $\\pr_1: S \\times T \\to T$ and $\\pr_2: S \\times T \\to T$ be the first projection and second projection respectively on $S \\times T$. Then $\\pr_1$ and $\\pr_2$ are both surjections."}225{"_id": "256", "title": "Sum Rule for Derivatives", "text": "Let $\\map f x, \\map j x, \\map k x$ be real functions defined on the open interval $I$. Let $\\xi \\in I$ be a point in $I$ at which both $j$ and $k$ are differentiable. Let $\\map f x = \\map j x + \\map k x$. Then $f$ is differentiable at $\\xi$ and: :$\\map {f'} \\xi = \\map {j'} \\xi + \\map {k'} \\xi$ It follows from the definition of derivative that if $j$ and $k$ are both differentiable on the interval $I$, then: :$\\forall x \\in I: \\map {f'} x = \\map {j'} x + \\map {k'} x$"}226{"_id": "257", "title": "Quotient Mapping is Surjection", "text": "Let $\\RR$ be an equivalence relation on $S$. Then the quotient mapping $q_\\RR: S \\to S / \\RR$ is a surjection."}227{"_id": "258", "title": "Trivial Quotient is a Bijection", "text": "Let $\\Delta_S$ be the diagonal relation on a set $S$. Let $q_{\\Delta_S}: S \\to S / \\Delta_S$ be the trivial quotient of $S$. Then $q_{\\Delta_S}: S \\to S / \\Delta_S$ is a bijection."}228{"_id": "260", "title": "Renaming Mapping is Well-Defined", "text": "Let $f: S \\to T$ be a mapping. Let $r: S / \\RR_f \\to \\Img f$ be the renaming mapping,  defined as: :$r: S / \\RR_f \\to \\Img f: \\map r {\\eqclass x {\\RR_f} } = \\map f x$ where: :$\\RR_f$ is the equivalence induced by the mapping $f$ :$S / \\RR_f$ is the quotient set of $S$ determined by $\\RR_f$ :$\\eqclass x {\\RR_f}$ is the equivalence class of $x$ under $\\RR_f$. The renaming mapping is always well-defined."}229{"_id": "261", "title": "Powers of Group Elements", "text": "Let $\\left({G, \\circ}\\right)$ be a group whose identity is $e$. Let $a \\in G$. Then the following results hold:"}230{"_id": "262", "title": "Renaming Mapping is Bijection", "text": "Let $f: S \\to T$ be a mapping. Let $r: S / \\RR_f \\to \\Img f$ be the renaming mapping, defined as: :$r: S / \\RR_f \\to \\Img f: \\map r {\\eqclass x {\\RR_f} } = \\map f x$ where: :$\\RR_f$ is the equivalence induced by the mapping $f$ :$S / \\RR_f$ is the quotient set of $S$ determined by $\\RR_f$ :$\\eqclass x {\\RR_f}$ is the equivalence class of $x$ under $\\RR_f$. The renaming mapping is a bijection."}231{"_id": "263", "title": "Composite of Mapping with Inverse", "text": "Let $f: S \\to T$ be a mapping. Then: :$\\forall x \\in S: \\map {f^{-1} \\circ f} x = \\eqclass x {\\RR_f}$ where: :$\\RR_f$ is the equivalence induced by $f$ :$\\eqclass x {\\RR_f}$ is the $\\RR_f$-equivalence class of $x$."}232{"_id": "264", "title": "Factoring Mapping into Surjection and Inclusion", "text": "Every mapping $f:S \\to T$ can be uniquely '''factored into''' a surjection $g$ followed by the inclusion mapping $i_T$. That is, $f = i_T \\circ g$ where: :$g: S \\to \\Img f: \\map g x = \\map f x$ :$i_T: \\Img f \\to T: \\map {i_T} x = x$ This can be illustrated using a commutative diagram as follows: ::$\\begin{xy}\\xymatrix@L+2mu@+1em {  S \\ar@{-->}[r]^*{g}      \\ar[rd]_*{f = i_T \\circ g} &  \\Img f \\ar@{-->}[d]^*{i_T} \\\\ &  T }\\end{xy}$"}233{"_id": "265", "title": "Quotient Theorem for Surjections", "text": "Let $f: S \\to T$ be a surjection. Then there is one and only one bijection $r: S / \\RR_f \\to T$ such that: :$r \\circ q_{\\RR_f} = f$ where: :$\\RR_f$ is the equivalence induced by $f$ :$r: S / \\RR_f \\to T$ is the renaming mapping :$q_{\\RR_f}: S \\to S / \\RR_f$ is the quotient mapping induced by $\\RR_f$. This can be illustrated using a commutative diagram as follows: ::$\\begin {xy} \\xymatrix@L + 2mu@ + 1em {  S \\ar@{-->}[rr]^*{f = r \\circ q_{\\RR_f} }    \\ar[dd]_*{q_{\\RR_f} } && T \\\\ \\\\ S / \\RR_f \\ar[urur]_*{r}  } \\end {xy}$"}234{"_id": "266", "title": "Quotient Theorem for Sets", "text": "A mapping $f: S \\to T$ can be uniquely '''factored into''' a surjection, followed by a bijection, followed by an injection. Thus: :$f = i \\circ r \\circ q_{\\RR_f}$ where: {{begin-eqn}} {{eqn | ll= q_{\\RR_f}:       | lo= S \\to S / \\RR_f:       | l = \\map {q_{\\RR_f} } s       | r = \\eqclass s {\\RR_f}       | c = Quotient Mapping }} {{eqn | ll= r:       | lo= S / \\RR_f \\to \\Img f:       | l = \\map r {\\eqclass s {\\RR_f} }       | r = \\map f s       | c = Renaming Mapping }} {{eqn | ll= i:       | lo= \\Img f \\to T:       | l = \\map i t       | r = t       | c = Inclusion Mapping }} {{end-eqn}} where: :$\\RR_f$ is the equivalence induced by $f$ :$S / \\RR_f$ is the quotient set of $S$ induced by $\\RR_f$ This can be illustrated using a commutative diagram as follows: ::$\\begin {xy} \\xymatrix@L + 2mu@ + 1em {  S \\ar@{-->}[rrr]^*{f = i_T \\circ r \\circ q_{\\RR_f} }    \\ar[d]_*{q_{\\RR_f} } & & & T \\\\ S / \\RR_f \\ar[rrr]_*{r} & & & \\Img f \\ar[u]_*{i_T} } \\end {xy}$"}235{"_id": "267", "title": "Empty Set is Element of Power Set", "text": "The empty set is an element of all power sets: :$\\forall S: \\O \\in \\powerset S$"}236{"_id": "268", "title": "Direct Image Mapping of Relation is Mapping", "text": "Let $S$ and $T$ be sets. Let $\\mathcal R \\subseteq S \\times T$ be a relation on $S \\times T$. Let $\\mathcal R^\\to: \\powerset S \\to \\powerset T$ be the direct image mapping of $\\mathcal R$: :$\\forall X \\in \\powerset S: \\map {\\mathcal R^\\to} X = \\set {t \\in T: \\exists s \\in X: \\tuple {s, t} \\in \\mathcal R}$ Then $\\mathcal R^\\to$ is indeed a mapping."}237{"_id": "269", "title": "Inverse Image Mapping of Relation is Mapping", "text": "Let $S$ and $T$ be sets. Let $\\mathcal R \\subseteq S \\times T$ be a relation on $S \\times T$. Let $\\mathcal R^\\gets$ be the inverse image mapping of $\\mathcal R$: :$\\mathcal R^\\gets: \\powerset T \\to \\powerset S: \\map {\\mathcal R^\\gets} Y = \\mathcal R^{-1} \\sqbrk Y$ Then $\\mathcal R^\\gets$ is indeed a mapping."}238{"_id": "270", "title": "Direct Image Mapping of Injection is Injection", "text": "Let $f: S \\to T$ be an injection. Then the direct image mapping of $f$: :$f^\\to: \\powerset S \\to \\powerset T$ is an injection."}239{"_id": "271", "title": "Direct Image Mapping of Surjection is Surjection", "text": "Let $f: S \\to T$ be a surjection. Then the direct image mapping of $f$: :$f^\\to: \\powerset S \\to \\powerset T$ is a surjection."}240{"_id": "272", "title": "Cantor's Theorem", "text": "There is no surjection from a set $S$ to its power set for any set $S$. That is, $S$ is strictly smaller than its power set."}241{"_id": "273", "title": "Surjection Induced by Powerset is Induced by Surjection", "text": "Let $\\mathcal R \\subseteq S \\times T$ be a relation. Let $\\mathcal R^\\to: \\powerset S \\to \\powerset T$ be the direct image mapping $\\mathcal R$. Let $\\mathcal R^\\to$ be a surjection. Let $X = \\Preimg {\\mathcal R}$, that is, the preimage of $\\mathcal R$. Then $\\mathcal R {\\restriction_X} \\subseteq X \\times T$, that is, the restriction of $\\mathcal R$ to $X$, is a surjection."}242{"_id": "274", "title": "Mapping is Injection if its Direct Image Mapping is Injection", "text": "Let $f: S \\to T$ be a mapping. Let $f^\\to: \\powerset S \\to \\powerset T$ be the direct image mapping of $f$. Let $f^\\to$ be an injection. Then $f: S \\to T$ is also an injection."}243{"_id": "275", "title": "Direct Image Mapping is Bijection iff Mapping is Bijection", "text": "Let $\\mathcal R \\subseteq S \\times T$ be a relation. Let $\\mathcal R^\\to: \\powerset S \\to \\powerset T$ be the direct image mapping of $\\mathcal R$. Then $\\mathcal R \\subseteq S \\times T$ is a bijection {{iff}} $\\mathcal R^\\to: \\powerset S \\to \\powerset T$ is a bijection."}244{"_id": "276", "title": "No Bijection from Set to its Power Set", "text": "Let $S$ be a set. Let $\\powerset S$ denote the power set of $S$. There is no bijection $f: S \\to \\powerset S$."}245{"_id": "278", "title": "Union of Power Sets", "text": "The union of the power sets of two sets $S$ and $T$ is a subset of the power set of their union: :$\\powerset S \\cup \\powerset T \\subseteq \\powerset {S \\cup T}$"}246{"_id": "279", "title": "Intersection of Power Sets", "text": "The intersection of the power sets of two sets $S$ and $T$ is equal to the power set of their intersection: :$\\powerset S \\cap \\powerset T = \\powerset {S \\cap T}$"}247{"_id": "280", "title": "Cantor-Bernstein-Schröder Theorem", "text": "If a subset of one set is equivalent to the other, and a subset of the other is equivalent to the first, then the two sets are themselves equivalent: :$\\forall S, T: T \\sim S_1 \\subseteq S \\land S \\sim T_1 \\subseteq T \\implies S \\sim T$"}248{"_id": "281", "title": "Trivial Ordering is Universally Compatible", "text": "Let $S$ be a set. Let $\\RR$ be the trivial ordering on $S$. Then $\\RR$ is universally compatible."}249{"_id": "282", "title": "Identity Mapping is Order Isomorphism", "text": "Let $\\left({S, \\preceq}\\right)$ be an ordered set. The identity mapping $I_S$ is an order isomorphism from $\\left({S, \\preceq}\\right)$ to itself."}250{"_id": "283", "title": "Inverse of Order Isomorphism is Order Isomorphism", "text": "Let $\\left({S, \\preceq_1}\\right)$ and $\\left({T, \\preceq_2}\\right)$ be ordered sets. Let $\\phi$ be a bijection from $\\left({S, \\preceq_1}\\right)$ to $\\left({T, \\preceq_2}\\right)$. Then: : $\\phi: \\left({S, \\preceq_1}\\right) \\to \\left({T, \\preceq_2}\\right)$ is an order isomorphism {{iff}}: : $\\phi^{-1}: \\left({T, \\preceq_2}\\right) \\to \\left({S, \\preceq_1}\\right)$ is also an order isomorphism."}251{"_id": "284", "title": "Composite of Order Isomorphisms is Order Isomorphism", "text": "Let $\\left({S_1, \\preceq_1}\\right)$, $\\left({S_2, \\preceq_2}\\right)$ and $\\left({S_3, \\preceq_3}\\right)$ be ordered sets. Let: : $\\phi: \\left({S_1, \\preceq_1}\\right) \\to \\left({S_2, \\preceq_2}\\right)$ and: : $\\psi: \\left({S_2, \\preceq_2}\\right) \\to \\left({S_3, \\preceq_3}\\right)$ be order isomorphisms. Then $\\psi \\circ \\phi: \\left({S_1, \\preceq_1}\\right) \\to \\left({S_3, \\preceq_3}\\right)$ is also an order isomorphism."}252{"_id": "285", "title": "Order Isomorphism is Equivalence Relation", "text": "Order isomorphism between ordered sets is an equivalence relation. So any given family of ordered sets can be partitioned into disjoint classes of isomorphic sets."}253{"_id": "286", "title": "Dual Ordering is Ordering", "text": "Let $\\left({S, \\preceq}\\right)$ be an ordered set. Let $\\succeq$ denote the dual ordering of $\\preceq$. Then $\\succeq$ is an ordering on $S$."}254{"_id": "288", "title": "Power Set is Complete Lattice", "text": "Let $S$ be a set. Let $\\left({\\mathcal P \\left({S}\\right), \\subseteq}\\right)$ be the relational structure defined on $\\mathcal P \\left({S}\\right)$ by the relation $\\subseteq$. Then $\\left({\\mathcal P \\left({S}\\right), \\subseteq}\\right)$ is a complete lattice."}255{"_id": "289", "title": "Totally Ordered Set is Lattice", "text": "Every totally ordered set is a lattice."}256{"_id": "290", "title": "Trichotomy Law (Ordering)", "text": "Let $\\struct {S, \\preceq}$ be an ordered set. Then $\\preceq$ is a total ordering {{iff}}: :$\\forall a, b \\in S: \\paren {a \\prec b} \\lor \\paren {a = b} \\lor \\paren {a \\succ b}$ That is, every element either strictly precedes, is the same as, or strictly succeeds, every other element. In other words, {{iff}} $\\prec$ is a trichotomy."}257{"_id": "291", "title": "Complement of Reflexive Relation", "text": "Let $\\RR \\subseteq S \\times S$ be a relation. Then $\\RR$ is reflexive {{iff}} its complement $\\relcomp {S \\times S} \\RR \\subseteq S \\times S$ is antireflexive. Likewise, $\\RR$ is antireflexive {{iff}} its complement $\\relcomp {S \\times S} \\RR \\subseteq S \\times S$ is reflexive."}258{"_id": "293", "title": "Strictly Increasing Mapping is Increasing", "text": "A mapping that is strictly increasing is an increasing mapping."}259{"_id": "294", "title": "Strictly Decreasing Mapping is Decreasing", "text": "A mapping that is strictly decreasing is a decreasing mapping."}260{"_id": "297", "title": "Strictly Monotone Mapping with Totally Ordered Domain is Injective", "text": "Let $\\struct {S, \\preceq_1}$ be a totally ordered set. Let $\\struct {T, \\preceq_2}$ be an ordered set. Let $\\phi: \\struct {S, \\preceq_1} \\to \\struct {T, \\preceq_2}$ be a strictly monotone mapping. Then $\\phi$ is injective."}261{"_id": "298", "title": "Mapping from Totally Ordered Set is Order Embedding iff Strictly Increasing", "text": "Let $\\struct {S, \\preceq_1}$ be a totally ordered set. Let $\\struct {T, \\preceq_2}$ be an ordered set. Let $\\phi: S \\to T$ be a mapping. Then $\\phi$ is an order embedding {{iff}} $\\phi$ is strictly increasing. That is: :$\\forall x, y \\in S: x \\preceq_1 y \\iff \\map \\phi x \\preceq_2 \\map \\phi y$ {{iff}}: :$\\forall x, y \\in S: x \\prec_1 y \\implies \\map \\phi x \\prec_2 \\map \\phi y$"}262{"_id": "299", "title": "Finite Totally Ordered Set is Well-Ordered", "text": "Every finite totally ordered set is well-ordered."}263{"_id": "300", "title": "Subset of Well-Ordered Set is Well-Ordered", "text": "Every non-empty subset of a well-ordered set is itself well-ordered."}264{"_id": "301", "title": "Element Commutes with Product of Commuting Elements", "text": "Let $(S, \\circ)$ be a semigroup. Let $x, y, z \\in S$. If $x$ commutes with both $y$ and $z$, then $x$ commutes with $y \\circ z$."}265{"_id": "302", "title": "Associative Idempotent Anticommutative", "text": "Let $\\circ$ be a binary operation on a set $S$. Let $\\circ$ be associative. Then $\\circ$ is anticommutative {{iff}}: : $(1): \\quad \\circ$ is idempotent and: : $(2): \\quad \\forall a, b \\in S: a \\circ b \\circ a = a$."}266{"_id": "303", "title": "Associative and Anticommutative", "text": "Let $\\circ$ be a binary operation on a set $S$. Let $\\circ$ be both associative and anticommutative. Then: :$\\forall x, y, z \\in S: x \\circ y \\circ z = x \\circ z$"}267{"_id": "304", "title": "Constant Operation is Commutative", "text": "Let $S$ be a set. Let $x \\sqbrk c y = c$ be a constant operation on $S$. Then $\\sqbrk c$ is a commutative operation: :$\\forall x, y \\in S: x \\sqbrk c y = y \\sqbrk c x$"}268{"_id": "305", "title": "Constant Operation is Associative", "text": "Let $S$ be a set. Let $x \\left[{c}\\right] y = c$ be a constant operation on $S$. Then $\\left[{c}\\right]$ is an associative operation: :$\\forall x, y, z \\in S: \\left({x \\left[{c}\\right] y}\\right) \\left[{c}\\right] z = x \\left[{c}\\right] \\left({y \\left[{c}\\right] z}\\right)$"}269{"_id": "306", "title": "Left Operation is Idempotent", "text": "The left operation is idempotent: :$\\forall x: x \\leftarrow x = x$"}270{"_id": "310", "title": "Left Operation is Associative", "text": "The left operation is associative: :$\\forall x, y, z: \\paren {x \\leftarrow y} \\leftarrow z = x \\leftarrow \\paren {y \\leftarrow z}$"}271{"_id": "311", "title": "Right Operation is Associative", "text": "The right operation is associative: :$\\forall x, y, z: \\paren {x \\rightarrow y} \\rightarrow z = x \\rightarrow \\paren {y \\rightarrow z}$"}272{"_id": "312", "title": "Max and Min are Commutative", "text": "The Max and Min operations are commutative: : $\\max \\left({x, y}\\right) = \\max \\left({y, x}\\right)$ : $\\min \\left({x, y}\\right) = \\min \\left({y, x}\\right)$"}273{"_id": "313", "title": "Max and Min are Associative", "text": "The Max and Min operations are associative: : $\\max \\left({\\max \\left({x, y}\\right), z}\\right) = \\max \\left({x, \\max \\left({y, z}\\right)}\\right)$ : $\\min \\left({\\min \\left({x, y}\\right), z}\\right) = \\min \\left({x, \\min \\left({y, z}\\right)}\\right)$ Thus we are justified in writing $\\max \\left({x, y, z}\\right)$ and $\\min \\left({x, y, z}\\right)$."}274{"_id": "314", "title": "Max and Min are Idempotent", "text": "The Max and Min operations are idempotent: : $\\max \\left({x, x}\\right) = x$ : $\\min \\left({x, x}\\right) = x$"}275{"_id": "315", "title": "Max and Min Operations are Distributive over Each Other", "text": "The Max and Min operations are distributive over each other: :$\\max \\set {x, \\min \\set {y, z} } = \\min \\set {\\max \\set {x, y}, \\max \\set {x, z} }$ :$\\max \\set {\\min \\set {x, y}, z} = \\min \\set {\\max \\set {x, z}, \\max \\set {y, z} }$ :$\\min \\set {x, \\max \\set {y, z} } = \\max \\set {\\min \\set {x, y}, \\min \\set {x, z} }$ :$\\min \\set {\\max \\set {x, y}, z} = \\max \\set {\\min \\set {x, z}, \\min \\set {y, z} }$"}276{"_id": "316", "title": "Equality of Algebraic Structures", "text": "Two algebraic structures $\\struct {S, \\circ}$ and $\\struct {T, *}$ are equal {{iff}}: :$S = T$ :$\\forall a, b \\in S: a \\circ b = a * b$"}277{"_id": "317", "title": "Restriction of Associative Operation is Associative", "text": "Let $\\struct {S, \\circ}$ be an semigroup. Let $T \\subseteq S$. Let $T$ be closed under $\\circ$. Then $\\struct {T, \\circ {\\restriction_T} }$ is also a semigroup, where $\\circ {\\restriction_T}$ is the restriction of $\\circ$ to $T$."}278{"_id": "318", "title": "Restriction of Commutative Operation is Commutative", "text": "Let $\\struct {S, \\circ}$ be an algebraic structure. Let $T \\subseteq S$. Let the operation $\\circ$ be commutative on $\\struct {S, \\circ}$. Then the restriction $\\circ {\\restriction_T}$ of $\\circ$ to $T$ is also commutative."}279{"_id": "319", "title": "Idempotent Magma Element forms Singleton Submagma", "text": "Let $\\struct {S, \\circ}$ be a magma. Let $x \\in S$ be an idempotent element of $\\struct {S, \\circ}$. Then  $\\struct {\\set x, \\circ}$ is a submagma of $\\struct {S, \\circ}$."}280{"_id": "320", "title": "Magma Subset Product with Self", "text": "Let $\\left({S, \\circ}\\right)$ be a magma. Let $T \\subseteq S$. Then $\\left({T, \\circ}\\right)$ is a magma {{iff}} $T \\circ T \\subseteq T$, where $T \\circ T$ is the subset product of $T$ with itself."}281{"_id": "321", "title": "Subset Product within Semigroup is Associative", "text": "Let $\\struct {S, \\circ}$ be a semigroup. Then the operation $\\circ_\\PP$ induced on the power set of $S$ is also associative."}282{"_id": "322", "title": "Subset Product within Commutative Structure is Commutative", "text": "Let $\\struct {S, \\circ}$ be a magma. If $\\circ$ is commutative, then the operation $\\circ_\\mathcal P$ induced on the power set of $S$ is also commutative."}283{"_id": "323", "title": "Subset of Subset Product", "text": "Let $\\struct {S, \\circ}$ be a magma. Let $\\powerset S$ be the power set of $S$. Let $X, Y, Z \\in \\powerset S$. Then: :$X \\subseteq Y \\implies \\paren {X \\circ Z} \\subseteq \\paren {Y \\circ Z}$ :$X \\subseteq Y \\implies \\paren {Z \\circ X} \\subseteq \\paren {Z \\circ Y}$ where $X \\circ Z$ etc. denotes subset product."}284{"_id": "324", "title": "Cancellable Element is Cancellable in Subset", "text": "Let $\\left ({S, \\circ}\\right)$ be an algebraic structure. Let $\\left ({T, \\circ}\\right) \\subseteq \\left ({S, \\circ}\\right)$. Let $x \\in T$ be cancellable in $S$. Then $x$ is also cancellable in $T$."}285{"_id": "325", "title": "Subsemigroup Closure Test", "text": "To show that an algebraic structure $\\left({T, \\circ}\\right)$ is a subsemigroup of a semigroup $\\struct {S, \\circ}$, we need to show only that: : $(1): \\quad T \\subseteq S$ : $(2): \\quad \\circ$ is a closed operation in $T$."}286{"_id": "326", "title": "Cancellable iff Regular Representations Injective", "text": "Let $\\struct {S, \\circ}$ be an algebraic structure. Then $a \\in S$ is cancellable {{iff}}: ::the left regular representation $\\map {\\lambda_a} x$ is injective and ::the right regular representation $\\map {\\rho_a} x$ is injective."}287{"_id": "328", "title": "More than one Left Identity then no Right Identity", "text": "Let $\\struct {S, \\circ}$ be an algebraic structure. If $\\struct {S, \\circ}$ has more than one left identity, then it has no right identity."}288{"_id": "329", "title": "Element under Left Operation is Right Identity", "text": "Let $\\struct {S, \\leftarrow}$ be an algebraic structure in which the operation $\\leftarrow$ is the left operation. Then no matter what $S$ is, $\\struct {S, \\leftarrow}$ is a semigroup all of whose elements are right identities. Thus it can be seen that any right identity in a semigroup is not necessarily unique."}289{"_id": "330", "title": "Element under Right Operation is Left Identity", "text": "Let $\\struct {S, \\rightarrow}$ be an algebraic structure in which the operation $\\rightarrow$ is the right operation. Then $\\struct {S, \\rightarrow}$ is a semigroup all of whose elements are left identities. Thus it can be seen that any left identity in a semigroup is not necessarily unique."}290{"_id": "331", "title": "Left and Right Identity are the Same", "text": "Let $\\struct {S, \\circ}$ be an algebraic structure. Let $e_L \\in S$ be a left identity, and $e_R \\in S$ be a right identity. Then: :$e_L = e_R$ that is, both the left identity and right identity are the same, and are therefore an identity $e$. Furthermore, $e$ is the only left identity and right identity for $\\circ$."}291{"_id": "332", "title": "Identity Property in Semigroup", "text": "Let $\\struct {S, \\circ}$ be a semigroup. Let $s \\in S$ be such that: :$\\forall a \\in S: \\exists x, y \\in S: s \\circ x = a = y \\circ s$ Then $\\struct {S, \\circ}$ has an identity."}292{"_id": "333", "title": "Identity of Monoid is Cancellable", "text": "The identity of a monoid is cancellable."}293{"_id": "334", "title": "Identity is only Idempotent Cancellable Element", "text": "Let $e_S$ is the identity of an algebraic structure $\\left({S, \\circ}\\right)$. Then $e_S$ is the only cancellable element of $\\left({S, \\circ}\\right)$ that is idempotent."}294{"_id": "335", "title": "Set of all Self-Maps is Monoid", "text": "Let $S$ be a set. Let $S^S$ be the set of all mappings from $S$ to itself. Let the operation $\\circ$ represent composition of mappings. Then the algebraic structure $\\struct {S^S, \\circ}$ is a monoid whose identity element is the identity mapping on $S$."}295{"_id": "336", "title": "Test for Submonoid", "text": "To show that $\\left({T, \\circ}\\right)$ is a submonoid of a monoid $\\left({S, \\circ}\\right)$, we need to show that: :$(1): \\quad T \\subseteq S$ :$(2): \\quad \\left({T, \\circ}\\right)$ is a magma (i.e. that it is closed) :$(3): \\quad \\left({T, \\circ}\\right)$ has an identity."}296{"_id": "337", "title": "Identity of Cancellable Monoid is Identity of Submonoid", "text": "Let $\\struct {S, \\circ}$ be a monoid, all of whose elements are cancellable. Let $\\struct {T, \\circ}$ be a submonoid of $\\struct {S, \\circ}$. Then the identity of $\\struct {T, \\circ}$ is the same element as the identity of $\\struct {S, \\circ}$."}297{"_id": "338", "title": "Cancellable Elements of Monoid form Submonoid", "text": "The cancellable elements of a monoid $\\struct {S, \\circ}$ form a submonoid of $\\struct {S, \\circ}$."}298{"_id": "339", "title": "Product of Semigroup Element with Left Inverse is Idempotent", "text": "Let $\\struct {S, \\circ}$ be a semigroup with a left identity $e_L$. Let $x \\in S$ such that $\\exists x_L: x_L \\circ x = e_L$, that is $x$ has a left inverse with respect to the left identity. Then: :$\\paren {x \\circ x_L} \\circ \\paren {x \\circ x_L} = x \\circ x_L$ That is, $x \\circ x_L$ is idempotent."}299{"_id": "340", "title": "Product of Semigroup Element with Right Inverse is Idempotent", "text": "Let $\\struct {S, \\circ}$ be a semigroup with a right identity $e_R$. Let $x \\in S$ such that $\\exists x_R: x \\circ x_R = e_R$, i.e. $x$ has a right inverse with respect to the right identity. Then: :$\\paren {x_R \\circ x} \\circ \\paren {x_R \\circ x} = x_R \\circ x$ That is, $x_R \\circ x$ is idempotent."}300{"_id": "341", "title": "Left Inverse for All is Right Inverse", "text": "Let $\\struct {S, \\circ}$ be a semigroup with a left identity $e_L$ such that: :$\\forall x \\in S: \\exists x_L: x_L \\circ x = e_L$ That is, every element of $S$ has a left inverse with respect to the left identity. Then $x \\circ x_L = e_L$, that is, $x_L$ is also a right inverse with respect to the left identity."}301{"_id": "342", "title": "Left Inverse and Right Inverse is Inverse", "text": "Let $\\struct {S, \\circ}$ be a monoid with identity element $e_S$. Let $x \\in S$ such that $x$ has both a left inverse and a right inverse. That is: :$\\exists x_L \\in S: x_L \\circ x = e_S$ :$\\exists x_R \\in S: x \\circ x_R = e_S$ Then $x_L = x_R$, that is, $x$ has an inverse. Furthermore, that element is the ''only'' inverse (both right and left) for $x$"}302{"_id": "343", "title": "Left and Right Inverses of Product", "text": "Let $\\struct {S, \\circ}$ be a monoid whose identity is $e_S$. Let $x, y \\in S$. Let: :$(1): \\quad x \\circ y$ have a left inverse for $\\circ$ :$(2): \\quad y \\circ x$ have a right inverse for $\\circ$. Then both $x$ and $y$ are invertible for $\\circ$."}303{"_id": "345", "title": "Equivalence of Definitions of Self-Inverse", "text": "Let $\\left({S, \\circ}\\right)$ be a monoid whose identity is $e_S$. Let $x \\in S$. {{TFAE|def = Self-Inverse Element|context = Abstract Algebra}}"}304{"_id": "346", "title": "Inverse of Identity Element is Itself", "text": "Let $\\struct {S, \\circ}$ be an algebraic structure with an identity element $e$. Let the inverse of $e$ be $e^{-1}$. Then: : $e^{-1} = e$ That is, $e$ is self-inverse."}305{"_id": "347", "title": "Invertible Element of Associative Structure is Cancellable", "text": "Let $\\left({S, \\circ}\\right)$ be an algebraic structure where $\\circ$ is associative. Let $\\left({S, \\circ}\\right)$ have an identity element $e_S$. An element of $\\left({S, \\circ}\\right)$ which is invertible is also cancellable."}306{"_id": "348", "title": "Regular Representation of Invertible Element is Permutation", "text": "Let $\\struct {S, \\circ}$ be a monoid. Let $a \\in S$ be invertible. Then the left regular representation $\\lambda_a$ and the right regular representation $\\rho_a$ are permutations of $S$."}307{"_id": "349", "title": "Group is Inverse Semigroup with Identity", "text": "A group is an inverse semigroup with an identity."}308{"_id": "350", "title": "Invertible Elements of Monoid form Subgroup of Cancellable Elements", "text": "Let $\\struct {S, \\circ}$ be an monoid whose identity is $e_S$. Let $C$ be the set of all cancellable elements of $S$. Let $T$ be the set of all invertible elements of $S$. Then $\\struct {T, \\circ}$ is a subgroup of $\\struct {C, \\circ}$."}309{"_id": "351", "title": "Structure Induced by Associative Operation is Associative", "text": "Let $\\struct {T, \\circ}$ be an algebraic structure, and let $S$ be a set. Let $\\struct {T^S, \\oplus}$ be the structure on $T^S$ induced by $\\circ$. Let $\\circ$ be associative. Then the pointwise operation $\\oplus$ induced on $T^S$ by $\\circ$ is also associative."}310{"_id": "352", "title": "Structure Induced by Commutative Operation is Commutative", "text": "Let $\\struct {T, \\circ}$ be an algebraic structure, and let $S$ be a set. Let $\\struct {T^S, \\oplus}$ be the structure on $T^S$ induced by $\\circ$. Let $\\circ$ be a commutative operation. Then the pointwise operation $\\oplus$ induced on $T^S$ by $\\circ$ is also commutative."}311{"_id": "353", "title": "Induced Structure Identity", "text": "Let $\\struct {T, \\circ}$ be an algebraic structure, and let $S$ be a set. Let $\\struct {T^S, \\oplus}$ be the structure on $T^S$ induced by $\\circ$. Let $e$ be an identity for $\\circ$. Then the constant mapping $f_e: S \\to T$ defined as: :$\\forall x \\in S: \\map {f_e} x = e$ is the identity for the pointwise operation $\\oplus$ induced on $T^S$ by $\\circ$."}312{"_id": "354", "title": "Structure Induced by Abelian Group Operation is Abelian Group", "text": "Let $\\struct {G, \\circ}$ be an abelian group whose identity is $e$. Let $S$ be a set. Let $\\struct {G^S, \\oplus}$ be the structure on $G^S$ induced by $\\circ$. Then $\\struct {G^S, \\oplus}$ is an abelian group."}313{"_id": "355", "title": "Power Set with Union is Commutative Monoid", "text": "Let $S$ be a set and let $\\powerset S$ be its power set. Then $\\struct {\\powerset S, \\cup}$ is a commutative monoid whose identity is $\\O$. The only invertible element of this structure is $\\O$. Thus (except in the degenerate case $S = \\O$) $\\struct {\\powerset S, \\cup}$ cannot be a group."}314{"_id": "356", "title": "Power Set with Intersection is Commutative Monoid", "text": "Let $S$ be a set and let $\\powerset S$ be its power set. Then $\\struct {\\powerset S, \\cap}$ is a commutative monoid whose identity is $S$. The only invertible element of this structure is $S$. Thus (except in the degenerate case $S = \\O$) $\\struct {\\powerset S, \\cap}$ cannot be a group."}315{"_id": "357", "title": "Diagonal Relation is Universally Compatible", "text": "The diagonal relation $\\Delta_S$ on a set $S$ is universally compatible with every operation on $S$."}316{"_id": "358", "title": "Trivial Relation is Universally Congruent", "text": "The trivial relation $\\mathcal R = S \\times S$ on a set $S$ is universally congruent with every closed operation on $S$."}317{"_id": "360", "title": "Quotient Structure is Well-Defined", "text": "Let $\\struct {S, \\circ}$ be an algebraic structure. Let $\\RR$ be a congruence relation on $\\struct {S, \\circ}$. Let $S / \\RR$ be the quotient set of $S$ by $\\RR$. Let $\\circ_\\RR$ be the operation induced on $S / \\RR$ by $\\circ$. Then $\\circ_\\RR$ is a well-defined operation in the quotient structure $\\struct {S / \\RR, \\circ_\\RR}$."}318{"_id": "362", "title": "External Direct Product Closure", "text": "Let $\\struct {S, \\circ_1}$ and $\\struct {T, \\circ_2}$ be algebraic structures. Let $\\struct {S \\times T, \\circ}$ be the external direct product of $\\struct {S, \\circ_1}$ and $\\struct {T, \\circ_2}$. Let $\\struct {S, \\circ_1}$ and $\\struct {T, \\circ_2}$ be closed. Then $\\struct {S \\times T, \\circ}$ is also closed."}319{"_id": "363", "title": "External Direct Product Associativity", "text": "Let $\\struct {S \\times T, \\circ}$ be the external direct product of the two algebraic structures $\\struct {S, \\circ_1}$ and $\\struct {T, \\circ_2}$. Let $\\circ_1$ and $\\circ_2$ be associative. Then $\\circ$ is also associative."}320{"_id": "364", "title": "External Direct Product of Semigroups", "text": "The external direct product of two semigroups is itself a semigroup."}321{"_id": "365", "title": "External Direct Product Commutativity", "text": "Let $\\left({S \\times T, \\circ}\\right)$ be the external direct product of the two algebraic structures $\\left({S, \\circ_1}\\right)$ and $\\left({T, \\circ_2}\\right)$. Let $\\circ_1$ and $\\circ_2$ be commutative operations. Then $\\circ$ is also a commutative operation."}322{"_id": "366", "title": "External Direct Product Identity", "text": "Let $\\struct {S \\times T, \\circ}$ be the external direct product of the two monoids $\\struct {S, \\circ_1}$ and $\\struct {T, \\circ_2}$. Let: : $e_S$ be the identity for $\\struct {S, \\circ_1}$ and: : $e_T$ be the identity for $\\struct {T, \\circ_2}$. Then $\\tuple {e_S, e_T}$ is the identity for $\\struct {S \\times T, \\circ}$."}323{"_id": "367", "title": "External Direct Product Inverses", "text": "Let $\\struct {S \\times T, \\circ}$ be the external direct product of the two monoids $\\struct {S, \\circ_1}$ and $\\struct {T, \\circ_2}$. Let: :$s^{-1}$ be an inverse of $s \\in \\struct {S, \\circ_1}$ and: :$t^{-1}$ be an inverse of $t \\in \\struct {T, \\circ_2}$. Then $\\tuple {s^{-1}, t^{-1} }$ is an inverse of $\\tuple {s, t} \\in \\struct {S \\times T, \\circ}$."}324{"_id": "368", "title": "Morphism Property Preserves Closure", "text": "Let $\\phi: \\left({S, \\circ_1, \\circ_2, \\ldots, \\circ_n}\\right) \\to \\left({T, *_1, *_2, \\ldots, *_n}\\right)$ be a mapping from one algebraic structure $\\left({S, \\circ_1, \\circ_2, \\ldots, \\circ_n}\\right)$ to another $\\left({T, *_1, *_2, \\ldots, *_n}\\right)$. Let $\\circ_k$ have the morphism property under $\\phi$ for some operation $\\circ_k$ in $\\left({S, \\circ_1, \\circ_2, \\ldots, \\circ_n}\\right)$. Then the following properties hold: : If $S' \\subseteq S$ is closed under $\\circ_k$, then $\\phi \\left[{S'}\\right]$ is closed under $*_k$ : If $T' \\subseteq T$ is closed under $*_k$, then $\\phi^{-1} \\left[{T'}\\right]$ is closed under $\\circ_k$ where $\\phi \\left[{S'}\\right]$ denotes the image of $S'$."}325{"_id": "370", "title": "Morphism Property Preserves Cancellability", "text": "Let: : $\\phi: \\left({S, \\circ_1, \\circ_2, \\ldots, \\circ_n}\\right) \\to \\left({T, *_1, *_2, \\ldots, *_n}\\right)$ be a mapping from one algebraic structure: : $\\left({S, \\circ_1, \\circ_2, \\ldots, \\circ_n}\\right)$ to another: : $\\left({T, *_1, *_2, \\ldots, *_n}\\right)$ Let $\\circ_k$ have the morphism property under $\\phi$ for some operation $\\circ_k$ in $\\left({S, \\circ_1, \\circ_2, \\ldots, \\circ_n}\\right)$. Then if an element $a \\in S$ is either left cancellable or right cancellable under $\\circ_k$, then $\\phi \\left({a}\\right)$ is correspondingly left cancellable or right cancellable under $*_k$. Thus, the morphism property is seen to preserve cancellability."}326{"_id": "371", "title": "Quotient Mapping on Structure is Canonical Epimorphism", "text": "Let $\\mathcal R$ be a congruence relation on an algebraic structure $\\struct {S, \\circ}$. Then the quotient mapping from $\\struct {S, \\circ}$ to the quotient structure $\\struct {S / \\mathcal R, \\circ_\\mathcal R}$ is an epimorphism: :$q_\\mathcal R: \\struct {S, \\circ} \\to \\struct {S / \\mathcal R, \\circ_\\mathcal R}: \\forall x, y \\in S: \\map {q_\\mathcal R} {x \\circ y} = \\map {q_\\mathcal R} x \\circ_\\mathcal R \\map {q_\\mathcal R} y$ This is sometimes called the '''canonical epimorphism''' from $\\struct {S, \\circ}$ to $\\struct {S / \\mathcal R, \\circ_\\mathcal R}$."}327{"_id": "372", "title": "Restriction of Homomorphism to Image is Epimorphism", "text": "Let $S$ and $T$ be algebraic structures. Let $\\phi: S \\to T$ be a homomorphism. Then a surjective restriction of $\\phi$ can be produced by limiting the codomain of $\\phi$ to its image $\\operatorname{Im} \\left({\\phi}\\right)$."}328{"_id": "373", "title": "Epimorphism Preserves Associativity", "text": "Let $\\struct {S, \\circ}$ and $\\struct {T, *}$ be algebraic structures. Let $\\phi: \\struct {S, \\circ} \\to \\struct {T, *}$ be an epimorphism. Let $\\circ$ be an associative operation. Then $*$ is also an associative operation."}329{"_id": "374", "title": "Epimorphism Preserves Semigroups", "text": "Let $\\left({S, \\circ}\\right)$ and $\\left({T, *}\\right)$ be algebraic structures. Let $\\phi: \\left({S, \\circ}\\right) \\to \\left({T, *}\\right)$ be an epimorphism. Let $\\left({S, \\circ}\\right)$ be a semigroup. Then $\\left({T, *}\\right)$ is also a semigroup."}330{"_id": "375", "title": "Homomorphism Preserves Subsemigroups", "text": "Let $\\struct {S, \\circ}$ and $\\struct {T, *}$ be semigroups. Let $\\phi: \\struct {S, \\circ} \\to \\struct {T, *}$ be a homomorphism. Let $S'$ be a subsemigroup of $S$. Then $\\phi \\paren {S'}$ is a subsemigroup of $T$."}331{"_id": "376", "title": "Epimorphism Preserves Commutativity", "text": "Let $\\left({S, \\circ}\\right)$ and $\\left({T, *}\\right)$ be algebraic structures. Let $\\phi: \\left({S, \\circ}\\right) \\to \\left({T, *}\\right)$ be an epimorphism. Let $\\circ$ be a commutative operation. Then $*$ is also a commutative operation."}332{"_id": "377", "title": "Epimorphism Preserves Identity", "text": "Let $\\struct {S, \\circ}$ and $\\struct {T, *}$ be algebraic structures. Let $\\phi: \\struct {S, \\circ} \\to \\struct {T, *}$ be an epimorphism. Let $\\struct {S, \\circ}$ have an identity element $e_S$. Then $\\struct {T, *}$ has the identity element $\\map \\phi {e_S}$."}333{"_id": "378", "title": "Epimorphism Preserves Inverses", "text": "Let $\\left({S, \\circ}\\right)$ and $\\left({T, *}\\right)$ be algebraic structures. Let $\\phi: \\left({S, \\circ}\\right) \\to \\left({T, *}\\right)$ be an epimorphism. Let $\\left({S, \\circ}\\right)$ have an identity $e_S$. Let $x^{-1}$ be an inverse element of $x$ for $\\circ$. Then $\\phi \\left({x^{-1}}\\right)$ is an inverse element of $\\phi \\left({x}\\right)$ for $*$. That is: : $\\phi \\left({x^{-1}}\\right) = \\left({\\phi \\left({x}\\right)}\\right)^{-1}$"}334{"_id": "379", "title": "Homomorphism with Cancellable Codomain Preserves Identity", "text": "Let $\\struct{S, \\circ}$ and $\\struct{T, *}$ be algebraic structures. Let $\\phi: \\struct{S, \\circ} \\to \\struct{T, *}$ be a homomorphism. Let $\\struct{S, \\circ}$ have an identity $e_S$. Let $\\struct{T, *}$ have an identity $e_T$. Let every element of $\\struct{T, *}$ be cancellable. Then $\\map \\phi {e_S}$ is the identity $e_T$."}335{"_id": "380", "title": "Homomorphism with Identity Preserves Inverses", "text": "Let $\\left({S, \\circ}\\right)$ and $\\left({T, *}\\right)$ be algebraic structures. Let $\\phi: \\left({S, \\circ}\\right) \\to \\left({T, *}\\right)$ be a homomorphism. Let $\\left({S, \\circ}\\right)$ have an identity $e_S$. Let $\\left({T, *}\\right)$ also have an identity $e_T = \\phi \\left({e_S}\\right)$. If $x^{-1}$ is an inverse of $x$ for $\\circ$, then $\\phi \\left({x^{-1}}\\right)$ is an inverse of $\\phi \\left({x}\\right)$ for $*$. That is: : $\\phi \\left({x^{-1}}\\right) = \\left({\\phi \\left({x}\\right)}\\right)^{-1}$"}336{"_id": "381", "title": "Homomorphism to Group Preserves Identity", "text": "Let $\\struct {S, \\circ}$ be an algebraic structure. Let $\\struct {T, *}$ be a group. Let $\\phi: \\struct {S, \\circ} \\to \\struct {T, *}$ be a homomorphism. Let $\\struct {S, \\circ}$ have an identity $e_S$. Then: :$\\map \\phi {e_S} = e_T$"}337{"_id": "382", "title": "Homomorphism of External Direct Products", "text": "Let: : $\\left({S_1 \\times S_2, \\circ}\\right)$ be the external direct product of two algebraic structures $\\left({S_1, \\circ_1}\\right)$ and $\\left({S_2, \\circ_2}\\right)$ : $\\left({T_1 \\times T_2, *}\\right)$ be the external direct product of two algebraic structures $\\left({T_1, *_1}\\right)$ and $\\left({T_2, *_2}\\right)$ : $\\phi_1$ be a homomorphism from $\\left({S_1, \\circ_1}\\right)$ onto $\\left({T_1, *_1}\\right)$ : $\\phi_2$ be a homomorphism from $\\left({S_2, \\circ_2}\\right)$ onto $\\left({T_2, *_2}\\right)$. Then the mapping $\\phi_1 \\times \\phi_2: \\left({S_1 \\times S_2, \\circ}\\right) \\to \\left({T_1 \\times T_2, *}\\right)$ defined as: :$\\left({\\phi_1 \\times \\phi_2}\\right) \\left({\\left({x, y}\\right)}\\right) = \\left({\\phi_1 \\left({x}\\right), \\phi_2 \\left({y}\\right)}\\right)$ is a homomorphism from $\\left({S_1 \\times S_2, \\circ}\\right)$ to $\\left({T_1 \\times T_2, *}\\right)$."}338{"_id": "383", "title": "Monomorphism Image is Isomorphic to Domain", "text": "The image of a monomorphism is isomorphic to its domain. That is, if $\\phi: S_1 \\to S_2$ is a monomorphism, then: :$\\phi: S_1 \\to \\Img \\phi$ is an isomorphism."}339{"_id": "384", "title": "Inverse of Algebraic Structure Isomorphism is Isomorphism", "text": "Let $\\left({S, \\circ}\\right)$ and $\\left({T, *}\\right)$ be algebraic structures. Let $\\phi: \\left({S, \\circ}\\right) \\to \\left({T, *}\\right)$ be a mapping. Then $\\phi$ is an isomorphism iff $\\phi^{-1}: \\left({T, *}\\right) \\to \\left({S, \\circ}\\right)$ is also an isomorphism."}340{"_id": "385", "title": "Isomorphism is Equivalence Relation", "text": "Isomorphism is an equivalence on a set of magmas. This result applies to all magmas: rings, groups, R-algebraic structures etc."}341{"_id": "386", "title": "Projection is Epimorphism", "text": "Let $\\struct {S, \\circ}$ be the external direct product of the algebraic structures $\\struct {S_1, \\circ_1}$ and $\\struct {S_2, \\circ_2}$. Then: :$\\pr_1$ is an epimorphism from $\\struct {S, \\circ}$ to $\\struct {S_1, \\circ_1}$ :$\\pr_2$ is an epimorphism from $\\struct {S, \\circ}$ to $\\struct {S_2, \\circ_2}$ where $\\pr_1$ and $\\pr_2$ are the first and second projection respectively of $\\struct {S, \\circ}$."}342{"_id": "387", "title": "Quotient Theorem for Epimorphisms", "text": "Let $\\struct {S, \\circ}$ and $\\struct {T, *}$ be algebraic structures. Let $\\phi: \\struct {S, \\circ} \\to \\struct {T, *}$ be an epimorphism. Let $\\mathcal R_\\phi$ be the equivalence induced by $\\phi$. Let $S / \\mathcal R_\\phi$ be the quotient of $S$ by $\\mathcal R_\\phi$. Let $q_{\\mathcal R_\\phi}: S \\to S / \\mathcal R_\\phi$ be the quotient mapping induced by $\\mathcal R_\\phi$. Let $\\struct {S / \\mathcal R_\\phi}, {\\circ_{\\mathcal R_\\phi} }$ be the quotient structure defined by $\\mathcal R_\\phi$. Then: :The induced equivalence $\\mathcal R_\\phi$ is a congruence relation for $\\circ$ :There is one and only one isomorphism $\\psi: \\struct {S / \\mathcal R_\\phi}, {\\circ_{\\mathcal R_\\phi} } \\to \\struct {T, *}$ which satisfies $\\psi \\bullet q_{\\mathcal R_\\phi} = \\phi$. where, in order not to cause notational confusion, $\\bullet$ is used as the symbol to denote composition of mappings."}343{"_id": "388", "title": "Homomorphism on Induced Structure", "text": "Let $\\struct {S, \\circ}$ be an algebraic structure. Let $\\struct {T, \\oplus}$ be a commutative semigroup. Let $T^S$ be the set of all mappings from $S$ to $T$. Let $f$ and $g$ be homomorphisms from $S$ into $T$. Let $f \\oplus' g$ be the pointwise operation on $T^S$ induced by $\\oplus$. Then $f \\oplus' g$ is a homomorphism from $\\struct {S, \\circ}$ into $\\struct {T, \\oplus}$."}344{"_id": "389", "title": "Inverse Mapping in Induced Structure", "text": "Let $\\left({S, \\circ}\\right)$ be an algebraic structure. Let $\\left({T, \\oplus}\\right)$ be an abelian group. Let $f$ be a homomorphism from $S$ into $T$. Let $f^*$ be the induced structure inverse of $f$. Then $f^*$ is a homomorphism from $\\left({S, \\circ}\\right)$ into $\\left({T, \\oplus}\\right)$."}345{"_id": "390", "title": "Set of Homomorphisms is Subgroup of All Mappings", "text": "Let $\\struct {S, \\circ}$ be an algebraic structure. Let $\\struct {T, \\oplus}$ be an abelian group. Let $\\struct {T^S, \\oplus}$ be the algebraic structure on $T^S$ induced by $\\oplus$. Then the set of all homomorphisms from $\\struct {S, \\circ}$ into $\\struct {T, \\oplus}$ is a subgroup of $\\struct {T^S, \\oplus}$."}346{"_id": "391", "title": "Transplanting Theorem", "text": "Let $\\left({S, \\circ}\\right)$ be an algebraic structure. Let $f: S \\to T$ be a bijection. Then there exists one and only one operation $\\oplus$ such that $f: \\left({S, \\circ}\\right) \\to \\left({T, \\oplus}\\right)$ is an isomorphism. The operation $\\oplus$ is defined by: :$\\forall x, y \\in T: x \\oplus y = f \\left({f^{-1} \\left({x}\\right) \\circ f^{-1} \\left({y}\\right)}\\right)$ The operation $\\oplus$ is called the '''transplant of $\\circ$ under $f$'''."}347{"_id": "392", "title": "Exists Bijection to a Disjoint Set", "text": "Let $S$ and $T$ be sets. Then there exists a bijection from $T$ onto a set $T'$ disjoint from $S$."}348{"_id": "393", "title": "Embedding Theorem", "text": "Let: :$(1): \\quad \\left({T_2, \\oplus_2}\\right)$ be a submagma of $\\left({S_2, *_2}\\right)$ :$(2): \\quad f: \\left({T_1, \\oplus_1}\\right) \\to \\left({T_2, \\oplus_2}\\right)$ be an isomorphism then there exists: :$(1): \\quad$ a magma $\\left({S_1, *_1}\\right)$ which algebraically contains $\\left({T_1, \\oplus_1}\\right)$ :$(2): \\quad g: \\left({S_1, *_1}\\right) \\to \\left({S_2, *_2}\\right)$ where $g$ is an isomorphism which extends $f$. {{Expand|Add the corollary that extends this theorem to structures with two operations.}}"}349{"_id": "396", "title": "Subset Product defining Inverse Completion of Commutative Semigroup is Commutative Semigroup", "text": "Let $\\left({S, \\circ}\\right)$ be a commutative semigroup. Let $\\left ({C, \\circ}\\right) \\subseteq \\left({S, \\circ}\\right)$ be the subsemigroup of cancellable elements of $\\left({S, \\circ}\\right)$. Let $\\left({T, \\circ'}\\right)$ be an inverse completion of $\\left({S, \\circ}\\right)$. Then: :$S \\circ' C^{-1}$ is a commutative semigroup where $S \\circ' C^{-1}$ is the subset product of $S$ with $C^{-1}$ under $\\circ'$ in $T$."}350{"_id": "397", "title": "Inverse Completion of Commutative Semigroup is Inverse Completion of Itself", "text": "Let $\\left({S, \\circ}\\right)$ be a commutative semigroup. Let $\\left ({C, \\circ}\\right) \\subseteq \\left({S, \\circ}\\right)$ be the subsemigroup of cancellable elements of $\\left({S, \\circ}\\right)$. Let $\\left({T, \\circ'}\\right)$ be an inverse completion of $\\left({S, \\circ}\\right)$. Then $\\left({T, \\circ'}\\right)$ is its own inverse completion."}351{"_id": "398", "title": "Identity of Inverse Completion of Commutative Monoid", "text": "Let $\\left({S, \\circ}\\right)$ be a commutative monoid whose identity is $e$. Let $\\left ({C, \\circ}\\right) \\subseteq \\left({S, \\circ}\\right)$ be the subsemigroup of cancellable elements of $\\left({S, \\circ}\\right)$. Let $\\left({T, \\circ'}\\right)$ be an inverse completion of $\\left({S, \\circ}\\right)$. Then $e \\in T$ is the identity for $\\circ'$."}352{"_id": "399", "title": "Inverse Completion of Commutative Semigroup is Abelian Group", "text": "Let $\\struct {S, \\circ}$ be a commutative semigroup Let all the elements of $\\struct {S, \\circ}$ be cancellable. Then an inverse completion of $\\struct {S, \\circ}$ is an abelian group."}353{"_id": "400", "title": "Taylor's Theorem", "text": "Every infinitely differentiable function can be approximated by a series of polynomials."}354{"_id": "401", "title": "Inverse Completion Theorem", "text": "Every commutative semigroup containing cancellable elements admits an inverse completion."}355{"_id": "402", "title": "Extension Theorem for Homomorphisms", "text": "Let $\\struct {S, \\circ}$ be a commutative semigroup with cancellable elements Let $\\struct {C, \\circ} \\subseteq \\struct {S, \\circ}$ be the subsemigroup of all cancellable elements of $S$ Let $\\struct {S', \\circ'}$ be an inverse completion of $\\struct {S, \\circ}$ Let $\\phi$ be a (semigroup) homomorphism from $\\struct {S, \\circ}$ into a semigroup $\\struct {T, *}$ such that $\\map \\phi y$ is invertible for all $y \\in C$. Then: :$(1): \\quad$ There is one and only one homomorphism $\\psi$ from $\\struct {S', \\circ'}$ into $\\struct {T, *}$ extending $\\phi$ :$(2): \\quad \\forall x \\in S, y \\in C: \\map \\psi {x \\circ' y^{-1} } = \\map \\phi * \\paren {\\map \\phi y}^{-1}$ :$(3): \\quad$ If $\\phi$ is a monomorphism, then so is $\\psi$."}356{"_id": "403", "title": "Extension Theorem for Isomorphisms", "text": "Let the following conditions be fulfilled: : Let $\\left({S, \\circ}\\right)$ be a commutative semigroup with cancellable elements : Let $\\phi$ be an isomorphism from $\\left({S, \\circ}\\right)$ into a semigroup $\\left({T, *}\\right)$ : Let $\\left({S', \\circ'}\\right)$ be an inverse completion of $\\left({S, \\circ}\\right)$ : Let $\\left({T', \\circ'}\\right)$ be an inverse completion of $\\left({T, \\circ}\\right)$. Then there is a unique isomorphism $\\phi': S' \\to T'$ extending $\\phi$."}357{"_id": "404", "title": "Inverse Completion is Unique", "text": "An inverse completion of a commutative semigroup is unique up to isomorphism."}358{"_id": "405", "title": "Extension Theorem for Distributive Operations", "text": "Let $\\struct {R, *}$ be a commutative semigroup, all of whose elements are cancellable. Let $\\struct {T, *}$ be an inverse completion of $\\struct {R, *}$. Let $\\circ$ be an operation on $R$ which distributes over $*$. Then: :$(1): \\quad$ There is a unique operation $\\circ'$ on $T$ which distributes over $*$ in $T$ and induces on $R$ the operation $\\circ$ :$(2): \\quad$ If $\\circ$ is associative, then so is $\\circ'$ :$(3): \\quad$ If $\\circ$ is commutative, then so is $\\circ'$ :$(4): \\quad$ If $e$ is an identity for $\\circ$, then $e$ is also an identity for $\\circ'$ :$(5): \\quad$ Every element cancellable for $\\circ$ is also cancellable for $\\circ'$."}359{"_id": "406", "title": "Monoid is not Empty", "text": "A monoid cannot be empty."}360{"_id": "407", "title": "Group is not Empty", "text": "A group cannot be empty."}361{"_id": "408", "title": "Identity is only Idempotent Element in Group", "text": "Every group has exactly one idempotent element: the identity."}362{"_id": "409", "title": "Group Product Identity therefore Inverses", "text": "Let $g$ and $h$ be elements of a group $G$ whose identity element is $e$. Then if either: :$g h = e$ or: :$h g = e$ it follows that: :$g = h^{-1}$ and: :$h = g^{-1}$"}363{"_id": "410", "title": "Self-Inverse Elements Commute iff Product is Self-Inverse", "text": "Let $\\struct {G, \\circ}$ be a group. Let $x, y \\in \\struct {G, \\circ}$, such that $x$ and $y$ are self-inverse. Then $x$ and $y$ commute {{iff}} $x \\circ y$ is also self-inverse."}364{"_id": "411", "title": "Power Set of Group under Induced Operation is Semigroup", "text": "Let $\\struct {G, \\circ}$ be a group. Let $\\struct {\\powerset G, \\circ_\\mathcal P}$ be the algebraic structure consisting of the power set of $G$ and the operation induced on $\\powerset G$ by $\\circ$. Then $\\struct {\\powerset G, \\circ_\\mathcal P}$ is a semigroup."}365{"_id": "412", "title": "Inverse of Product of Subsets of Group", "text": "Let $\\struct {G, \\circ}$ be a group. Let $X, Y \\subseteq G$. Then: :$\\paren {X \\circ Y}^{-1} = Y^{-1} \\circ X^{-1}$ where $X^{-1}$ is the inverse of $X$."}366{"_id": "413", "title": "Regular Representations in Group are Permutations", "text": "Let $\\struct {G, \\circ}$ be a group. Let $a \\in G$ be any element of $G$. Then the left regular representation $\\lambda_a$ and the right regular representation $\\rho_a$ are permutations of $G$."}367{"_id": "414", "title": "Set Equivalence of Regular Representations", "text": "If $S$ is a finite subset of a group $G$, then: :$\\card {a \\circ S} = \\card S = \\left|{S \\circ a}\\right|$ That is, $a \\circ S$, $S$ and $S \\circ a$ are equivalent: $a \\circ S \\sim S \\sim S \\circ a$."}368{"_id": "415", "title": "Composition of Regular Representations", "text": "Let $\\left({S, *}\\right)$ be a semigroup. Let $\\lambda_x, \\rho_x$ be the left and right regular representations of $\\left({S, *}\\right)$ with respect to $x$. Let $\\lambda_x \\circ \\lambda_y$, $\\rho_x \\circ \\rho_y$ etc. be defined as the composition of the mappings $\\lambda_x$ and $\\lambda_y$ etc. Then $\\forall x, y \\in S$, the following results hold:"}369{"_id": "416", "title": "All Elements Self-Inverse then Abelian", "text": "Let $\\struct {G, \\circ}$ be a group. Suppose that every element of $G$ is self-inverse. Then $G$ is abelian."}370{"_id": "417", "title": "Commutation Property in Group", "text": "Let $\\struct {G, \\circ}$ be a group. Then $x$ and $y$ commute {{iff}} $x \\circ y \\circ x^{-1} = y$."}371{"_id": "418", "title": "Identity Mapping is Automorphism", "text": "The identity mapping $I_S: \\left({S, \\circ}\\right) \\to \\left({S, \\circ}\\right)$ on the algebraic structure $\\left({S, \\circ}\\right)$ is an automorphism. Its image is $S$."}372{"_id": "419", "title": "Group Homomorphism of Product with Inverse", "text": "Let $\\phi: \\struct {G, \\circ} \\to \\struct {H, *}$ be a group homomorphism. Then: :$(1): \\quad \\forall x, y \\in G: \\map \\phi {x \\circ y^{-1} } = \\map \\phi x * \\paren {\\map \\phi y}^{-1}$ :$(2): \\quad \\forall x, y \\in G: \\map \\phi {y^{-1} \\circ x} = \\paren {\\map \\phi y}^{-1} * \\map \\phi x$"}373{"_id": "420", "title": "Mapping to Square is Endomorphism iff Abelian", "text": "Let $\\struct {G, \\circ}$ be a group. Let $\\phi: G \\to G$ be defined as: :$\\forall g \\in G: \\map \\phi g = g \\circ g$ Then $\\struct {G, \\circ}$ is abelian {{iff}} $\\phi$ is a (group) endomorphism."}374{"_id": "421", "title": "Induced Group Product is Homomorphism iff Commutative", "text": "Let $\\struct {G, \\circ}$ be a group. Let $H_1, H_2$ be subgroups of $G$. Let $\\phi: H_1 \\times H_2 \\to G$ be defined such that: :$\\forall \\tuple {h_1, h_2} \\in H_1 \\times H_2: \\map \\phi {h_1, h_2} = h_1 \\circ h_2$ Then $\\phi$ is a homomorphism {{iff}} every element of $H_1$ commutes with every element of $H_2$."}375{"_id": "422", "title": "Isomorphism of Abelian Groups", "text": "Let $\\phi: \\struct {G, \\circ} \\to \\struct {H, *}$ be a group isomorphism. Then $\\struct {G, \\circ}$ is abelian {{iff}} $\\struct {H, *}$ is abelian."}376{"_id": "424", "title": "Group Example: x inv c y", "text": "Let $\\struct {G, \\circ}$ be a group. Let $c \\in G$. We define a new operation $*$ on $G$ as: :$\\forall x, y \\in G: x * y = x \\circ c^{-1} \\circ y$ Then $\\struct {G, *}$ is a group."}377{"_id": "425", "title": "Symmetric Difference on Power Set forms Abelian Group", "text": "Let $S$ be a set such that $\\O \\subset S$ (that is, $S$ is non-empty). Let $A * B$ be defined as the symmetric difference between $A$ and $B$. Let $\\powerset S$ denote the power set of $S$. Then the algebraic structure $\\struct {\\powerset S, *}$ is an abelian group."}378{"_id": "426", "title": "Group is Subgroup of Itself", "text": "Let $\\struct {G, \\circ}$ be a group. Then: :$\\struct {G, \\circ} \\le \\struct {G, \\circ}$ That is, a group is always a subgroup of itself."}379{"_id": "427", "title": "Identity of Subgroup", "text": "Let $G$ be a group whose identity is $e$. Let $H$ be a subgroup of group $G$. Then the identity of $H$ is also $e$."}380{"_id": "428", "title": "Inverses in Subgroup", "text": "Let $G$ be a group. Let $H$ be a subgroup of $G$. Then for each $h \\in H$, the inverse of $h$ in $H$ is the same as the inverse of $h$ in $G$."}381{"_id": "429", "title": "Subgroup of Abelian Group is Abelian", "text": "A subgroup of an abelian group is itself abelian."}382{"_id": "430", "title": "Intersection of Subgroups is Subgroup", "text": "The intersection of two subgroups of a group is itself a subgroup of that group: :$\\forall H_1, H_2 \\le \\struct {G, \\circ}: H_1 \\cap H_2 \\le G$ It also follows that $H_1 \\cap H_2 \\le H_1$ and $H_1 \\cap H_2 \\le H_2$."}383{"_id": "431", "title": "Union of Subgroups", "text": "Let $\\struct {G, \\circ}$ be a group. Let $H, K \\le G$ be subgroups of $G$. Let neither $H \\subseteq K$ nor $K \\subseteq H$. Then $H \\cup K$ is ''not'' a subgroup of $G$."}384{"_id": "432", "title": "Elements of Group with Equal Images under Homomorphisms form Subgroup", "text": "Let $\\struct {G, \\circ}$ and $\\struct {H, *}$ be groups. Let $f: G \\to H$ and $g: G \\to H$ be group homomorphisms. Then the set: :$S = \\set {x \\in G: \\map f x = \\map g x}$ is a subgroup of $G$."}385{"_id": "433", "title": "Product of Subgroup with Itself", "text": "Let $\\struct {G, \\circ}$ be a group. Then: : $\\forall H \\le \\struct {G, \\circ}: H \\circ H = H$"}386{"_id": "434", "title": "Inverse of Subgroup", "text": "Let $\\struct {G, \\circ}$ be a group. Let $H$ be a subgroup of $G$. Then: :$H^{-1} = H$ where $H^{-1}$ is the inverse of $H$."}387{"_id": "435", "title": "Subset Product of Subgroups", "text": "Let $\\struct {G, \\circ}$ be a group. Let $H, K$ be subgroups of $G$. Then $H \\circ K$ is a subgroup of $G$ {{iff}} $H$ and $K$ are permutable. That is: $H \\circ K$ is a subgroup of $G$ {{iff}}: :$H \\circ K = K \\circ H$ where $H \\circ K$ denotes subset product."}388{"_id": "436", "title": "Group Homomorphism Preserves Subgroups", "text": "Let $\\struct {G_1, \\circ}$ and $\\struct {G_2, *}$ be groups. Let $\\phi: \\struct {G_1, \\circ} \\to \\struct {G_2, *}$ be a group homomorphism.  Then: :$H \\le G_1 \\implies \\phi \\sqbrk H \\le G_2$ where: :$\\phi \\sqbrk H$ denotes the image of $H$ under $\\phi$ :$\\le$ denotes subgroup. That is, group homomorphism preserves subgroups."}389{"_id": "437", "title": "Image of Group Homomorphism is Subgroup", "text": "Let $\\phi: G_1 \\to G_2$ be a group homomorphism. Then: :$\\Img \\phi \\le G_2$ where $\\le$ denotes the relation of being a subgroup."}390{"_id": "438", "title": "Conjugacy is Equivalence Relation", "text": "Conjugacy of group elements is an equivalence relation."}391{"_id": "439", "title": "Kernel of Group Homomorphism is Subgroup", "text": "The kernel of a group homomorphism is a subgroup of its domain: :$\\map \\ker \\phi \\le \\Dom \\phi$"}392{"_id": "441", "title": "Commutative Semigroup is Entropic Structure", "text": "A commutative semigroup is an entropic structure."}393{"_id": "442", "title": "Abelian Group Induces Entropic Structure", "text": "Let $\\struct {G, \\circ}$ be an abelian group. Let the operation $*$ be defined on $G$ such that: :$\\forall x, y \\in G: x * y = x \\circ y^{-1}$ Then $\\struct {G, *}$ is an entropic structure."}394{"_id": "443", "title": "Cancellable Semiring with Unity is Additive Semiring", "text": "Let $\\struct {S, *, \\circ}$ be a cancellable semiring with unity $1_S$. Then the distributand $*$ is commutative. That is to say, $\\struct {S, *, \\circ}$ is also an additive semiring."}395{"_id": "444", "title": "Ring is not Empty", "text": "A ring cannot be empty."}396{"_id": "445", "title": "Ring Product with Zero", "text": "Let $\\struct {R, +, \\circ}$ be a ring whose zero is $0_R$. Then: :$\\forall x \\in R: 0_R \\circ x = 0_R = x \\circ 0_R$ That is, the zero is a zero element for the ring product, thereby justifying its name."}397{"_id": "446", "title": "Product with Ring Negative", "text": "Let $\\struct {R, +, \\circ}$ be a ring. Then: :$\\forall x, y \\in \\struct  {R, +, \\circ}: \\paren {-x} \\circ y = -\\paren {x \\circ y} = x \\circ \\paren {-y}$ where $\\paren {-x}$ denotes the negative of $x$."}398{"_id": "447", "title": "Unity of Ring is Unique", "text": "A ring can have no more than one unity."}399{"_id": "449", "title": "Null Ring iff Zero and Unity Coincide", "text": "The null ring is the only ring in which the unity and zero coincide."}400{"_id": "450", "title": "Unity is Unit", "text": "The unity in a ring is a unit."}401{"_id": "451", "title": "Unity and Negative form Subgroup of Units", "text": "Let $\\struct {R, +, \\circ}$ be a ring with unity. Then: :$\\struct {\\set {1_R, -1_R}, \\circ} \\le U_R$ That is, the set consisting of the unity and its negative forms a subgroup of the group of units."}402{"_id": "452", "title": "Negative of Product Inverse", "text": "Let $\\left({R, +, \\circ}\\right)$ be a ring with unity. Let $z \\in U_R$, where $U_R$ is the set of units. Then: :$\\left({- z}\\right)^{-1} = - \\left({z^{-1}}\\right)$. where $z^{-1}$ is the ring product inverse of $z$."}403{"_id": "453", "title": "Product of Negative with Product Inverse", "text": "Let $\\left({R, +, \\circ}\\right)$ be a ring with unity. Let $z \\in U_R$, where $U_R$ is the set of units. Then: : $(1): \\quad \\forall x \\in R: -\\left({x \\circ z^{-1}}\\right) = \\left({- x}\\right) \\circ z^{-1} = x \\circ \\left({\\left({- z}\\right)^{-1}}\\right)$ : $(2): \\quad \\forall x \\in R: -\\left({z^{-1} \\circ x}\\right) = z^{-1} \\circ \\left({- x}\\right) = \\left({\\left({- z}\\right)^{-1}}\\right) \\circ x$"}404{"_id": "454", "title": "Negative of Division Product", "text": "Let $\\struct {R, +, \\circ}$ be a commutative ring with unity. Let $\\struct {U_R, \\circ}$ be the group of units of $\\struct {R, +, \\circ}$. Then: :$\\displaystyle \\forall x \\in R: -\\frac x z = \\frac {-x} z = \\frac x {-z}$ where $\\dfrac x z$ is defined as $x \\circ \\paren {z^{-1} }$, that is the division product of $x$ by $z$."}405{"_id": "455", "title": "Addition of Division Products", "text": "Let $\\struct {R, +, \\circ}$ be a commutative ring with unity. Let $\\struct {U_R, \\circ}$ be the group of units of $\\struct {R, +, \\circ}$. Let $a, c \\in R, b, d \\in U_R$. Then: :$\\dfrac a b + \\dfrac c d = \\dfrac {a \\circ d + b \\circ c} {b \\circ d}$ where $\\dfrac x z$ is defined as $x \\circ \\paren {z^{-1} }$, that is, $x$ divided by $z$. The operation $+$ is well-defined. That is: :$\\dfrac a b = \\dfrac {a'} {b'}, \\dfrac c d = \\dfrac {c'} {d'} \\implies \\dfrac a b + \\dfrac c d = \\dfrac {a'} {b'} + \\dfrac {c'} {d'}$ {{questionable|This is an existing operation and thus already well-defined. See talk.}}"}406{"_id": "456", "title": "Equality of Division Products", "text": "Let $\\struct {R, +, \\circ}$ be a commutative ring with unity. Let $\\struct {U_R, \\circ}$ be the group of units of $\\struct {R, +, \\circ}$. Let $a, b \\in R, c, d \\in U_R$. Then: :$\\dfrac a c = \\dfrac b d \\iff a \\circ d = b \\circ c$ where $\\dfrac x z$ is defined as $x \\circ \\paren {z^{-1} }$, that is, $x$ divided by $z$."}407{"_id": "457", "title": "Product of Division Products", "text": "Let $\\struct {R, +, \\circ}$ be a commutative ring with unity. Let $\\struct {U_R, \\circ}$ be the group of units of $\\struct {R, +, \\circ}$. Let $a, b \\in R, c, d \\in U_R$. Then: :$\\dfrac a c \\circ \\dfrac b d = \\dfrac {a \\circ b} {c \\circ d}$ where $\\dfrac x z$ is defined as $x \\circ \\paren {z^{-1} }$, that is, $x$ divided by $z$."}408{"_id": "458", "title": "Inverse of Division Product", "text": "Let $\\struct {R, +, \\circ}$ be a commutative ring with unity. Let $\\struct {U_R, \\circ}$ be the group of units of $\\struct {R, +, \\circ}$. Let $a, b \\in U_R$. Then: :$\\paren {\\dfrac a b}^{-1} = \\dfrac {1_R} {\\paren {a / b}} = \\dfrac b a$ where $\\dfrac x z$ is defined as $x \\circ \\paren {z^{-1} }$, that is, $x$ divided by $z$."}409{"_id": "459", "title": "Zero Product with Proper Zero Divisor is with Zero Divisor", "text": "Let $\\struct {R, +, \\circ}$ be a ring. Let $x \\in R$ be a proper zero divisor of $R$. Then: :$\\paren {x \\divides 0_R} \\land \\paren {x \\circ y = 0_R} \\land \\paren {y \\ne 0_R} \\implies y \\divides 0_R$ That is, if $x$ is a proper zero divisor, then whatever non-zero element you form the product with it by to get zero must itself be a zero divisor."}410{"_id": "460", "title": "Unit Not Zero Divisor", "text": "A unit of a ring is not a zero divisor."}411{"_id": "461", "title": "Zero Divisor Product is Zero Divisor", "text": "The ring product of a zero divisor with any ring element is a zero divisor."}412{"_id": "462", "title": "Product is Zero Divisor means Zero Divisor", "text": "If the ring product of two elements of a ring is a zero divisor, then one of the two elements must be a zero divisor."}413{"_id": "463", "title": "Ring Element is Zero Divisor iff not Cancellable", "text": "Let $\\struct {R, +, \\circ}$ be a ring which is not null. Let $z \\in R^*$. Then $z$ is a zero divisor {{iff}} $z$ is not cancellable for $\\circ$."}414{"_id": "464", "title": "Ring Less Zero is Semigroup for Product iff No Proper Zero Divisors", "text": "Let $\\struct {R, +, \\circ}$ be a non-null ring. Then $R$ has no zero divisors {{iff}} $\\struct {R^*, \\circ}$ is a semigroup."}415{"_id": "465", "title": "Idempotent Elements of Ring with No Proper Zero Divisors", "text": "Let $\\left({R, +, \\circ}\\right)$ be a non-null ring with no (proper) zero divisors. Let $x \\in R$. Then: :$x \\circ x = x \\iff x \\in \\left\\{{0_R, 1_R}\\right\\}$ That is, the only elements of $R$ that are idempotent are zero and unity."}416{"_id": "466", "title": "Non-Zero Elements of Division Ring form Group", "text": "Let $\\struct {R, +, \\circ}$ be a division ring. Then $\\struct {R^*, \\circ}$ is a group."}417{"_id": "467", "title": "Null Ring and Ring Itself Subrings", "text": "In any ring $R$, the null ring and $R$ itself are subrings of $R$."}418{"_id": "468", "title": "Subring Test", "text": "Let $S$ be a subset of a ring $\\struct {R, +, \\circ}$. Then $\\struct {S, +, \\circ}$ is a subring of $\\struct {R, +, \\circ}$ {{iff}} these all hold: :$(1): \\quad S \\ne \\O$ :$(2): \\quad \\forall x, y \\in S: x + \\paren {-y} \\in S$ :$(3): \\quad \\forall x, y \\in S: x \\circ y \\in S$"}419{"_id": "469", "title": "Subdomain Test", "text": "Let $S$ be a subset of an integral domain $\\struct {R, +, \\circ}$. Then $\\struct {S, +\\restriction_S, \\circ \\restriction_S}$ is a subdomain of $\\struct {R, +, \\circ}$ {{iff}} these conditions hold: :$(1): \\quad$ $\\struct {S, + \\restriction_S, \\circ \\restriction_S}$ is a subring of $\\struct {R, +, \\circ}$ :$(2): \\quad$ The unity of $R$ is also in $S$, that is $1_R = 1_S$."}420{"_id": "470", "title": "Centralizer of Ring Subset is Subring", "text": "Let $S$ be a subset of a ring $\\struct {R, +, \\circ}$ Then $\\map {C_R} S$, the centralizer of $S$ in $R$, is a subring of $R$."}421{"_id": "471", "title": "Center of Ring is Commutative Subring", "text": "The center $\\map Z R$ of a ring $R$ is a commutative subring of $R$."}422{"_id": "472", "title": "Ideal is Subring", "text": "Let $\\left({R, +, \\circ}\\right)$ be a ring, and let $J$ be an ideal of $R$. Then $J$ is a subring of $R$."}423{"_id": "473", "title": "Ring is Ideal of Itself", "text": "Let $\\struct {R, +, \\circ}$ be a ring. Then $R$ is an ideal of $R$."}424{"_id": "474", "title": "Ideal of Unit is Whole Ring", "text": "Let $\\struct {R, +, \\circ}$ be a ring with unity. Let $J$ be an ideal of $R$. If $J$ contains a unit of $R$, then $J = R$."}425{"_id": "475", "title": "Test for Ideal", "text": "Let $J$ be a subset of a ring $\\struct {R, +, \\circ}$. Then $J$ is an ideal of $\\struct {R, +, \\circ}$ {{iff}} these all hold: : $(1): \\quad J \\ne \\O$ : $(2): \\quad \\forall x, y \\in J: x + \\paren {-y} \\in J$ : $(3): \\quad \\forall j \\in J, r \\in R: r \\circ j \\in J, j \\circ r \\in J$"}426{"_id": "476", "title": "Epimorphism Preserves Rings", "text": "Let $\\struct {R_1, +_1, \\circ_1}$ be a ring, and $\\struct {R_2, +_2, \\circ_2}$ be a closed algebraic structure. Let $\\phi: R_1 \\to R_2$ be an epimorphism. Then $\\struct {R_2, +_2, \\circ_2}$ is a ring."}427{"_id": "477", "title": "Ring Homomorphism of Addition is Group Homomorphism", "text": "Let $\\phi: \\left({R_1, +_1, \\circ_1}\\right) \\to \\left({R_2, +_2, \\circ_2}\\right)$ be a ring homomorphism. Then $\\phi: \\left({R_1, +_1}\\right) \\to \\left({R_2, +_2}\\right)$ is a group homomorphism."}428{"_id": "478", "title": "Element of Integral Domain Divides Zero", "text": "Let $\\struct {D, +, \\circ}$ be an integral domain whose zero is $0_D$. Then every element of $D$ is a divisor of zero: :$\\forall x \\in D: x \\divides 0_D$"}429{"_id": "479", "title": "Unity Divides All Elements", "text": "Let $\\struct {D, +, \\circ}$ be an integral domain whose unity is $1_D$. Then unity is a divisor of every element of $D$: :$\\forall x \\in D: 1_D \\divides x$ Also: :$\\forall x \\in D: -1_D \\divides x$"}430{"_id": "480", "title": "Element of Integral Domain is Divisor of Itself", "text": "Let $\\struct {D, +, \\circ}$ be an integral domain whose unity is $1_D$. Then every element of $D$ is a divisor of itself: :$\\forall x \\in D: x \\divides x$"}431{"_id": "481", "title": "Unit of Integral Domain divides all Elements", "text": "Let $\\struct {D, +, \\circ}$ be an integral domain whose unity is $1_D$. Let $\\struct {U_D, \\circ}$ be the group of units of $\\struct {D, +, \\circ}$. Then: :$\\forall x \\in D: \\forall u \\in U_D: u \\divides x$ That is, every unit of $D$ is a divisor of every element of $D$."}432{"_id": "484", "title": "Divisor Relation in Integral Domain is Transitive", "text": "Let $\\struct {D, +, \\circ}$ be an integral domain. Let $x, y, z \\in D$. Then: :$x \\divides y \\land y \\divides z \\implies x \\divides z$"}433{"_id": "485", "title": "Integers form Unique Factorization Domain", "text": "The integers $\\struct {\\Z, +, \\times}$ form a unique factorization domain."}434{"_id": "486", "title": "Trivial Ordering Compatibility in Boolean Ring", "text": "Let $\\struct {S, +, \\circ}$ be a Boolean ring. Then the trivial ordering is the only ordering on $S$ compatible with both its operations."}435{"_id": "488", "title": "Field is Integral Domain", "text": "Every field is an integral domain."}436{"_id": "489", "title": "Finite Integral Domain is Galois Field", "text": "A finite integral domain is a Galois field."}437{"_id": "491", "title": "Center of Division Ring is Subfield", "text": "Let $\\struct {K, +, \\circ}$ be an division ring. Then $\\map Z K$, the center of $K$, is a subfield of $K$."}438{"_id": "492", "title": "Ideals of Field", "text": "Let $\\struct {R, +, \\circ}$ be a commutative ring with unity whose zero is $0_R$ and whose unity is $1_R$. Then $\\struct {R, +, \\circ}$ is a field {{iff}} the only ideals of $\\struct {R, +, \\circ}$ are $\\struct {R, +, \\circ}$ and $\\set {0_R}$."}439{"_id": "493", "title": "Epimorphism from Division Ring to Ring", "text": "Let $\\left({K, +, \\circ}\\right)$ be a division ring whose zero is $0_K$. Let $\\left({R, +, \\circ}\\right)$ be a ring whose zero is $0_R$. Let $\\phi: K \\to R$ be a ring epimorphism. Then one of the following applies: :$(1): \\quad R$ is a null ring :$(2): \\quad R$ is a division ring and $\\phi$ is a ring isomorphism."}440{"_id": "494", "title": "Congruence Class Modulo Subgroup is Coset", "text": "Let $G$ be a group. Let $H$ be a subgroup of $G$."}441{"_id": "495", "title": "Cosets are Equivalent", "text": "All left cosets of a group $G$ with respect to a subgroup $H$ are equivalent. That is, any two left cosets are in one-to-one correspondence. The same applies to right cosets. As a special case of this: :$\\forall x \\in G: \\order {x H} = \\order H = \\order {H x}$ where $H$ is a subgroup of $G$."}442{"_id": "496", "title": "Left and Right Coset Spaces are Equivalent", "text": "Let $\\struct {G, \\circ}$ be a group. Let $H$ be a subgroup of $G$. Let: :$x H$ denote the left coset of $H$ by $x$ :$H y$ denote the right coset of $H$ by $y$. Then: :$\\order {\\set {x H: x \\in G} } = \\order {\\set {H y: y \\in G} }$"}443{"_id": "497", "title": "Cosets in Abelian Group", "text": "Let $G$ be an abelian group. Then every right coset modulo $H$ is a left coset modulo $H$. That is: :$\\forall x \\in G: x H = H x$ In an abelian group, therefore, we can talk about '''congruence modulo $H$''' and not worry about whether it is left or right."}444{"_id": "498", "title": "Conjugate of Set by Identity", "text": "Let $\\struct {G, \\circ}$ be a group whose identity is $e$. Let $S \\subseteq G$. Then the conjugate of $S$ by $e$ is $S$: :$S^e = S$"}445{"_id": "499", "title": "Conjugate of Set by Group Product", "text": "Let $\\struct {G, \\circ}$ be a group. Let $S \\subseteq G$. Let $S^a$ denote the $G$-conjugate of $S$ by $a$ as: :$S^a := \\set {y \\in G: \\exists x \\in S: y = a \\circ x \\circ a^{-1} } = a \\circ S \\circ a^{-1}$ Then: :$\\paren {S^a}^b = S^{b \\circ a}$"}446{"_id": "500", "title": "Conjugate of Subgroup is Subgroup", "text": "Let $G$ be a group. Let $H \\le G$ be a subgroup of $G$. Then the conjugate of $H$ by $a$ is a subgroup of $G$: :$\\forall H \\le G, a \\in G: H^a \\le G$"}447{"_id": "501", "title": "Inner Automorphisms form Normal Subgroup of Automorphism Group", "text": "Let $G$ be a group. Then the set $\\Inn G$ of all inner automorphisms of $G$ is a normal subgroup of the automorphism group $\\Aut G$ of $G$: :$\\Inn G \\lhd \\Aut G$"}448{"_id": "503", "title": "Subgroup of Abelian Group is Normal", "text": "Every subgroup of an abelian group is normal."}449{"_id": "505", "title": "Intersection with Normal Subgroup is Normal", "text": "Let $G$ be a group. Let $H$ be a subgroup of $G$, and let $N$ be a normal subgroup of $G$. Then $H \\cap N$ is a normal subgroup of $H$."}450{"_id": "506", "title": "Kernel is Normal Subgroup of Domain", "text": "Let $\\phi$ be a group homomorphism. Then the kernel of $\\phi$ is a normal subgroup of the domain of $\\phi$: :$\\map \\ker \\phi \\lhd \\Dom \\phi$"}451{"_id": "507", "title": "Coset Product is Well-Defined", "text": "Let $\\struct {G, \\circ}$ be a group. Let $N$ be a normal subgroup of $G$. Let $a, b \\in G$. Then the coset product: :$\\paren {a \\circ N} \\circ \\paren {b \\circ N} = \\paren {a \\circ b} \\circ N$ is well-defined."}452{"_id": "508", "title": "Quotient Group of Abelian Group is Abelian", "text": "Let $G$ be an abelian group. Let $N \\le G$. Then the quotient group $G / N$ is abelian."}453{"_id": "509", "title": "Quotient Theorem for Group Epimorphisms", "text": "Let $\\struct {G, \\oplus}$ and $\\struct {H, \\odot}$ be groups. Let $\\phi: \\struct {G, \\oplus} \\to \\struct {H, \\odot}$ be a group epimorphism. Let $e_G$ and $e_H$ be the identities of $G$ and $H$ respectively. Let $K = \\map \\ker \\phi$ be the kernel of $\\phi$. There is one and only one group isomorphism $\\psi: G / K \\to H$ satisfying: :$\\psi \\circ q_K = \\phi$ where $q_K$ is the quotient epimorphism from $G$ to $G / K$."}454{"_id": "510", "title": "Congruence Relation induces Normal Subgroup", "text": "Let $\\left({G, \\circ}\\right)$ be a group whose identity is $e$. Let $\\mathcal R$ be a congruence relation for $\\circ$. Let $H = \\left[\\!\\left[{e}\\right]\\!\\right]_\\mathcal R$, where $\\left[\\!\\left[{e}\\right]\\!\\right]_\\mathcal R$ is the equivalence class of $e$ under $\\mathcal R$. Then: : $(1): \\quad \\left({H, \\circ \\restriction_H}\\right)$ is a normal subgroup of $G$ : $(2): \\quad \\mathcal R$ is the equivalence relation $\\mathcal R_H$ defined by $H$ : $(3): \\quad \\left({G / \\mathcal R, \\circ_\\mathcal R}\\right)$ is the subgroup $\\left({G / H, \\circ_H}\\right)$ of the semigroup $\\left({\\mathcal P \\left({G}\\right), \\circ_\\mathcal P}\\right)$."}455{"_id": "511", "title": "Preimage of Normal Subgroup of Quotient Group under Quotient Epimorphism is Normal", "text": "Let $G$ be a group. Let $H \\lhd G$ where $\\lhd$ denotes that $H$ is a normal subgroup of $G$. Let $K \\lhd G/H$ and $L = q_H^{-1} \\left[{K}\\right]$, where: :$q_H: G \\to G/H$ is the quotient epimorphism from $G$ to the quotient group $G/H$ :$q_H^{-1} \\left[{K}\\right]$ is the preimage of $K$ under $q_H$. Then: :$L \\lhd G$"}456{"_id": "512", "title": "Trivial Quotient Group is Quotient Group", "text": "Let $G$ be a group. Then the trivial quotient group: :$G / \\set {e_G} \\cong G$ where: :$\\cong$ denotes group isomorphism :$e_G$ denotes the identity element of $G$ is a quotient group."}457{"_id": "513", "title": "Correspondence Theorem (Group Theory)", "text": "Let $G$ be a group. Let $N \\lhd G$ be a normal subgroup of $G$. Then every subgroup of the quotient group $G / N$ is of the form $H / N = \\set {h N: h \\in H}$, where $N \\le H \\le G$. Conversely, if $N \\le H \\le G$ then $H / N \\le G / N$. The correspondence between subgroups of $G / N$ and subgroups of $G$ containing $N$ is a bijection. This bijection maps normal subgroups of $G / N$ onto normal subgroups of $G$ which contain $N$."}458{"_id": "514", "title": "Centralizer of Group Element is Subgroup", "text": "Let $\\struct {G, \\circ}$ be a group and let $a \\in G$. Then $\\map {C_G} a$, the centralizer of $a$ in $G$, is a subgroup of $G$."}459{"_id": "515", "title": "Centralizer in Subgroup is Intersection", "text": "Let $G$ be a group. Let $H$ be a subgroup of $G$. Then: :$\\forall x \\in G: \\map {C_H} x = \\map {C_G} x \\cap H$ That is, the centralizer of an element in a subgroup is the intersection of that subgroup with the centralizer of the element in the group."}460{"_id": "516", "title": "Kernel of Inner Automorphism Group is Center", "text": "Let the mapping $\\kappa: G \\to \\Inn G$ from a group $G$ to its inner automorphism group $\\Inn G$ be defined as: :$\\map \\kappa a = \\kappa_a$ where $\\kappa_a$ is the inner automorphism of $G$ given by $a$. Then $\\kappa$ is a group epimorphism, and its kernel is the center of $G$: :$\\map \\ker \\kappa = \\map Z G$"}461{"_id": "517", "title": "Group equals Center iff Abelian", "text": "Let $G$ be a group. Then $G$ is abelian {{iff}} $\\map Z G = G$, that is, {{iff}} $G$ equals its center."}462{"_id": "518", "title": "Normalizer is Subgroup", "text": "Let $G$ be a group. The normalizer of a subset $S \\subseteq G$ is a subgroup of $G$. :$S \\subseteq G \\implies \\map {N_G} S \\le G$"}463{"_id": "519", "title": "Subgroup is Normal Subgroup of Normalizer", "text": "Let $G$ be a group. A subgroup $H \\le G$ is a normal subgroup of its normalizer: :$H \\le G \\implies H \\lhd \\map {N_G} H$"}464{"_id": "520", "title": "Normalizer of Subgroup is Largest Subgroup containing that Subgroup as Normal Subgroup", "text": "Let $G$ be a group. Let $H$ be a subgroup of $G$. Then $\\map {N_G} H$, the normalizer of $H$ in $G$, is the largest subgroup of $G$ containing $H$ as a normal subgroup."}465{"_id": "521", "title": "Normal Subgroup iff Normalizer is Group", "text": "Let $G$ be a group. Let $H$ be a subgroup of $G$. Then $H$ is normal in $G$ {{iff}} the normalizer of $H$ is equal to $G$: :$H \\lhd G \\iff \\map {N_G} H = G$"}466{"_id": "522", "title": "Normalizer of Conjugate is Conjugate of Normalizer", "text": "The normalizer of a conjugate is the conjugate of the normalizer: :$S \\subseteq G \\implies \\map {N_G} {S^a} = \\paren {\\map {N_G} S}^a$"}467{"_id": "525", "title": "Quotient Ring is Ring/Quotient Ring Product is Well-Defined", "text": "Let $\\struct {R, +, \\circ}$ be a ring whose zero is $0_R$ and whose unity is $1_R$. Let $J$ be an ideal of $R$. Let $\\struct {R / J, +, \\circ}$ be the quotient ring of $R$ by $J$. Then $\\circ$ is well-defined on $R / J$, that is: :$x_1 + J = x_2 + J, y_1 + J = y_2 + J \\implies x_1 \\circ y_1 + J = x_2 \\circ y_2 + J$"}468{"_id": "526", "title": "Quotient Ring of Commutative Ring is Commutative", "text": "Let $\\struct {R, +, \\circ}$ be a ring whose zero is $0_R$ and whose unity is $1_R$. Let $J$ be an ideal of $R$. Let $\\struct {R / J, +, \\circ}$ be the quotient ring defined by $J$. If $\\struct {R, +, \\circ}$ is a commutative ring, then so is $\\struct {R / J, +, \\circ}$."}469{"_id": "527", "title": "Quotient Ring of Ring with Unity is Ring with Unity", "text": "Let $\\struct {R, +, \\circ}$ be a ring with unity whose zero is $0_R$ and whose unity is $1_R$. Let $J$ be an ideal of $R$. Let $\\struct {R / J, +, \\circ}$ be the quotient ring defined by $J$. Then $\\struct {R / J, +, \\circ}$ is a ring with unity, and its unity is $1_R + J$."}470{"_id": "528", "title": "Ring Epimorphism with Trivial Kernel is Isomorphism", "text": "Let $\\phi: \\left({R_1, +_1, \\circ_1}\\right) \\to \\left({R_2, +_2, \\circ_2}\\right)$ be a ring epimorphism. Then $\\phi$ is a ring isomorphism iff $\\ker \\left({\\phi}\\right) = \\left\\{ {0_{R_1} }\\right\\}$."}471{"_id": "529", "title": "Quotient Epimorphism is Epimorphism/Ring", "text": "Let $\\struct {R, +, \\circ}$ be a ring whose zero is $0_R$ and whose unity is $1_R$. Let $J$ be an ideal of $R$. Let $\\struct {R / J, +, \\circ}$ be the quotient ring defined by $J$. Let $\\phi: R \\to R / J$ be the quotient (ring) epimorphism from $R$ to $R / J$: :$x \\in R: \\map \\phi x = x + J$ Then $\\phi$ is a ring epimorphism whose kernel is $J$."}472{"_id": "530", "title": "Ring Homomorphism Preserves Subrings", "text": "Let $\\phi: \\struct {R_1, +_1, \\circ_1} \\to \\struct {R_2, +_2, \\circ_2}$ be a ring homomorphism. If $S$ is a subring of $R_1$, then $\\phi \\sqbrk S$ is a subring of $R_2$."}473{"_id": "531", "title": "Kernel of Ring Homomorphism is Subring", "text": "Let $\\phi: \\struct {R_1, +_1, \\circ_1} \\to \\struct {R_2, +_2, \\circ_2}$ be a ring homomorphism. Then the kernel of $\\phi$ is a subring of $R_1$."}474{"_id": "532", "title": "Kernel of Ring Homomorphism is Ideal", "text": "Let $\\phi: \\struct {R_1, +_1, \\circ_1} \\to \\struct {R_2, +_2, \\circ_2}$ be a ring homomorphism. Then the kernel of $\\phi$ is an ideal of $R_1$."}475{"_id": "533", "title": "Ideals of Division Ring", "text": "Let $\\struct {R, +, \\circ}$ be a division ring whose zero is $0_R$. The only ideals of $\\struct {R, +, \\circ}$ are $\\set {0_R}$ and $R$ itself. That is, $\\struct {R, +, \\circ}$ has no non-null proper ideals."}476{"_id": "534", "title": "Quotient Ring of Kernel of Ring Epimorphism", "text": "Let $\\phi: \\struct {R_1, +_1, \\circ_1} \\to \\struct {R_2, +_2, \\circ_2}$ be a ring epimorphism. Let $K = \\map \\ker \\phi$. Then there is a unique ring isomorphism $g: R_1 / K \\to R_2$ such that: :$g \\circ q_K = \\phi$ $\\phi$ is an isomorphism {{iff}} $K = \\set {0_{R_1} }$."}477{"_id": "535", "title": "Ring Epimorphism Preserves Ideals", "text": "Let $\\phi: \\struct {R_1, +_1, \\circ_1} \\to \\struct {R_2, +_2, \\circ_2}$ be a ring epimorphism. Let $J$ be an ideal of $R_1$. Then $\\phi \\sqbrk J$ is an ideal of $R_2$."}478{"_id": "536", "title": "Preimage of Image of Subring under Ring Homomorphism", "text": "Let $\\phi: \\left({R_1, +_1, \\circ_1}\\right) \\to \\left({R_2, +_2, \\circ_2}\\right)$ be a ring homomorphism. Let $K = \\ker \\left({\\phi}\\right)$, where $\\ker \\left({\\phi}\\right)$ is the kernel of $\\phi$.  Let $J$ be a subring of $R_1$. Then: :$\\phi^{-1} \\left[{\\phi \\left[{J}\\right]}\\right] = J + K$"}479{"_id": "537", "title": "Preimage of Subring under Ring Homomorphism is Subring", "text": "Let $\\phi: \\left({R_1, +_1, \\circ_1}\\right) \\to \\left({R_2, +_2, \\circ_2}\\right)$ be a ring homomorphism. Let $S_2$ be a subring of $R_2$. Then $S_1 = \\phi^{-1} \\left[{S_2}\\right]$ is a subring of $R_1$ such that $\\ker \\left({\\phi}\\right) \\subseteq S_1$."}480{"_id": "538", "title": "Preimage of Ideal under Ring Homomorphism is Ideal", "text": "Let $\\phi: \\left({R_1, +_1, \\circ_1}\\right) \\to \\left({R_2, +_2, \\circ_2}\\right)$ be a ring homomorphism. Let $S_2$ be an ideal of $R_2$. Then $S_1 = \\phi^{-1} \\left[{S_2}\\right]$ is an ideal of $R_1$ such that $\\ker \\left({\\phi}\\right) \\subseteq S_1$."}481{"_id": "539", "title": "Image of Preimage of Subring under Ring Epimorphism", "text": "Let $\\phi: \\struct {R_1, +_1, \\circ_1} \\to \\struct {R_2, +_2, \\circ_2}$ be a ring epimorphism. Let $S_2$ be a subring of $R_2$. Then: :$\\phi \\sqbrk {\\phi^{-1} \\sqbrk {S_2} } = S_2$"}482{"_id": "541", "title": "Properties of Ordered Ring", "text": "Let $\\struct {R, +, \\circ, \\le}$ be an ordered ring whose zero is $0_R$ and whose unity is $1_R$. Let $U_R$ be the group of units of $R$. Let $x, y, z \\in \\struct {R, +, \\circ, \\le}$. Then the following properties hold: : $(1): \\quad x < y \\iff x + z < y + z$. Hence $x \\le y \\iff x + z \\le y + z$ (because $\\struct {R, +, \\le}$ is an ordered group). : $(2): \\quad x < y \\iff 0 < y + \\paren {-x}$. Hence $x \\le y \\iff 0 \\le y + \\paren {-x}$ : $(3): \\quad 0 < x \\iff \\paren {-x} < 0$. Hence $0 \\le x \\iff \\paren {-x} \\le 0$ : $(4): \\quad x < 0 \\iff 0 < \\paren {-x}$. Hence $x \\le 0 \\iff 0 \\le \\paren {-x}$ : $(5): \\quad \\forall n \\in \\Z_{>0}: x > 0 \\implies n \\cdot x > 0$ : $(6): \\quad x \\le y, 0 \\le z: x \\circ z \\le y \\circ z, z \\circ x \\le z \\circ y$ : $(7): \\quad x \\le y, z \\le 0: y \\circ z \\le x \\circ z, z \\circ y \\le z \\circ x$"}483{"_id": "542", "title": "Positive Elements of Ordered Ring", "text": "Let $\\struct {R, +, \\circ, \\le}$ be an ordered ring with unity whose zero is $0_R$ and whose unity is $1_R$. Let $P$ be the set of positive elements of $R$ , that is, $P = R_{\\ge 0}$. Then: : $(1): \\quad P + P \\subseteq P$ : $(2): \\quad P \\cap \\paren {-P} = \\set {0_R}$ : $(3): \\quad P \\circ P \\subseteq P$ If $\\le$ is a total ordering, that is, if $\\struct {R, +, \\circ, \\le}$ is a totally ordered ring, then: : $(4): \\quad P \\cup \\paren {-P} = R$ The converse is also true: Let $\\struct {R, +, \\circ}$ be a ring. Let $P \\subseteq R$ such that $(1)$, $(2)$ and $(3)$ are satisfied. Then there is one and only one ordering $\\le$ compatible with the ring structure of $R$ such that $P = R_{\\ge 0}$. Also, if $(4)$ is also satisfied, then $\\le$ is a total ordering."}484{"_id": "543", "title": "Symmetric Difference with Intersection forms Ring", "text": "Let $S$ be a set. Then $\\struct {\\powerset S, *, \\cap}$ is a commutative ring with unity, in which the unity is $S$. This ring is not an integral domain."}485{"_id": "544", "title": "Subfield Test", "text": "Let $\\struct {F, +, \\circ}$ be a field whose zero is $0_F$. Let $K$ be a subset of $F$. Then $\\struct {K, +, \\circ}$ is a subfield of $\\struct {F, +, \\circ}$ {{iff}} these all hold: :$(1): \\quad K^* \\ne \\O$ :$(2): \\quad \\forall x, y \\in K: x + \\paren {-y} \\in K$ :$(3): \\quad \\forall x, y \\in K: x \\circ y \\in K$ :$(4): \\quad x \\in K^* \\implies x^{-1} \\in K^*$ where $K^*$ denotes $K \\setminus \\set {0_F}$."}486{"_id": "545", "title": "Field of Quotients of Subdomain", "text": "Let $\\struct {F, +, \\circ}$ be a field whose unity is $1_F$. Let $\\struct {D, +, \\circ}$ be a subdomain of $\\struct {F, +, \\circ}$ whose unity is $1_D$. Let: :$K = \\set {\\dfrac x y: x \\in D, y \\in D^*}$ where $\\dfrac x y$ is the division product of $x$ by $y$. Then $\\struct {K, +, \\circ}$ is a field of quotients of $\\struct {D, +, \\circ}$."}487{"_id": "546", "title": "Existence of Field of Quotients", "text": "Let $\\struct {D, +, \\circ}$ be an integral domain. Then there exists a field of quotients of $\\struct {D, +, \\circ}$."}488{"_id": "547", "title": "Quotient Theorem for Monomorphisms", "text": "Let $K, L$ be fields of quotients of integral domains $\\struct {R, +_R, \\circ_R}, \\struct {S, +_S, \\circ_S}$ respectively. Let $\\phi: R \\to S$ be a monomorphism. Then there is one and only one monomorphism $\\psi: K \\to L$ extending $\\phi$, and: :$\\forall x \\in R, y \\in R^*: \\map \\psi {\\dfrac x y} = \\dfrac {\\map \\phi x} {\\map \\phi y}$ Also, if $\\phi$ is a ring isomorphism, then so is $\\psi$."}489{"_id": "548", "title": "Field of Quotients is Unique", "text": "Let $\\struct {D, +, \\circ}$ be an integral domain. Let $K, L$ be field of quotients of $\\struct {D, +, \\circ}$. Then there is one and only one (field) isomorphism $\\phi: K \\to L$ satisfying: :$\\forall x \\in D: \\map \\phi x = x$"}490{"_id": "549", "title": "Divided by Positive Element of Field of Quotients", "text": "Let $\\struct {K, +, \\circ}$ be the field of quotients of a totally ordered integral domain $\\struct {D, +, \\circ, \\le}$. Then: :$\\forall z \\in K: \\exists x, y \\in D: z = \\dfrac x y, y \\in D_{>0}$"}491{"_id": "550", "title": "Total Ordering on Field of Quotients is Unique", "text": "Let $\\struct {K, +, \\circ}$ be a field of quotients of an ordered integral domain $\\struct {D, +, \\circ, \\le}$. Then there is one and only one total ordering $\\le'$ on $K$ which is compatible with its ring structure and induces on $D$ its given total ordering $\\le$. That ordering is the one defined by: :$P = \\set {\\dfrac x y \\in K: x \\in D_+, y \\in D_+^*}$"}492{"_id": "551", "title": "Order Embedding between Quotient Fields is Unique", "text": "Let $\\struct {R_1, +_1, \\circ_1, \\le_1}$ and $\\struct {S, +_2, \\circ_2, \\le_2}$ be totally ordered integral domains. Let $K, L$ be totally ordered fields of quotients of $\\struct {R_1, +_1, \\circ_1, \\le_1}$ and $\\struct {S, +_2, \\circ_2, \\le_2}$ respectively. Let $\\phi: R \\to S$ be a order embedding. Then there is one and only one order embedding $\\psi: K \\to L$ extending $\\phi$. Also: :$\\forall x \\in R, y \\in R_{\\ne 0}: \\map \\psi {\\dfrac x y} = \\dfrac {\\map \\phi x} {\\map \\phi y}$ If $\\phi: R \\to S$ is an order isomorphism, then so is $\\psi$."}493{"_id": "552", "title": "Strict Ordering Preserved under Product with Cancellable Element", "text": "Let $\\left({S, \\circ, \\preceq}\\right)$ be an ordered semigroup. Let $x, y, z \\in S$ be such that: :$(1): \\quad z$ is cancellable for $\\circ$ :$(2): \\quad x \\prec y$ Then: : $x \\circ z \\prec y \\circ z$ : $z \\circ x \\prec z \\circ y$"}494{"_id": "553", "title": "Ordering of Inverses in Ordered Monoid", "text": "Let $\\left({S, \\circ, \\preceq}\\right)$ be an ordered monoid whose identity is $e$. Let $x, y \\in S$ be invertible. Then: :$x \\prec y \\iff y^{-1} \\prec x^{-1}$"}495{"_id": "555", "title": "Monomorphism from Total Ordering", "text": "Let the following conditions hold: : $(1): \\quad$ Let $\\left({S, \\circ, \\preceq}\\right)$ and $\\left({T, *, \\preccurlyeq}\\right)$ be ordered semigroups. : $(2): \\quad$ Let $\\phi: S \\to T$ be a mapping. : $(3): \\quad$ Let $\\preceq$ be a total ordering on $S$. Then $\\phi \\left({S, \\circ, \\preceq}\\right) \\to \\left({T, *, \\preccurlyeq}\\right)$ is a (structure) monomorphism iff: : $(1): \\quad \\phi$ is strictly increasing from $\\left({S, \\preceq}\\right)$ into $\\left({T, \\preccurlyeq}\\right)$; : $(2): \\quad \\phi$ is a homomorphism from $\\left({S, \\circ}\\right)$ into $\\left({T, *}\\right)$."}496{"_id": "556", "title": "Extension Theorem for Total Orderings", "text": "Let the following conditions be fulfilled: :$(1):\\quad$ Let $\\struct {S, \\circ, \\preceq}$ be a totally ordered commutative semigroup :$(2):\\quad$ Let all the elements of $\\struct {S, \\circ, \\preceq}$ be cancellable :$(3):\\quad$ Let $\\struct {T, \\circ}$ be an inverse completion of $\\struct {S, \\circ}$. Then: :$(1):\\quad$ The relation $\\preceq'$ on $T$ satisfying $\\forall x_1, x_2, y_1, y_2 \\in S: x_1 \\circ \\paren {y_1}^{-1} \\preceq' x_2 \\circ \\paren {y_2}^{-1} \\iff x_1 \\circ y_2 \\preceq x_2 \\circ y_1$ is a well-defined relation :$(2):\\quad$ $\\preceq'$ is the only total ordering on $T$ compatible with $\\circ$ :$(3):\\quad$ $\\preceq'$ is the only total ordering on $T$ that induces the given ordering $\\preceq$ on $S$."}497{"_id": "557", "title": "Cancellability in Naturally Ordered Semigroup", "text": "Let $\\left({S, \\circ, \\preceq}\\right)$ be a naturally ordered semigroup. Then:"}498{"_id": "558", "title": "Strict Lower Closure of Sum with One", "text": "Let $\\left({S, \\circ, \\preceq}\\right)$ be a naturally ordered semigroup. Then: :$\\forall n \\in \\left({S, \\circ, \\preceq}\\right): \\left({n \\circ 1}\\right)^\\prec = n^\\prec \\cup \\left\\{{n}\\right\\}$ where $n^\\prec$ is defined as the strict lower closure of $n$, that is, the set of elements strictly preceding $n$."}499{"_id": "559", "title": "Closed Interval of Naturally Ordered Semigroup with Successor equals Union with Successor", "text": "Let $\\left({S, \\circ, \\preceq}\\right)$ be a naturally ordered semigroup. Then: :$\\forall m, n \\in \\left({S, \\circ, \\preceq}\\right): m \\preceq n \\implies \\left[{m \\,.\\,.\\, n \\circ 1}\\right] = \\left[{m \\,.\\,.\\, n}\\right] \\cup \\left\\{{n \\circ 1}\\right\\}$ where $\\left[{m \\,.\\,.\\, n}\\right]$ is the closed interval between $m$ and $n$."}500{"_id": "561", "title": "Homomorphism of Powers", "text": "Let $\\struct {T_1, \\odot}$ and $\\struct {T_2, \\oplus}$ be semigroups. Let $\\phi: \\struct {T_1, \\odot} \\to \\struct {T_2, \\oplus}$ be a (semigroup) homomorphism."}501{"_id": "562", "title": "Naturally Ordered Semigroup is Unique", "text": "Let $\\struct {S, \\circ, \\preceq}$ and $\\struct {S', \\circ', \\preceq'}$ be naturally ordered semigroups. Let: :$0'$ be the smallest element of $S'$ :$1'$ be the smallest element of $S' \\setminus \\set {0'} = S'^*$. Then the mapping $g: S \\to S'$ defined as: :$\\forall a \\in S: \\map g a = \\circ'^a 1'$ is an isomorphism from $\\struct {S, \\circ, \\preceq}$ to $\\struct {S', \\circ', \\preceq'}$. This isomorphism is unique. Thus, up to isomorphism, there is only one naturally ordered semigroup."}502{"_id": "563", "title": "Consecutive Subsets of N", "text": "Let $\\N_k$ denote the initial segment of the natural numbers determined by $k$: :$\\N_k = \\left\\{{0, 1, 2, 3, \\ldots, k - 1}\\right\\}$ Then: :$\\N_k = \\N_{k + 1} \\setminus \\left\\{{k}\\right\\}$ In particular: :$\\N_{k - 1} = \\N_k \\setminus \\left\\{{k - 1}\\right\\}$"}503{"_id": "564", "title": "Well-Ordering Principle", "text": "Every non-empty subset of $\\N$ has a smallest (or '''first''') element. This is called the '''well-ordering principle'''. The '''well-ordering principle''' also holds for $\\N_{\\ne 0}$."}504{"_id": "565", "title": "Equality of Natural Numbers", "text": "Let $m, n \\in \\N$. Then: : $\\N_m \\sim \\N_n \\iff m = n$ where $\\sim$ denotes set equivalence and $\\N_n$ denotes the set of all natural numbers less than $n$."}505{"_id": "567", "title": "Cardinality Less One", "text": "Let $S$ be a finite set. Let: :$\\left\\lvert{S}\\right\\rvert = n + 1$ where $\\left\\lvert{S}\\right\\rvert$ is the cardinality of $S$. Let $a \\in S$. Then: :$\\left\\lvert{S \\setminus \\left\\{ {a}\\right\\} }\\right\\rvert = n$ where $\\setminus$ denotes set difference."}506{"_id": "568", "title": "Cardinality of Empty Set", "text": ":$\\card S = 0 \\iff S = \\O$ That is, the empty set is finite, and has a cardinality of zero."}507{"_id": "569", "title": "Cardinality of Subset of Finite Set", "text": "Let $A$ and $B$ be finite sets such that $A \\subseteq B$. Let :$\\card B = n$ where $\\card {\\, \\cdot \\,}$ denotes cardinality. Then $\\card A \\le n$."}508{"_id": "570", "title": "Cardinality of Surjection", "text": "Let $S$ be a set. Let: :$\\card S = n$ where $\\card S$ denotes the cardinality of $S$. Let $f: S \\to T$ be a surjection. Then $\\card T \\le n$. The equality: :$\\card T = n$ occurs {{iff}} $f$ is a bijection."}509{"_id": "571", "title": "Equivalence of Mappings between Sets of Same Cardinality", "text": "Let $S$ and $T$ be finite sets such that $\\card S = \\card T$. Let $f: S \\to T$ be a mapping. Then the following statements are equivalent: : $(1): \\quad f$ is bijective : $(2): \\quad f$ is injective : $(3): \\quad f$ is surjective."}510{"_id": "572", "title": "Natural Numbers are Infinite", "text": "The set $\\N$ of natural numbers is infinite."}511{"_id": "573", "title": "Finite Non-Empty Subset of Totally Ordered Set has Smallest and Greatest Elements", "text": "Let $\\left({S, \\preceq}\\right)$ be a totally ordered set. Then every finite $T$ such that $\\varnothing \\subset T \\subseteq S$ has both a smallest and a greatest element."}512{"_id": "574", "title": "Unique Isomorphism between Finite Totally Ordered Sets", "text": "Let $S$ and $T$ be finite sets such that: :$\\card S = \\card T$ Let $\\struct {S, \\preceq}$ and $\\struct {T, \\preccurlyeq}$ be totally ordered sets. Then there is exactly one order isomorphism from $\\struct {S, \\preceq}$ to $\\struct {T, \\preccurlyeq}$."}513{"_id": "575", "title": "Isomorphism to Closed Interval", "text": "Let $m, n \\in \\N$ such that $m < n$. Then: : $\\left|{\\left[{m + 1 \\,.\\,.\\, n}\\right]}\\right| = n - m$ Let $h: \\N_{n - m} \\to \\left[{m + 1 \\,.\\,.\\, n}\\right]$ be the mapping defined as: :$\\forall x \\in \\N_{n - m}: h \\left({x}\\right) = x + m + 1$ Let the orderings on $\\left[{m + 1 \\,.\\,.\\, n}\\right]$ and $\\N_{n - m}$ be those induced by the ordering of $\\N$. Then $h$ a unique order isomorphism."}514{"_id": "576", "title": "Regular Representation wrt Cancellable Element on Finite Semigroup is Bijection", "text": "Let $\\left({S, \\circ}\\right)$ be a finite semigroup. Let $a \\in S$ be cancellable. Then: :the left regular representation $\\lambda_a$ and: :the right regular representation $\\rho_a$ of $\\left({S, \\circ}\\right)$ with respect to $a$ are both bijections."}515{"_id": "577", "title": "Power Set of Natural Numbers is not Countable", "text": "The power set $\\powerset \\N$ of the natural numbers $\\N$ is not countable."}516{"_id": "578", "title": "Subset of Countably Infinite Set is Countable", "text": "Every subset of a countably infinite set is countable."}517{"_id": "580", "title": "Infinite Set has Countably Infinite Subset", "text": "Every infinite set has a countably infinite subset."}518{"_id": "581", "title": "No Bijection between Finite Set and Proper Subset", "text": "A finite set can not be in one-to-one correspondence with one of its proper subsets. That is, a finite set is not Dedekind-infinite."}519{"_id": "582", "title": "Infinite Set is Equivalent to Proper Subset", "text": "A set is infinite {{iff}} it is equivalent to one of its proper subsets."}520{"_id": "583", "title": "Cartesian Product of Countable Sets is Countable", "text": "The cartesian product of two countable sets is countable."}521{"_id": "584", "title": "Composition of Sequence with Mapping", "text": "Let $\\left \\langle {a_j} \\right \\rangle_{j \\in B}$ be a sequence. Let $\\sigma: A \\to B$ be a mapping, where $A \\subseteq \\N$. Then $\\left \\langle {a_j} \\right \\rangle \\circ \\sigma$ is a sequence whose value at each $k \\in A$ is $a_{\\sigma \\left({k}\\right)}$. Thus $\\left \\langle {a_j} \\right \\rangle \\circ \\sigma$ is denoted $\\left \\langle {a_{\\sigma \\left({k}\\right)}} \\right \\rangle_{k \\in A}$."}522{"_id": "585", "title": "General Operation from Binary Operation", "text": "Let $\\left({S, \\oplus}\\right)$ be a magma. Then there a unique sequence $\\left \\langle {\\oplus_k} \\right \\rangle_{k \\mathop \\ge 1}$ such that: :$(1): \\quad \\forall n \\in \\N_{>0}: \\oplus_n$ is an $n$-ary operation on $S$ such that: :$(2): \\quad \\forall \\left({a_1, \\ldots, a_k}\\right) \\in S^k: \\oplus_k \\left({a_1, \\ldots, a_k}\\right) = \\begin{cases} a: & k = 1 \\\\ \\oplus_n \\left({a_1, \\ldots, a_n}\\right) \\oplus a_{n+1}: & k = n + 1 \\end{cases}$ In particular, $\\oplus_2$ is the same as the given binary operation $\\oplus$. The $n$th term $\\oplus_n$ of the sequence $\\left \\langle {\\oplus} \\right \\rangle$ is called the '''$n$-ary operation defined by $\\oplus$'''."}523{"_id": "586", "title": "Strictly Increasing Sequence induces Partition", "text": "Let $\\left \\langle {r_k} \\right \\rangle_{0 \\mathop \\le k \\mathop \\le n}$ be a strictly increasing finite sequence of natural numbers. Let: :$\\forall k \\in \\left[{1 \\,.\\,.\\, n}\\right]: A_k := \\left[{r_{k-1} + 1 \\,.\\,.\\, r_k}\\right]$ Then: :$\\left\\{{A_k: k \\in \\left[{1 \\,.\\,.\\, n}\\right]}\\right\\}$ is a partition of $\\left[{r_0 + 1 \\,.\\,.\\, r_n}\\right]$."}524{"_id": "587", "title": "Fundamental Principle of Counting", "text": "Let $A$ be a finite set. Let $\\sequence {B_n}$ be a sequence of distinct finite subsets of $A$ which form a partition of $A$. Let $p_k = \\size {B_k}$ for each $k \\in \\closedint 1 n$. Then: :$\\displaystyle \\size A = \\sum_{k \\mathop = 1}^n p_k$ That is, the sum of the numbers of elements in the subsets of a partition of a set is equal to the total number of elements in the set."}525{"_id": "588", "title": "Odd Number Theorem", "text": ":$\\displaystyle \\sum_{j \\mathop = 1}^n \\paren {2 j - 1} = n^2$ That is, the sum of the first $n$ odd numbers is the $n$th square number."}526{"_id": "589", "title": "General Associativity Theorem", "text": "If an operation is associative on $3$ entities, then it is associative on any number of them."}527{"_id": "590", "title": "General Commutativity Theorem", "text": "Let $\\struct {S, \\circ}$ be a semigroup. Let $\\family {a_k}_{1 \\mathop \\le k \\mathop \\le n}$ be a sequence of elements of $S$. Suppose that: :$\\forall i, j \\in \\closedint 1 n: a_i \\circ a_j = a_j \\circ a_i$ Then for every permutation $\\sigma: \\N_n \\to \\N_n$: :$a_{\\map \\sigma 1} \\circ \\cdots \\circ a_{\\map \\sigma n} = a_1 \\circ \\cdots \\circ a_n$ where $\\N_n$ is used here to denote the initial segment of $\\N_{>0}$: :$\\N_n = \\set {1, 2, \\ldots, n}$"}528{"_id": "591", "title": "General Distributivity Theorem", "text": ":$\\displaystyle \\paren {\\sum_{i \\mathop = 1}^m a_i} * \\paren {\\sum_{j \\mathop = 1}^n b_j} = \\sum_{ {1 \\mathop \\le i \\mathop \\le m} \\atop {1 \\mathop \\le j \\mathop \\le n} } \\paren {a_i * b_j}$"}529{"_id": "592", "title": "Associativity on Indexing Set", "text": "Let $\\left({S, \\circ}\\right)$ be a commutative semigroup. Let $\\left \\langle {x_\\alpha} \\right \\rangle_{\\alpha \\mathop \\in A}$ be a family of terms of $S$ indexed by a finite non-empty set $A$. Let $\\left \\langle {B_k} \\right \\rangle_{1 \\mathop \\le k \\mathop \\le n}$ be a family of distinct subsets of $A$ forming a partition of $A$. Then: : $\\displaystyle \\prod_{k \\mathop = 1}^n \\left({\\prod_{a \\mathop \\in B_k} x_\\alpha}\\right) = \\prod_{\\alpha \\mathop \\in A} x_\\alpha$"}530{"_id": "593", "title": "External Direct Product of Groups is Group", "text": "Let $\\struct {G_1, \\circ_1}$ and $\\struct {G_2, \\circ_2}$ be groups whose identity elements are $e_1$ and $e_2$ respectively. Let $\\struct {G_1 \\times G_2, \\circ}$ be the external direct product of $G_1$ and $G_2$. Then $\\struct {G_1 \\times G_2, \\circ}$ is a group whose identity element is $\\tuple {e_1, e_2}$."}531{"_id": "594", "title": "External Direct Product of Projection with Canonical Injection", "text": "Let $\\struct {S_1, \\circ_1}$ and $\\struct {S_2, \\circ_2}$ be algebraic structures with identity elements $e_1$ and $e_2$ respectively. Let $\\struct {S_1 \\times S_2, \\circ}$ be the external direct product of $\\struct {S_1, \\circ_1}$ and $\\struct {S_2, \\circ_2}$ Let: :$\\pr_1: \\struct {S_1 \\times S_2, \\circ} \\to \\struct {S_1, \\circ_1}$ be the first projection from $\\struct {S_1 \\times S_2, \\circ}$ to $\\struct {S_1, \\circ_1}$ :$\\pr_2: \\struct {S_1 \\times S_2, \\circ} \\to \\struct {S_2, \\circ_2}$ be the second projection from $\\struct {S_1 \\times S_2, \\circ}$ to $\\struct {S_2, \\circ_2}$. Let: :$\\inj_1: \\struct {S_1, \\circ_1} \\to \\struct {S_1 \\times S_2, \\circ}$ be the canonical injection from $\\struct {S_1, \\circ_1}$ to $\\struct {S_1 \\times S_2, \\circ}$ :$\\inj_2: \\struct {S_2, \\circ_2} \\to \\struct {S_1 \\times S_2, \\circ}$ be the canonical injection from $\\struct {S_2, \\circ_2}$ to $\\struct {S_1 \\times S_2, \\circ}$. Then: :$(1): \\quad \\pr_1 \\circ \\inj_1 = I_{S_1}$ :$(2): \\quad \\pr_2 \\circ \\inj_2 = I_{S_2}$ where $I_{S_1}$ and $I_{S_2}$ are the identity mappings on $S_1$ and $S_2$ respectively."}532{"_id": "595", "title": "Inverse Completion of Natural Numbers", "text": "There exists an inverse completion of the natural numbers under addition."}533{"_id": "596", "title": "L'Hôpital's Rule", "text": "Let $f$ and $g$ be real functions which are continuous on the closed interval $\\closedint a b$ and differentiable on the open interval $\\openint a b$. Let: :$\\forall x \\in \\openint a b: \\map {g'} x \\ne 0$ where $g'$ denotes the derivative of $g$ {{WRT|Differentiation}} $x$. Let: :$\\map f a = \\map g a = 0$ Then: :$\\displaystyle \\lim_{x \\mathop \\to a^+} \\frac {\\map f x} {\\map g x} = \\lim_{x \\mathop \\to a^+} \\frac {\\map {f'} x} {\\map {g'} x}$ provided that the second limit exists."}534{"_id": "597", "title": "Natural Numbers under Addition form Commutative Monoid", "text": "The algebraic structure $\\struct {\\N, +}$ consisting of the set of natural numbers $\\N$ under addition $+$ is a commutative monoid whose identity is zero."}535{"_id": "598", "title": "Natural Numbers form Commutative Semiring", "text": "The semiring of natural numbers $\\struct {\\N, +, \\times}$ forms a commutative semiring."}536{"_id": "600", "title": "Integer Multiplication is Closed", "text": "The set of integers is closed under multiplication: :$\\forall a, b \\in \\Z: a \\times b \\in \\Z$"}537{"_id": "601", "title": "Integer Multiplication is Commutative", "text": "The operation of multiplication on the set of integers $\\Z$ is commutative: :$\\forall x, y \\in \\Z: x \\times y = y \\times x$"}538{"_id": "602", "title": "Integer Multiplication is Associative", "text": "The operation of multiplication on the set of integers $\\Z$ is associative: :$\\forall x, y, z \\in \\Z: x \\times \\paren {y \\times z} = \\paren {x \\times y} \\times z$"}539{"_id": "603", "title": "Integer Multiplication Distributes over Addition", "text": "The operation of multiplication on the set of integers $\\Z$ is distributive over addition: :$\\forall x, y, z \\in \\Z: x \\times \\paren {y + z} = \\paren {x \\times y} + \\paren {x \\times z}$ :$\\forall x, y, z \\in \\Z: \\paren {y + z} \\times x = \\paren {y \\times x} + \\paren {z \\times x}$"}540{"_id": "604", "title": "Construction of Inverse Completion", "text": "This page consists of a series of linked theorems, each of which builds towards one result. To access the proofs for the individual theorems, click on the links which form the titles of each major section."}541{"_id": "605", "title": "Integer Multiplication Identity is One", "text": "The identity of integer multiplication is $1$: :$\\exists 1 \\in \\Z: \\forall a \\in \\Z: a \\times 1 = a = 1 \\times a$"}542{"_id": "606", "title": "Integer Multiplication has Zero", "text": "The set of integers under multiplication $\\struct {\\Z, \\times}$ has a zero element, which is $0$."}543{"_id": "607", "title": "Ring of Integers has no Zero Divisors", "text": "The integers have no zero divisors: :$\\forall x, y, \\in \\Z: x \\times y = 0 \\implies x = 0 \\lor y = 0$"}544{"_id": "608", "title": "Integers form Integral Domain", "text": "The integers $\\Z$ form an integral domain under addition and multiplication."}545{"_id": "610", "title": "Natural Numbers are Non-Negative Integers", "text": "Let $m \\in \\Z$. Then: :$(1): \\quad m \\in \\N \\iff 0 \\le m$ :$(2): \\quad m \\in \\N_{> 0} \\iff 0 < m$ :$(3): \\quad m \\notin \\N \\iff -m \\in \\N_{> 0}$ That is, the natural numbers are precisely those integers which are greater than or equal to zero."}546{"_id": "611", "title": "Subtraction on Integers is Extension of Natural Numbers", "text": "Integer subtraction is an extension of the definition of subtraction on the natural numbers."}547{"_id": "613", "title": "Multiplicative Ordering on Integers", "text": "Let $x, y, z \\in \\Z$ such that $z > 0$. Then: :$x < y \\iff z x < z y$ :$x \\le y \\iff z x \\le z y$"}548{"_id": "614", "title": "Invertible Integers under Multiplication", "text": "The only invertible elements of $\\Z$ for multiplication (that is, units of $\\Z$) are $1$ and $-1$."}549{"_id": "615", "title": "Index Laws for Monoids", "text": "These results are an extension of the results in Index Laws for Semigroup in which the domain of the indices is extended to include all integers. Let $\\struct {S, \\circ}$ be a monoid whose identity is $e$. Let $a \\in S$ be invertible for $\\circ$. Let $n \\in \\N$. Let $a^n$ be the $n$th power of $a$: :$a^n = \\begin{cases} e : & n = 0 \\\\ a^{n - 1} \\circ a : & n > 0 \\\\ \\paren {a^{-n}}^{-1} : & n < 0 \\end{cases}$ Then we have the following results:"}550{"_id": "617", "title": "Totally Ordered Abelian Group Isomorphism", "text": "Let $\\left({\\Z', +', \\le'}\\right)$ be a totally ordered abelian group. Let $0'$ be the identity of $\\left({\\Z', +', \\le'}\\right)$. Let $\\N' = \\left\\{{x \\in \\Z': x \\ge' 0'}\\right\\}$. Let $\\Z'$ contain at least two elements. Let $\\N'$ be well-ordered for the ordering induced on $\\N'$ by $\\le'$. Then the mapping $g: \\Z \\to \\Z'$ defined by: :$\\forall n \\in \\Z: g \\left({n}\\right) = \\left({+'}\\right)^n 1'$ is an isomorphism from $\\left({\\Z, +, \\le}\\right)$ onto $\\left({\\Z', +', \\le'}\\right)$, where $1'$ is the smallest element of $\\N' \\setminus \\left\\{{0'}\\right\\}$."}551{"_id": "618", "title": "Integers under Addition form Totally Ordered Group", "text": "Then the ordered structure $\\struct {\\Z, +, \\le}$ is a totally ordered group."}552{"_id": "619", "title": "Integers form Totally Ordered Ring", "text": "The structure $\\struct {\\Z, +, \\times, \\le}$ is a totally ordered ring."}553{"_id": "620", "title": "Congruences on Rational Numbers", "text": "There are only two congruence relations on the field of rational numbers $\\left({\\Q, +, \\times}\\right)$: :$(1): \\quad$ The diagonal relation $\\Delta_\\Q$ :$(2): \\quad$ The trivial relation $\\Q \\times \\Q$."}554{"_id": "621", "title": "Number of Elements in Partition", "text": "Let $S$ be a set. Let there be a partition on $S$ of $n$ subsets, each of which has $m$ elements. Then: :$\\card S = n m$"}555{"_id": "622", "title": "Cardinality of Complement", "text": "Let $T \\subseteq S$ such that $\\card S = n, \\card T = m$. Then: :$\\card {\\relcomp S T} = \\card {S \\setminus T} = n - m$ where: :$\\relcomp S T$ denotes the complement of $T$ relative to $S$ :$S \\setminus T$ denotes the difference between $S$ and $T$."}556{"_id": "623", "title": "Cardinality of Cartesian Product", "text": "Let $S \\times T$ be the cartesian product of two finite sets $S$ and $T$. Then: :$\\card {S \\times T} = \\card S \\times \\card T$ where $\\card S$ denotes cardinality."}557{"_id": "624", "title": "Cardinality of Set of All Mappings", "text": "Let $S$ and $T$ be sets. The cardinality of the set of all mappings from $S$ to $T$ (that is, the total number of mappings from $S$ to $T$) is: :$\\card {T^S} = \\card T^{\\card S}$"}558{"_id": "626", "title": "Cardinality of Power Set of Finite Set", "text": "Let $S$ be a set such that: :$\\card S = n$ where $\\card S$ denotes the cardinality of $S$, Then: :$\\card {\\powerset S} = 2^n$ where $\\powerset S$ denotes the power set of $S$."}559{"_id": "627", "title": "Cardinality of Set of Injections", "text": "Let $S$ and $T$ be finite sets. The number of injections from $S$ to $T$, where $\\card S = m, \\card T = n$ is often denoted ${}^m P_n$, and is: :${}^m P_n = \\begin{cases} \\dfrac {n!} {\\paren {n - m}!} & : m \\le n \\\\ 0 & : m > n \\end{cases}$"}560{"_id": "628", "title": "Cardinality of Set of Bijections", "text": "Let $S$ and $T$ be sets such that $\\size S = \\size T = n$. Then there are $n!$ bijections from $S$ to $T$."}561{"_id": "629", "title": "Cardinality of Set of Subsets", "text": "Let $S$ be a set such that $\\card S = n$. Let $m \\le n$. Then the number of subsets $T$ of $S$ such that $\\card T = m$ is: : ${}^m C_n = \\dfrac {n!} {m! \\paren {n - m}!}$"}562{"_id": "630", "title": "Set of Integers Bounded Below by Integer has Smallest Element", "text": "Let $\\Z$ be the set of integers. Let $\\le$ be the ordering on the integers. Let $\\O \\subset S \\subseteq \\Z$ such that $S$ is bounded below in $\\struct {\\Z, \\le}$. Then $S$ has a smallest element."}563{"_id": "631", "title": "Set of Integers Bounded Above by Integer has Greatest Element", "text": "Let $\\Z$ be the set of integers. Let $\\le$ be the ordering on the integers. Let $\\O \\subset S \\subseteq \\Z$ such that $S$ is bounded above in $\\struct {\\Z, \\le}$. Then $S$ has a greatest element."}564{"_id": "632", "title": "Principle of Least Counterexample", "text": "Suppose $P \\paren n$ is a condition on $n \\in \\set {x \\in \\Z: x \\ge m \\in \\Z}$. Suppose next that: $\\neg \\paren {\\forall n \\ge m: P \\paren n}$. (That is, not all $n \\ge m$ satisfy $P \\paren n$.) Then there is a '''least counterexample''', that is a smallest integral value of $n$ for which $\\neg P \\paren n$."}565{"_id": "633", "title": "Absolute Value is Bounded Below by Zero", "text": "Let $x \\in \\R$ be a real number. Then the absolute value $\\size x$ of $x$ is bounded below by $0$."}566{"_id": "634", "title": "Division Theorem", "text": "For every pair of integers $a, b$ where $b \\ne 0$, there exist unique integers $q, r$ such that $a = q b + r$ and $0 \\le r < \\size b$: :$\\forall a, b \\in \\Z, b \\ne 0: \\exists! q, r \\in \\Z: a = q b + r, 0 \\le r < \\size b$"}567{"_id": "635", "title": "Odd Integer 2n + 1", "text": "Let $m$ be an odd integer. Then there exists exactly one integer $n$ such that $2 n + 1 = m$."}568{"_id": "636", "title": "Integer Divisor Results", "text": "Let $m, n \\in \\Z$ be integers. Let $m \\divides n$ denote that $m$ is a divisor of $n$. The following results all hold:"}569{"_id": "637", "title": "Zero Divides Zero", "text": "Let $n \\in \\Z$ be an integer. Then: :$0 \\divides n \\implies n = 0$ That is, zero is the only integer divisible by zero."}570{"_id": "638", "title": "Absolute Value of Integer is not less than Divisors", "text": "A (non-zero) integer is greater than or equal to its divisors in magnitude: :$\\forall c \\in \\Z_{\\ne 0}: a \\divides c \\implies a \\le \\size a \\le \\size c$"}571{"_id": "639", "title": "Divisor Relation on Positive Integers is Partial Ordering", "text": "The divisor relation is a partial ordering of $\\Z_{>0}$."}572{"_id": "640", "title": "Common Divisor in Integral Domain Divides Linear Combination", "text": "Let $\\struct {D, +, \\times}$ be an integral domain. Let $c$ be a common divisor of two elements $a$ and $b$ of $D$. That is: :$a, b, c \\in D: c \\divides a \\land c \\divides b$ Then: :$\\forall p, q \\in D: c \\divides \\paren {p \\times a + q \\times b}$"}573{"_id": "642", "title": "Greatest Common Divisor is at least 1", "text": "Let $a, b \\in \\Z$ be integers. The greatest common divisor of $a$ and $b$ is at least $1$: :$\\forall a, b \\in \\Z_{\\ne 0}: \\gcd \\set {a, b} \\ge 1$"}574{"_id": "643", "title": "GCD of Integer and Divisor", "text": "Let $a, b \\in \\Z_{>0}$, i.e. integers such that $a, b > 0$. Then: : $a \\divides b \\implies \\gcd \\set {a, b} = a$"}575{"_id": "644", "title": "GCD for Negative Integers", "text": ":$\\gcd \\set {a, b} = \\gcd \\set {\\size a, b} = \\gcd \\set {a, \\size b} = \\gcd \\set {\\size a, \\size b}$ Alternatively, this can be put: :$\\gcd \\set {a, b} = \\gcd \\set {-a, b} = \\gcd \\set {a, -b} = \\gcd \\set {-a, -b}$   which follows directly from the above."}576{"_id": "645", "title": "GCD with Zero", "text": "Let $a \\in \\Z$ be an integer such that $a \\ne 0$. Then: :$\\gcd \\left\\{{a, 0}\\right\\} = \\left\\lvert{a}\\right\\rvert$ where $\\gcd$ denotes greatest common divisor (GCD)."}577{"_id": "646", "title": "Set of Integer Combinations equals Set of Multiples of GCD", "text": "The set of all integer combinations of $a$ and $b$ is precisely the set of all integer multiples of the GCD of $a$ and $b$: :$\\gcd \\set {a, b} \\divides c \\iff \\exists x, y \\in \\Z: c = x a + y b$"}578{"_id": "647", "title": "GCD with Remainder", "text": "Let $a, b \\in \\Z$. Let $q, r \\in \\Z$ such that $a = q b + r$. Then: :$\\gcd \\set {a, b} = \\gcd \\set {b, r}$ where $\\gcd \\set {a, b}$ is the greatest common divisor of $a$ and $b$."}579{"_id": "648", "title": "Integer Combination of Coprime Integers", "text": "Two integers are coprime {{iff}} there exists an integer combination of them equal to $1$: :$\\forall a, b \\in \\Z: a \\perp b \\iff \\exists m, n \\in \\Z: m a + n b = 1$"}580{"_id": "650", "title": "Integers Divided by GCD are Coprime", "text": "Any pair of integers, not both zero, can be reduced to a pair of coprime ones by dividing them by their GCD: :$\\gcd \\set {a, b} = d \\iff \\dfrac a d, \\dfrac b d \\in \\Z \\land \\gcd \\set {\\dfrac a d, \\dfrac b d} = 1$ That is: :$\\dfrac a {\\gcd \\set {a, b} } \\perp \\dfrac b {\\gcd \\set {a, b} }$"}581{"_id": "651", "title": "Product of Coprime Factors", "text": "Let $a, b, c \\in \\Z$ such that $a$ and $b$ are coprime. Let both $a$ and $b$ be divisors of $c$. Then $a b$ is also a divisor of $c$. That is: :$a \\perp b \\land a \\divides c \\land b \\divides c \\implies a b \\divides c$"}582{"_id": "652", "title": "Existence of Lowest Common Multiple", "text": "Let $a, b \\in \\Z: a b \\ne 0$. The lowest common multiple of $a$ and $b$, denoted $\\lcm \\set {a, b}$, always exists."}583{"_id": "653", "title": "Product of GCD and LCM", "text": ":$\\lcm \\set {a, b} \\times \\gcd \\set {a, b} = \\size {a b}$ where: :$\\lcm \\set {a, b}$ denotes the lowest common multiple of $a$ and $b$ :$\\gcd \\set {a, b}$ denotes the greatest common divisor of $a$ and $b$."}584{"_id": "654", "title": "Congruent to Zero if Modulo is Divisor", "text": "Let $a, z \\in \\R$. Then $a$ is congruent to $0$ modulo $z$ {{iff}} $a$ is an integer multiple of $z$. :$\\exists k \\in \\Z: k z = a \\iff a \\equiv 0 \\pmod z$ If $z \\in \\Z$, then further: :$z \\divides a \\iff a \\equiv 0 \\pmod z$"}585{"_id": "655", "title": "Integer is Congruent Modulo Divisor to Remainder", "text": "Let $a \\in \\Z$. Let $a$ have a remainder $r$ on division by $m$. Then: : $a \\equiv r \\pmod m$ where the notation denotes that $a$ and $r$ are congruent modulo $m$."}586{"_id": "656", "title": "Integer is Congruent to Integer less than Modulus", "text": "Let $m \\in \\Z$. Then each integer is congruent (modulo $m$) to precisely one of the integers $0, 1, \\ldots, m - 1$."}587{"_id": "657", "title": "Modulo Addition is Well-Defined", "text": "Let $m \\in \\Z$ be an integer. Let $\\Z_m$ be the set of integers modulo $m$. The modulo addition operation on $\\Z_m$, defined by the rule: :$\\eqclass a m +_m \\eqclass b m = \\eqclass {a + b} m$ is a well-defined operation. That is: :If $a \\equiv b \\pmod m$ and $x \\equiv y \\pmod m$, then $a + x \\equiv b + y \\pmod m$."}588{"_id": "658", "title": "Modulo Multiplication is Well-Defined", "text": "The multiplication modulo $m$ operation on $\\Z_m$, the set of integers modulo $m$, defined by the rule: :$\\eqclass x m \\times_m \\eqclass y m = \\eqclass {x y} m$ is a well-defined operation. That is: :If $a \\equiv b \\pmod m$ and $x \\equiv y \\pmod m$, then $a x \\equiv b y \\pmod m$."}589{"_id": "659", "title": "Congruence of Product", "text": "Let $a, b, z \\in \\R$. Let $a$ be congruent to $b$ modulo $z$, that is: :$a \\equiv b \\pmod z$ Then: :$\\forall m \\in \\Z: m a \\equiv m b \\pmod z$"}590{"_id": "660", "title": "Congruence of Powers", "text": "Let $a, b \\in \\R$ and $m \\in \\Z$. Let $a$ be congruent to $b$ modulo $m$, that is: :$a \\equiv b \\pmod m$ Then: :$\\forall n \\in \\Z_{\\ge 0}: a^n \\equiv b^n \\pmod m$"}591{"_id": "662", "title": "Modulo Addition is Associative", "text": "Addition modulo $m$ is associative: :$\\forall \\eqclass x m, \\eqclass y m, \\eqclass z m \\in \\Z_m: \\paren {\\eqclass x m +_m \\eqclass y m} +_m \\eqclass z m = \\eqclass x m +_m \\paren {\\eqclass y m +_m \\eqclass z m}$ where $\\Z_m$ is the set of integers modulo $m$. That is: :$\\forall x, y, z \\in \\Z: \\paren {x + y} + z \\equiv x + \\paren {y + z} \\pmod m$"}592{"_id": "663", "title": "Modulo Addition is Commutative", "text": "Modulo addition is commutative: :$\\forall x, y, z \\in \\Z: x + y \\pmod m = y + x \\pmod m$"}593{"_id": "664", "title": "Modulo Addition has Identity", "text": "Let $m \\in \\Z$ be an integer. Then addition modulo $m$ has an identity: :$\\forall \\eqclass x m \\in \\Z_m: \\eqclass x m +_m \\eqclass 0 m = \\eqclass x m = \\eqclass 0 m +_m \\eqclass x m$ That is: :$\\forall a \\in \\Z: a + 0 \\equiv a \\equiv 0 + a \\pmod m$"}594{"_id": "665", "title": "Modulo Addition has Inverses", "text": "Let $m \\in \\Z$ be an integer. Then addition modulo $m$ has inverses: For each element $\\eqclass x m \\in \\Z_m$, there exists the element $\\eqclass {-x} m \\in \\Z_m$ with the property: :$\\eqclass x m +_m \\eqclass {-x} m = \\eqclass 0 m = \\eqclass {-x} m +_m \\eqclass x m$ where $\\Z_m$ is the set of integers modulo $m$. That is: :$\\forall a \\in \\Z: a + \\paren {-a} \\equiv 0 \\equiv \\paren {-a} + a \\pmod m$"}595{"_id": "666", "title": "Modulo Multiplication is Closed", "text": "Multiplication modulo $m$ is closed on the set of integers modulo $m$: :$\\forall \\eqclass x m, \\eqclass y m \\in \\Z_m: \\eqclass x m \\times_m \\eqclass y m \\in \\Z_m$."}596{"_id": "667", "title": "Modulo Multiplication is Associative", "text": "Multiplication modulo $m$ is associative: :$\\forall \\eqclass x m, \\eqclass y m, \\eqclass z m \\in \\Z_m: \\paren {\\eqclass x m \\times_m \\eqclass y m} \\times_m \\eqclass z m = \\eqclass x m \\times_m \\paren {\\eqclass y m \\times_m \\eqclass z m}$ That is: :$\\forall x, y, z \\in \\Z_m: \\paren {x \\cdot_m y} \\cdot_m z = x \\cdot_m \\paren {y \\cdot_m z}$"}597{"_id": "668", "title": "Modulo Multiplication is Commutative", "text": "Multiplication modulo $m$ is commutative: :$\\forall \\eqclass x m, \\eqclass y m \\in \\Z_m: \\eqclass x m \\times_m \\eqclass y m = \\eqclass y m \\times_m \\eqclass x m$"}598{"_id": "669", "title": "Modulo Multiplication has Identity", "text": "Multiplication modulo $m$ has an identity: :$\\forall \\eqclass x m \\in \\Z_m: \\eqclass x m \\times_m \\eqclass 1 m = \\eqclass x m = \\eqclass 1 m \\times_m \\eqclass x m$"}599{"_id": "670", "title": "Modulo Multiplication Distributes over Modulo Addition", "text": "Multiplication modulo $m$ is distributive over addition modulo $m$: :$\\forall \\eqclass x m, \\eqclass y m, \\eqclass z m \\in \\Z_m$: :: $\\eqclass x m \\times_m \\paren {\\eqclass y m +_m \\eqclass z m} = \\paren {\\eqclass x m \\times_m \\eqclass y m} +_m \\paren {\\eqclass x m \\times_m \\eqclass z m}$ :: $\\paren {\\eqclass x m +_m \\eqclass y m} \\times_m \\eqclass z m = \\paren {\\eqclass x m \\times_m \\eqclass z m} +_m \\paren {\\eqclass y m \\times_m \\eqclass z m}$ where $\\Z_m$ is the set of integers modulo $m$. That is, $\\forall x, y, z, m \\in \\Z$: : $x \\paren {y + z} \\equiv x y + x z \\pmod m$ : $\\paren {x + y} z \\equiv x z + y z \\pmod m$"}600{"_id": "671", "title": "Intersection of Congruence Classes", "text": "Let $\\mathcal R_m$ denote congruence modulo $m$ on the set of integers $\\Z$. Then: :$\\mathcal R_m \\cap \\mathcal R_n = \\mathcal R_{\\lcm \\set {m, n} }$ where $\\lcm \\set {m, n}$ is the lowest common multiple of $m$ and $n$. In the language of modulo arithmetic, this is equivalent to: :$a \\equiv b \\pmod m, a \\equiv b \\pmod n \\implies a \\equiv b \\pmod {\\lcm \\set {m, n} }$"}601{"_id": "672", "title": "Mappings Between Residue Classes", "text": "Let $\\eqclass a m$ be the residue class of $a$ (modulo $m$). Let $\\phi: \\Z_m \\to \\Z_n$ be a mapping given by: :$\\map \\phi {\\eqclass x m} = \\eqclass x n$ Then $\\phi$ is well defined {{iff}} $m$ is a divisor of $n$."}602{"_id": "674", "title": "Prime Number has 4 Integral Divisors", "text": "Let $p$ be an integer. Then $p$ is a prime number {{iff}} $p$ has exactly four integral divisors: $1, -1, p, -p$."}603{"_id": "675", "title": "Prime not Divisor implies Coprime", "text": "Let $p, a \\in \\Z$. If $p$ is a prime number then: : $p \\nmid a \\implies p \\perp a$ where: : $p \\nmid a$ denotes that $p$ does not divide $a$ : $p \\perp a$ denotes that $p$ and $a$ are coprime. It follows directly that if $p$ and $q$ are primes, then: : $p \\divides q \\implies p = q$ : $p \\ne q \\implies p \\perp q$."}604{"_id": "676", "title": "Composite Number has Two Divisors Less Than It", "text": "Let $n \\in \\Z_{> 1}$ such that $n \\notin \\mathbb P$. Then: :$\\exists a, b \\in \\Z: 1 < a < n, 1 < b < n: n = a b$ That is, a non-prime number greater than $1$ can be expressed as the product of two positive integers strictly greater than $1$ and less than $n$. Note that these two numbers are not necessarily distinct."}605{"_id": "678", "title": "Exponents of Primes in Prime Decomposition are Less iff Divisor", "text": "Let $a, b \\in \\Z_{>0}$. Then $a \\divides b$ {{iff}}: :$(1): \\quad$ every prime $p_i$ in the prime decomposition of $a$ appears in the prime decomposition of $b$ and: :$(2): \\quad$ the exponent of each $p_i$ in $a$ is less than or equal to its exponent in $b$."}606{"_id": "679", "title": "Not Coprime means Common Prime Factor", "text": "Let $a, b \\in \\Z$. If $d \\divides a$ and $d \\divides b$ such that $d > 1$, then $a$ and $b$ have a common divisor which is prime."}607{"_id": "680", "title": "Set of Divisors of Integer", "text": "Let $n \\in \\Z_{>1}$. Let $n$ be expressed in its prime decomposition: :$n = p_1^{k_1} p_2^{k_2} \\dotsm p_r^{k_r}$ where $p_1 < p_2 < \\dotsb < p_r$ are distinct primes and $k_1, k_2, \\ldots, k_r$ are positive integers. The set of divisors of $n$ is: :$\\set {p_1^{h_1} p_2^{h_2} \\dotsm p_r^{h_r}: 0 \\le h_i \\le k_i, i = 1, 2, \\ldots, r}$"}608{"_id": "681", "title": "Sum Less Minimum is Maximum", "text": "For all numbers $a, b$ where $a, b$ in $\\N, \\Z, \\Q$ or $\\R$: :$a + b - \\min \\left({a, b}\\right) = \\max \\left({a, b}\\right)$"}609{"_id": "682", "title": "Sum Less Maximum is Minimum", "text": "For all numbers $a, b$ where $a, b$ in $\\N, \\Z, \\Q$ or $\\R$: :$a + b - \\max \\left({a, b}\\right) = \\min \\left({a, b}\\right)$."}610{"_id": "683", "title": "GCD and LCM from Prime Decomposition", "text": "Let $m, n \\in \\Z$. Let: :$m = p_1^{k_1} p_2^{k_2} \\dotsm p_r^{k_r}$ :$n = p_1^{l_1} p_2^{l_2} \\dotsm p_r^{l_r}$ :$p_i \\divides m \\lor p_i \\divides n, 1 \\le i \\le r$. That is, the primes given in these prime decompositions may be divisors of ''either'' of the numbers $m$ or $n$. Note that if one of the primes $p_i$ does not appear in the prime decompositions of either one of $m$ or $n$, then its corresponding index $k_i$ or $l_i$ will be zero. Then the following results apply: :$\\gcd \\set {m, n} = p_1^{\\min \\set {k_1, l_1} } p_2^{\\min \\set {k_2, l_2} } \\ldots p_r^{\\min \\set {k_r, l_r} }$ :$\\lcm \\set {m, n} = p_1^{\\max \\set {k_1, l_1} } p_2^{\\max \\set {k_2, l_2} } \\ldots p_r^{\\max \\set {k_r, l_r} }$"}611{"_id": "684", "title": "GCD and LCM Distribute Over Each Other", "text": "Let $a, b, c \\in \\Z$. Then: : $\\lcm \\set {a, \\gcd \\set {b, c} } = \\gcd \\set {\\lcm \\set {a, b}, \\lcm \\set {a, c} }$ : $\\gcd \\set {a, \\lcm \\set {b, c} } = \\lcm \\set {\\gcd \\set {a, b}, \\gcd \\set {a, c} }$ That is, greatest common divisor and lowest common multiple are distributive over each other."}612{"_id": "685", "title": "Tau Function from Prime Decomposition", "text": "Let $n$ be an integer such that $n \\ge 2$. Let the prime decomposition of $n$ be: :$n = p_1^{k_1} p_2^{k_2} \\cdots p_r^{k_r}$ Let $\\map \\tau n$ be the tau function of $n$. Then: :$\\displaystyle \\map \\tau n = \\prod_{j \\mathop = 1}^r \\paren {k_j + 1}$"}613{"_id": "686", "title": "N less than M to the N", "text": ":$\\forall m, n \\in \\Z_{>0}: m > 1 \\implies n < m^n$"}614{"_id": "687", "title": "Basis Representation Theorem", "text": "Let $b \\in \\Z: b > 1$. For every $n \\in \\Z_{> 0}$, there exists one and only one sequence $\\sequence {r_j}_{0 \\mathop \\le j \\mathop \\le t}$ such that: : $(1): \\quad \\displaystyle n = \\sum_{k \\mathop = 0}^t r_k b^k$ : $(2): \\quad \\displaystyle \\forall k \\in \\closedint 0 t: r_k \\in \\N_b$ : $(3): \\quad r_t \\ne 0$ This unique sequence is called the '''representation of $n$ to the base $b$''', or, informally, we can say '''$n$ is (written) in base $b$'''."}615{"_id": "688", "title": "Sum of Geometric Sequence", "text": "Let $x$ be an element of one of the standard number fields: $\\Q, \\R, \\C$ such that $x \\ne 1$. Let $n \\in \\N_{>0}$. Then: :$\\displaystyle \\sum_{j \\mathop = 0}^{n - 1} x^j = \\frac {x^n - 1} {x - 1}$"}616{"_id": "689", "title": "Congruence of Sum of Digits to Base Less 1", "text": "Let $x \\in \\Z$, and $b \\in \\N, b > 1$. Let $x$ be written in base $b$: :$x = \\sqbrk {r_m r_{m - 1} \\ldots r_2 r_1 r_0}_b$ Then: :$\\displaystyle \\map {s_b} x = \\sum_{j \\mathop = 0}^m r_j \\equiv x \\pmod {b - 1}$ where $\\map {s_b} x$ is the digit sum of $x$ in base $b$ notation. That is, the digit sum of any integer $x$ in base $b$ notation is congruent to $x$ modulo $b - 1$."}617{"_id": "690", "title": "Euler Phi Function of Integer", "text": "Let $n \\in \\Z_{>0}$, that is, a (strictly) positive integer. Let $\\phi: \\Z_{>0} \\to \\Z_{>0}$ be the Euler $\\phi$-function. Then for any $n \\in \\Z_{>0}$, we have: :$\\map \\phi n = n \\paren {1 - \\dfrac 1 {p_1} } \\paren {1 - \\dfrac 1 {p_2} } \\cdots \\paren {1 - \\dfrac 1 {p_r} }$ where $p_1, p_2, \\ldots, p_r$ are the distinct primes dividing $n$. Or, more compactly: :$\\displaystyle \\map \\phi n = n \\prod_{p \\mathop \\divides n} \\paren {1 - \\frac 1 p}$ where $p \\divides n$ denotes the primes which divide $n$."}618{"_id": "691", "title": "Euler Phi Function is Multiplicative", "text": "The Euler $\\phi$ function is a multiplicative function: :$m \\perp n \\implies \\map \\phi {m n} = \\map \\phi m \\, \\map \\phi n$ where $m, n \\in \\Z_{>0}$."}619{"_id": "692", "title": "Euler Phi Function of Prime Power", "text": "Let $p^n$ be a prime power for some prime number $p > 1$. Then: :$\\map \\phi {p^n} = p^n \\paren {1 - \\dfrac 1 p} = \\paren {p - 1} p^{n - 1}$ where $\\phi: \\Z_{>0} \\to \\Z_{>0}$ is the Euler $\\phi$ function."}620{"_id": "694", "title": "Möbius Function is Multiplicative", "text": "The Möbius function $\\mu$  is a multiplicative function: :$m \\perp n \\implies \\map \\mu {m n} = \\map \\mu m \\map \\mu n$ where $m, n \\in \\Z_{>0}$."}621{"_id": "695", "title": "Sum of Möbius Function over Divisors", "text": "Let $n \\in \\Z_{>0}$ be a strictly positive integer. Then: :$\\displaystyle \\sum_{d \\mathop \\divides n} \\map \\mu d \\frac n d = \\map \\phi n$ where: :$\\displaystyle \\sum_{d \\mathop \\divides n}$ denotes the sum over all of the divisors of $n$ :$\\map \\phi n$ is the Euler $\\phi$ function, the number of integers less than $n$ that are prime to $n$ :$\\map \\mu d$ is the Möbius function. Equivalently, this says that: :$\\phi = \\mu * I_{\\Z_{>0} }$ where: :$*$ denotes Dirichlet convolution :$I_{\\Z_{>0} }$ denotes the identity mapping on $\\Z_{>0}$, that is: ::$\\forall n \\in \\Z_{>0}: I_{\\Z_{>0} }: n \\mapsto n$ {{wtd|Add a link to a page proving this equivalence.}}"}622{"_id": "696", "title": "Binomial Coefficient of Prime", "text": "Let $p$ be a prime number. Then: :$\\forall k \\in \\Z: 0 < k < p: \\dbinom p k \\equiv 0 \\pmod p$ where $\\dbinom p k$ is defined as a binomial coefficient."}623{"_id": "697", "title": "Prime Power of Sum Modulo Prime", "text": "Let $p$ be a prime number. Then: :$\\forall n \\in \\N_{> 0}: \\paren {a + b}^{p^n} \\equiv a^{p^n} + b^{p^n} \\pmod p$"}624{"_id": "698", "title": "Binomial Coefficient involving Power of Prime", "text": ": $\\dbinom {p^n k} {p^n} \\equiv k \\pmod p$ where $\\dbinom {p^n k} {p^n}$ is a binomial coefficient."}625{"_id": "699", "title": "Cassini's Identity", "text": ":$F_{n + 1} F_{n - 1} - F_n^2 = \\paren {-1}^n$"}626{"_id": "700", "title": "Rational Numbers are Countably Infinite", "text": "The set $\\Q$ of rational numbers is countably infinite."}627{"_id": "701", "title": "Real Numbers form Ordered Field", "text": "The set of real numbers $\\R$ forms an ordered field under addition and multiplication: $\\struct {\\R, +, \\times, \\le}$."}628{"_id": "702", "title": "Rational Numbers form Subfield of Real Numbers", "text": "The (ordered) field $\\struct {\\Q, +, \\times, \\le}$ of rational numbers forms a subfield of the field of real numbers $\\struct {\\R, +, \\times, \\le}$. That is, the field of real numbers $\\struct {\\R, +, \\times, \\le}$ is an extension of the rational numbers $\\struct {\\Q, +, \\times, \\le}$."}629{"_id": "703", "title": "Definition:Rational Number/Canonical Form", "text": "Let $r \\in \\Q$ be a rational number. The '''canonical form of $r$''' is the expression $\\dfrac p q$, where: :$r = \\dfrac p q: p \\in \\Z, q \\in \\Z_{>0}, p \\perp q$ where $p \\perp q$ denotes that $p$ and $q$ have no common divisor except $1$."}630{"_id": "704", "title": "Ordering Properties of Real Numbers", "text": "=== Trichotomy Law === {{:Trichotomy Law for Real Numbers}} === Ordering is Transitive === {{:Real Number Ordering is Transitive}} === Ordering is Compatible with Addition === {{:Real Number Ordering is Compatible with Addition}} === Ordering is Compatible with Multiplication === {{:Real Number Ordering is Compatible with Multiplication}}"}631{"_id": "705", "title": "Order is Preserved on Positive Reals by Squaring", "text": ":$x < y \\iff x^2 < y^2$"}632{"_id": "706", "title": "Real Plus Epsilon", "text": "Let $a, b \\in \\R$, such that: :$\\forall \\epsilon \\in \\R_{>0}: a < b + \\epsilon$ where $\\R_{>0}$ is the set of strictly positive real numbers. That is: :$\\epsilon > 0$ Then: :$a \\le b$"}633{"_id": "707", "title": "Difference of Two Squares", "text": "Let $\\struct {R, +, \\circ}$ be a commutative ring whose zero is $0_R$. Let $x, y \\in R$. Then: : $x \\circ x + \\paren {- \\paren {y \\circ y} } = \\paren {x + y} \\circ \\paren {x + \\paren {-y} }$ When $R$ is one of the standard sets of numbers, that is $\\Z, \\Q, \\R$, and so on, then this translates into: :$x^2 - y^2 = \\paren {x + y} \\paren {x - y}$"}634{"_id": "709", "title": "Mediant is Between", "text": "Let $a, b, c, d$ be ''any'' real numbers such that $b > 0, d > 0$. Let $r = \\dfrac a b < \\dfrac c d = s$. Then: :$r < \\dfrac {a + c} {b + d} < s$"}635{"_id": "710", "title": "Real Numbers Between Epsilons", "text": "Let $a, b \\in \\R$ such that $\\forall \\epsilon \\in \\R_{>0}: a - \\epsilon < b < a + \\epsilon$. Then $a = b$."}636{"_id": "711", "title": "Exists Integer Below Any Real Number", "text": "Let $x$ be a real number. Then there exists an integer less than $x$: :$\\forall x \\in \\R: \\exists n \\in \\Z: n < x$"}637{"_id": "712", "title": "Archimedean Principle", "text": "Let $x$ be a real number. Then there exists a natural number greater than $x$. :$\\forall x \\in \\R: \\exists n \\in \\N: n > x$ That is, the set of natural numbers is unbounded above."}638{"_id": "713", "title": "Real Number is between Floor Functions", "text": ":$\\forall x \\in \\R: \\floor x \\le x < \\floor {x + 1}$"}639{"_id": "714", "title": "Real Number is between Ceiling Functions", "text": ":$\\forall x \\in \\R: \\left \\lceil {x - 1} \\right \\rceil \\le x < \\left \\lceil {x} \\right \\rceil$"}640{"_id": "715", "title": "Real Number minus Floor", "text": ":$x - \\floor x \\in \\hointr 0 1$"}641{"_id": "716", "title": "Ceiling minus Real Number", "text": ":$\\forall x \\in \\R: \\left \\lceil {x} \\right \\rceil - x \\in \\left[{0 \\,.\\,.\\, 1}\\right)$"}642{"_id": "717", "title": "Real Number is Floor plus Difference", "text": ":There exists an integer $n \\in \\Z$ such that for some $t \\in \\hointr 0 1$: ::$x = n + t$ {{iff}}: :$n = \\floor x$"}643{"_id": "718", "title": "Floor plus One", "text": "Let $x \\in \\R$. Then: :$\\left \\lfloor {x + 1} \\right \\rfloor = \\left \\lfloor {x} \\right \\rfloor + 1$ where $\\left \\lfloor {x} \\right \\rfloor$ is the floor function of $x$."}644{"_id": "719", "title": "Real Number is Integer iff equals Floor", "text": ":$x = \\floor x \\iff x \\in \\Z$"}645{"_id": "720", "title": "Sum of Floor and Floor of Negative", "text": "Let $x \\in \\R$. Then: :$\\floor x + \\floor {-x} = \\begin{cases} 0 & : x \\in \\Z \\\\ -1 & : x \\notin \\Z \\end{cases}$ where $\\floor x$ denotes the floor of $x$."}646{"_id": "722", "title": "Ceiling defines Equivalence Relation", "text": "Let $\\mathcal R$ be the relation defined on $\\R$ such that: :$\\forall x, y, \\in \\R: \\left({x, y}\\right) \\in \\mathcal R \\iff \\left \\lceil {x}\\right \\rceil = \\left \\lceil {y}\\right \\rceil$ where $\\left \\lceil {x}\\right \\rceil$ is the ceiling of $x$. Then $\\mathcal R$ is an equivalence, and $\\forall n \\in \\Z$, the $\\mathcal R$-class of $n$ is the half-open interval $\\left({n - 1 \\,.\\,.\\, n}\\right]$."}647{"_id": "723", "title": "Real Number is Ceiling minus Difference", "text": "Let $n$ be a integer. {{TFAE}} :$(1): \\quad$ There exists $t \\in \\hointr 0 1$ such that $x = n - t$ :$(2): \\quad n = \\ceiling x$"}648{"_id": "724", "title": "Cauchy's Inequality", "text": ":$\\displaystyle \\sum {r_i^2} \\sum {s_i^2} \\ge \\left({\\sum {r_i s_i}}\\right)^2$"}649{"_id": "725", "title": "Cancellable Finite Semigroup is Group", "text": "Let $\\struct {S, \\circ}$ be a non-empty finite semigroup in which all elements are cancellable. Then $\\struct {S, \\circ}$ is a group."}650{"_id": "726", "title": "Finite Semigroup Equal Elements for Different Powers", "text": "Let $\\left({S, \\circ}\\right)$ be a finite semigroup. Then: : $\\forall x \\in S: \\exists m, n \\in \\N: m \\ne n: x^m = x^n$"}651{"_id": "727", "title": "Element has Idempotent Power in Finite Semigroup", "text": "Let $\\struct {S, \\circ}$ be a finite semigroup. For every element in $\\struct {S, \\circ}$, there is a power of that element which is idempotent. That is: :$\\forall x \\in S: \\exists i \\in \\N: x^i = x^i \\circ x^i$"}652{"_id": "728", "title": "Powers of Ring Elements", "text": "Let $\\struct {R, +, \\circ}$ be a ring whose zero is $0_R$. Let $n \\cdot x$ be an integral multiple of $x$: :$n \\cdot x = \\begin {cases} 0_R & : n = 0 \\\\ x & : n = 1 \\\\ \\paren {n - 1} \\cdot x + x & : n > 1 \\end {cases}$ that is $n \\cdot x = x + x + \\cdots \\paren n \\cdots x$. For $n < 0$ we use: :$-n \\cdot x = n \\cdot \\paren {-x}$ Then: :$\\forall n \\in \\Z: \\forall x \\in R: \\paren {n \\cdot x} \\circ x = n \\cdot \\paren {x \\circ x} = x \\circ \\paren {n \\cdot x}$"}653{"_id": "729", "title": "Power of Conjugate equals Conjugate of Power", "text": "Let $\\struct {G, \\circ}$ be a group whose identity is $e$. Let $x, y \\in G$ such that $\\exists a \\in G: x \\circ a = a \\circ y$. That is, let $x$ and $y$ be conjugate. Then: : $\\forall n \\in \\Z: y^n = \\paren {a^{-1} \\circ x \\circ a}^n = a^{-1} \\circ x^n \\circ a$ It follows directly that: : $\\exists b \\in G: \\forall n \\in \\Z: y^n = b \\circ x^n \\circ b^{-1}$ In particular: : $y^{-1} = \\paren {a^{-1} \\circ x \\circ a}^{-1} = a^{-1} \\circ x^{-1} \\circ a$"}654{"_id": "730", "title": "Product of Conjugates equals Conjugate of Products", "text": "Let $\\struct {G, \\circ}$ be a group. Then: :$\\forall a, x, y \\in G: \\paren {a \\circ x \\circ a^{-1} } \\circ \\paren {a \\circ y \\circ a^{-1} } = a \\circ \\paren {x \\circ y} \\circ a^{-1}$ That is, the product of conjugates is equal to the conjugate of the product."}655{"_id": "731", "title": "Power of Product with Inverse", "text": "Let $G$ be a group whose identity is $e$. Let $a, b \\in G: a b = b a^{-1}$. Then: : $\\forall n \\in \\Z: a^n b = b a^{-n}$"}656{"_id": "732", "title": "Powers of Elements in Group Direct Product", "text": "Let $\\left({G, \\circ_1}\\right)$ and $\\left({H, \\circ_2}\\right)$ be group whose identities are $e_G$ and $e_H$. Let $\\left({G \\times H, \\circ}\\right)$ be the group direct product (either external or internal) of $G$ and $H$. Then: : $\\forall n \\in \\Z: \\forall g \\in G, h \\in H: \\left({g, h}\\right)^n = \\left({g^n, h^n}\\right)$"}657{"_id": "733", "title": "Powers of Commutative Elements in Semigroups", "text": "Let $\\left ({S, \\circ}\\right)$ be a semigroup. Let $a, b \\in S$ both be cancellable elements of $S$. Then the following results hold:"}658{"_id": "734", "title": "Powers of Commutative Elements in Monoids", "text": "These results are an extension of the results in Powers of Commutative Elements in Semigroups in which the domain of the indices is extended to include all integers. Let $\\left ({S, \\circ}\\right)$ be a monoid whose identity is $e_S$. Let $a, b \\in S$ be invertible elements for $\\circ$ that also commute. Then the following results hold."}659{"_id": "735", "title": "Powers of Commutative Elements in Groups", "text": "Let $\\struct {G, \\circ}$ be a group. Let $a, b \\in G$ such that $a$ and $b$ commute. Then the following results hold:"}660{"_id": "736", "title": "General Morphism Property for Semigroups", "text": "Let $\\struct {S, \\circ}$ and $\\struct {T, *}$ be semigroups. Let $\\phi: S \\to T$ be a homomorphism. Then: :$\\forall s_k \\in S: \\map \\phi {s_1 \\circ s_2 \\circ \\cdots \\circ s_n} = \\map \\phi {s_1} * \\map \\phi {s_2} * \\cdots * \\map \\phi {s_n}$ Hence it follows that: :$\\forall n \\in \\N_{>0}: \\forall s \\in S: \\map \\phi {s^n} = \\paren {\\map \\phi s}^n$"}661{"_id": "737", "title": "Homomorphism of Power of Group Element", "text": "Let $\\struct {G, \\circ}$ and $\\struct {H, \\ast}$ be groups. Let $\\phi: S \\to T$ be a group homomorphism. Then: : $\\forall n \\in \\Z: \\forall g \\in G: \\map \\phi {g^n} = \\paren {\\map \\phi g}^n$"}662{"_id": "739", "title": "Finite Subgroup Test", "text": "Let $\\struct {G, \\circ}$ be a group. Let $H$ be a non-empty finite subset of $G$. Then: :$H$ is a subgroup of $G$ {{iff}}: :$\\forall a, b \\in H: a \\circ b \\in H$ That is, a non-empty finite subset of $G$ is a subgroup {{iff}} it is closed."}663{"_id": "740", "title": "Powers of Element form Subgroup", "text": "Let $\\struct {G, \\circ}$ be a group. Then: :$\\forall a \\in G: H = \\set {a^n: n \\in \\Z} \\le G$ That is, the subset of $G$ comprising all elements possible as powers of $a \\in G$ is a subgroup of $G$."}664{"_id": "742", "title": "Homomorphism of Generated Group", "text": "Let $\\left({G, \\circ}\\right)$ and $\\left({H, \\circ}\\right)$ be groups. Let $\\phi: G \\to H$ and $\\psi: G \\to H$ be homomorphisms.  Let $\\left \\langle {S} \\right \\rangle = G$ be the group generated by $S$. Let $\\forall x \\in S: \\phi \\left({x}\\right) = \\psi \\left({x}\\right)$ Then $\\phi = \\psi$."}665{"_id": "743", "title": "Set of Words Generates Group", "text": "Let $S \\subseteq G$ where $G$ is a group. Let $\\hat S$ be defined as $S \\cup S^{-1}$, where $S^{-1}$ is the set of all the inverses of all the elements of $S$. Then $\\gen S = \\map W {\\hat S}$, where $\\map W {\\hat S}$ is the set of words of $\\hat S$."}666{"_id": "744", "title": "Subset Product is Subset of Generator", "text": "Let $\\struct {G, \\circ}$ be a group. Let $X, Y \\subseteq \\struct {G, \\circ}$. Then $X \\circ Y \\subseteq \\gen {X, Y}$ where: :$X \\circ Y$ is the Subset Product of $X$ and $Y$ in $G$. :$\\gen {X, Y}$ is the subgroup of $G$ generated by $X$ and $Y$."}667{"_id": "745", "title": "Order of Subset Product with Singleton", "text": "Let $\\struct {G, \\circ}$ be a group. Let $X, Y \\subseteq \\struct {G, \\circ}$ such that $X$ is a singleton: :$X = \\set x$ Then: :$\\order {X \\circ Y} = \\order Y = \\order {Y \\circ X}$ where $\\order S$ is defined as the order of $S$."}668{"_id": "746", "title": "Product of Subset with Intersection", "text": "Let $\\struct {G, \\circ}$ be an algebraic structure. Let $X, Y, Z \\subseteq G$. Then: :$X \\circ \\paren {Y \\cap Z} \\subseteq \\paren {X \\circ Y} \\cap \\paren {X \\circ Z}$ :$\\paren {Y \\cap Z} \\circ X \\subseteq \\paren {Y \\circ X} \\cap \\paren {Z \\circ X}$ where $X \\circ Y$ denotes the subset product of $X$ and $Y$."}669{"_id": "747", "title": "Order of Subgroup Product", "text": "Let $G$ be a group. Let $H$ and $K$ be subgroups of $G$. Then: :$\\order {H K} = \\dfrac {\\order H \\order K} {\\order {H \\cap K} }$ where: :$H K$ denotes subset product :$\\order H$ denotes the order of $H$."}670{"_id": "748", "title": "Index of Intersection of Subgroups", "text": "Let $G$ be a group. Let $H, K$ be subgroups of finite index of $G$. Then: :$\\index G {H \\cap K} \\le \\index G H \\index G K$ where $\\index G H$ denotes the index of $H$ in $G$. Note that here the symbol $\\le$ is being used with its meaning '''less than or equal to'''. Equality holds {{iff}} $H K = \\set {h k: h \\in H, k \\in K} = G$."}671{"_id": "749", "title": "Intersection of Subgroups of Prime Order", "text": "Let $G$ be a group whose identity is $e$. Let $H$ and $K$ be subsets of $G$ such that: :$\\order H = \\order K = p$ :$H \\ne K$ :$p$ is prime. Then: : $H \\cap K = \\set e$ That is, the intersection of two unequal subgroups of a group, both of whose order is the same prime, consists solely of the identity."}672{"_id": "750", "title": "Tower Law for Subgroups", "text": "Let $\\struct {G, \\circ}$ be a group. Let $H$ be a subgroup of $G$ with finite index. Let $K$ be a subgroup of $H$. Then: :$\\index G K = \\index G H \\index H K$ where $\\index G H$ denotes the index of $H$ in $G$."}673{"_id": "751", "title": "Morphism from Integers to Group", "text": "Let $G$ be a group whose identity is $e$. Let $g \\in G$. Let $\\phi: \\Z \\to G$ be the mapping defined as: :$\\forall n \\in \\Z: \\map \\phi n = g^n$. Then: : If $g$ has infinite order, then $\\phi$ is a group isomorphism from $\\struct {\\Z, +}$ to $\\gen g$. : If $g$ has finite order such that $\\order g = m$, then $\\phi$ is a group epimorphism from $\\struct {\\Z, +}$ to $\\gen g$ whose kernel is the principal ideal $\\paren m$. :Thus $\\gen g$ is isomorphic to $\\struct {\\Z, +}$, and $m$ is the smallest (strictly) positive integer such that $g^m = e$."}674{"_id": "752", "title": "Identity is Only Group Element of Order 1", "text": "In every group, the identity, and only the identity, has order $1$."}675{"_id": "753", "title": "Group Element is Self-Inverse iff Order 2", "text": "Let $\\struct {S, \\circ}$ be a group whose identity is $e$. An element $x \\in \\struct {S, \\circ}$ is self-inverse {{iff}}: :$\\order x = 2$"}676{"_id": "754", "title": "Powers of Infinite Order Element", "text": "Let $G$ be a group whose identity is $e$. Let $a \\in G$ have infinite order in $G$. Then: :$\\forall m, n \\in \\Z: m \\ne n \\implies a^m \\ne a^n$"}677{"_id": "755", "title": "Element of Finite Group is of Finite Order", "text": "In any finite group, each element has finite order."}678{"_id": "756", "title": "Inverse Element is Power of Order Less 1", "text": "Let $G$ be a group whose identity is $e$. Let $g \\in G$ be of finite order. Then: : $\\order g = n \\implies g^{n - 1} = g^{-1}$"}679{"_id": "757", "title": "Equal Powers of Finite Order Element", "text": ":$g^r = g^s \\iff k \\divides \\paren {r - s}$"}680{"_id": "758", "title": "Order of Element Divides Order of Finite Group", "text": "In a finite group, the order of a group element divides the order of its group: :$\\forall x \\in G: \\order x \\divides \\order G$"}681{"_id": "760", "title": "Element to Power of Group Order is Identity", "text": "Let $G$ be a group whose identity is $e$ and whose order is $n$. Then: :$\\forall g \\in G: g^n = e$"}682{"_id": "761", "title": "Boolean Group is Abelian", "text": "Let $G$ be a Boolean group. Then $G$ is abelian."}683{"_id": "762", "title": "Order of Group Element equals Order of Inverse", "text": "Let $G$ be a group whose identity is $e$. Then: : $\\forall x \\in G: \\order x = \\order {x^{-1} }$ where $\\order x$ denotes the order of $x$."}684{"_id": "764", "title": "Order of Conjugate Element equals Order of Element", "text": "Let $\\struct {G, \\circ}$ be a group whose identity is $e$. Then :$\\forall a, x \\in \\struct {G, \\circ}: \\order {x \\circ a \\circ x^{-1} } = \\order a$ where $\\order a$ denotes the order of $a$ in $G$."}685{"_id": "765", "title": "Order of Homomorphic Image of Group Element", "text": "Let $G$ and $H$ be groups whose identities are $e_G$ and $e_H$ respectively. Let $\\phi: G \\to H$ be a homomorphism. Let $g \\in G$ be of finite order. Then: :$\\forall g \\in G: \\order {\\map \\phi g} \\divides \\order g$ where $\\divides$ denotes divisibility."}686{"_id": "769", "title": "Non-Trivial Group has Non-Trivial Cyclic Subgroup", "text": "Let $G$ be a group whose identity element is $e$. Let $g \\in G$. If $g$ has infinite order, then $\\gen g$ is an infinite cyclic group. If $\\order g = n$, then $\\gen g$ is a cyclic group with $n$ elements. Thus, every group which is non-trivial has at least one cyclic subgroup which is also non-trivial. In the case that $G$ is itself cyclic, that cyclic subgroup may of course be itself."}687{"_id": "770", "title": "Epimorphism from Integers to Cyclic Group", "text": "Let $\\gen a = \\struct {G, \\circ}$ be a cyclic group. Let $f: \\Z \\to G$ be a mapping defined as: $\\forall n \\in \\Z: \\map f n = a^n$. Then $f$ is a (group) epimorphism from $\\struct {\\Z, +}$ onto $\\gen a$."}688{"_id": "771", "title": "Cyclic Group is Abelian", "text": "Let $G$ be a cyclic group. Then $G$ is abelian."}689{"_id": "772", "title": "Cyclic Groups of Same Order are Isomorphic", "text": "Two cyclic groups of the same order are isomorphic to each other."}690{"_id": "773", "title": "Order of Subgroup of Cyclic Group", "text": "Let $C_n = \\gen g$ be the cyclic group of order $n$ which is generated by $g$ whose identity is $e$. Let $a \\in C_n: a = g^i$. Let $H = \\gen a$. Then: :$\\order H = \\dfrac n {\\gcd \\set {n, i} }$ where: :$\\order H$ denotes the order of $H$ :$\\gcd \\set {n, i}$ denotes the greatest common divisor of $n$ and $i$."}691{"_id": "774", "title": "Number of Powers of Cyclic Group Element", "text": "Let $G$ be a cyclic group of order $n$, generated by $g$. Let $d \\divides n$. Then the element $g^{n/d}$ has $d$ distinct powers."}692{"_id": "775", "title": "Subgroup of Finite Cyclic Group is Determined by Order", "text": "Let $G = \\gen g$ be a cyclic group whose order is $n$ and whose identity is $e$. Let $d \\divides n$, where $\\divides$ denotes divisibility. Then there exists exactly one subgroup $G_d = \\gen {g^{n / d} }$ of $G$ with $d$ elements."}693{"_id": "777", "title": "Prime Group is Cyclic", "text": "Let $p$ be a prime number. Let $G$ be a group whose order is $p$. Then $G$ is cyclic."}694{"_id": "778", "title": "Group of Order less than 6 is Abelian", "text": "All groups with less than $6$ elements are abelian."}695{"_id": "779", "title": "Power of Generator of Cyclic Group is Generator iff Power is Coprime with Order", "text": "Let $C_n$ be the cyclic group of order $n$. Let $C_n = \\gen a$, that is, that $C_n$ is generated by $a$. Then: :$C_n = \\gen {a^k} \\iff k \\perp n$ That is, $C_n$ is also generated by $a^k$ {{iff}} $k$ is coprime to $n$."}696{"_id": "780", "title": "Order of Conjugate of Subgroup", "text": "Let $G$ be a group. Let $H$ be a subgroup of $G$ such that $H$ is of finite order. Then $\\order {H^a} = \\order H$."}697{"_id": "781", "title": "Subgroup of Index 2 is Normal", "text": "A subgroup of index $2$ is always normal."}698{"_id": "782", "title": "Intersection of Normal Subgroups is Normal", "text": "Let $G$ be a group. Let $I$ be an indexing set. Let $\\family {N_i}_{i \\mathop \\in I}$ be a non-empty indexed family of normal subgroups of $G$. Then $\\displaystyle \\bigcap_{i \\mathop \\in I} N_i$ is a normal subgroup of $G$."}699{"_id": "783", "title": "Union of Conjugacy Classes is Normal", "text": "Let $G$ be a group. Let $H \\le G$. Then $H$ is normal in $G$ {{iff}} $H$ is a union of conjugacy classes of $G$."}700{"_id": "784", "title": "Unique Subgroup of a Given Order is Normal", "text": "Let a group $G$ have only one subgroup of a given order. Then that subgroup is normal."}701{"_id": "786", "title": "Smallest Normal Subgroup containing Set", "text": "Let $S \\subseteq G$ where $G$ is a group. Then there exists a unique smallest normal subgroup of $G$ which contains $S$."}702{"_id": "787", "title": "Conjugate of Set with Inverse Closed for Inverses", "text": "Let $G$ be a group. Let $S \\subseteq G$. Let $\\hat S = S \\cup S^{-1}$, where $S^{-1}$ is the set of all the inverses of all the elements of $S$. Let $\\tilde S = \\set {a s a^{-1}: s \\in \\hat S, a \\in G}$. That is, $\\tilde S$ is the set containing all the conjugates of the elements of $S$ and all their inverses. Then: : $\\forall x \\in \\tilde S: x^{-1} \\in \\tilde S$"}703{"_id": "788", "title": "Conjugate of Set with Inverse is Closed", "text": "Let $G$ be a group. Let $S \\subseteq G$. Let $\\hat S = S \\cup S$. Let $\\tilde S = \\left\\{{a s a^{-1}: s \\in \\hat S, a \\in G}\\right\\}$. Let $W \\left({\\tilde S}\\right)$ be the set of words of $\\tilde S$. Then $\\forall w \\in W \\left({\\tilde S}\\right): \\forall a \\in G: a w a^{-1} \\in W \\left({\\tilde S}\\right)$."}704{"_id": "789", "title": "Generator of Normal Subgroup", "text": "Let $G$ be a group. Let $S \\subseteq G$. Let $\\hat S = S \\cup S^{-1}$, where $S^{-1}$ is the set of all the inverses of all the elements of $S$. Let $\\tilde S = \\left\\{{a s a^{-1}: s \\in \\hat S, a \\in G}\\right\\}$. Let $W \\left({\\tilde S}\\right)$ be the set of words of $\\tilde S$. Let $N$ be the smallest normal subgroup of $G$ that contains $S$. Then $N = \\left \\langle {S} \\right \\rangle = W \\left({\\tilde S}\\right)$."}705{"_id": "790", "title": "Subset Product with Normal Subgroup as Generator", "text": "Let $G$ be a group whose identity is $e$. Let: :$H$ be a subgroup of $G$ :$N$ be a normal subgroup of $G$. Then: :$N \\lhd \\gen {N, H} = N H = H N \\le G$ where: :$\\le$ denotes subgroup :$\\lhd$ denotes normal subgroup :$\\gen {N, H}$ denotes a subgroup generator :$N H$ denotes subset product."}706{"_id": "791", "title": "Subset Product of Normal Subgroups is Normal", "text": "Let $\\struct {G, \\circ}$ be a group. Let $N$ and $N'$ be normal subgroups of $G$. Then $N N'$ is also a normal subgroup of $G$."}707{"_id": "792", "title": "Prime Group is Simple", "text": "Groups of prime order are simple."}708{"_id": "793", "title": "Prime Group has no Proper Subgroups", "text": "A nontrivial group $G$ has no proper subgroups except the trivial group {{iff}} $G$ is finite and its order is prime."}709{"_id": "794", "title": "Quotient Group of Cyclic Group", "text": "Let $G$ be a cyclic group which is generated by $g$. Let $H$ be a subgroup of $G$. Then $g H$ generates $G / H$."}710{"_id": "795", "title": "Order of Element in Quotient Group", "text": "Let $G$ be a group, and let $H$ be a normal subgroup of $G$. Let $G / H$ be the quotient group of $G$ by $H$. The order of $a H \\in G / H$ divides the order of $a \\in G$."}711{"_id": "799", "title": "Quotient of Group by Center Cyclic implies Abelian", "text": "Let $G$ be a group. Let $\\map Z G$ be the center of $G$. Let $G / \\map Z G$ be the quotient group of $G$ by $\\map Z G$. Let $G / \\map Z G$ be cyclic. Then $G$ is abelian, so $G = \\map Z G$. That is, the group $G / \\map Z G$ cannot be a cyclic group which is non-trivial."}712{"_id": "800", "title": "Centralizer is Normal Subgroup of Normalizer", "text": "Let $G$ be a group. Let $H \\le G$ be a subgroup of $G$.  Let $\\map {C_G} H$ be the centralizer of $H$ in $G$. Let $\\map {N_G} H$ be the normalizer of $H$ in $G$. Let $\\Aut H$ be the automorphism group of $H$. Then: :$(1): \\quad \\map {C_G} H \\lhd \\map {N_G} H$ :$(2): \\quad \\map {N_G} H / \\map {C_G} H \\cong K$ where: :$\\map {N_G} H / \\map {C_G} H$ is the quotient group of $\\map {N_G} H$ by $\\map {C_G} H$ :$K$ is a subgroup of $\\Aut H$."}713{"_id": "801", "title": "Number of Distinct Conjugate Subsets is Index of Normalizer", "text": "Let $G$ be a group. Let $S$ be a subset of $G$. Let $\\map {N_G} S$ be the normalizer of $S$ in $G$. Let $\\index G {\\map {N_G} S}$ be the index of $\\map {N_G} S$ in $G$. The number of distinct subsets of $G$ which are conjugates of $S \\subseteq G$ is $\\index G {\\map {N_G} S}$."}714{"_id": "803", "title": "Element of Group Not Conjugate to Proper Subgroup", "text": "Let $G$ be a finite group. Let $H$ be a proper subgroup of $G$. Then there is at least one element of $G$ not contained in $H$ or in any of its conjugates."}715{"_id": "804", "title": "Second Isomorphism Theorem", "text": "=== Groups === {{:Second Isomorphism Theorem/Groups}} === Rings === {{:Second Isomorphism Theorem/Rings}} This result is also referred to by some sources as the '''first isomorphism theorem'''."}716{"_id": "805", "title": "Third Isomorphism Theorem", "text": "=== Groups === {{:Third Isomorphism Theorem/Groups}} === Rings === {{:Third Isomorphism Theorem/Rings}}"}717{"_id": "807", "title": "Inverse of Inner Automorphism", "text": "Let $G$ be a group. Let $x \\in G$. Let $\\kappa_x$ be the inner automorphism of $G$ given by $x$. Then: : $\\paren {\\kappa_x}^{-1} = \\kappa_{x^{-1} }$"}718{"_id": "809", "title": "Conjugates of Elements in Centralizer", "text": "Let $G$ be a group. Let $\\map {C_G} a$ be the centralizer of $a$ in $G$. Then $\\forall g, h \\in G: g a g^{-1} = h a h^{-1}$ {{iff}} $g$ and $h$ belong to the same left coset of $\\map {C_G} a$."}719{"_id": "810", "title": "Number of Conjugates is Number of Cosets of Centralizer", "text": "Let $G$ be a group. Let $\\map {C_G} a$ be the centralizer of $a$ in $G$. Then the number of different conjugates of $a$ in $G$ equals the number of different (left) cosets of $\\map {C_G} a$: :$\\card {\\conjclass a} = \\index G {\\map {C_G} a}$ where: :$\\conjclass a$ is the conjugacy class of $a$ in $G$ :$\\index G {\\map {C_G} a}$ is the index of $\\map {C_G} a$ in $G$. Consequently: :$\\card {\\conjclass a} \\divides \\order G$"}720{"_id": "811", "title": "Size of Conjugacy Class is Index of Normalizer", "text": "Let $G$ be a group. Let $x \\in G$. Let $\\conjclass x$ be the conjugacy class of $x$ in $G$. Let $\\map {N_G} x$ be the normalizer of $x$ in $G$. Let $\\index G {\\map {N_G} x}$ be the index of $\\map {N_G} x$ in $G$. The number of elements in $\\conjclass x$ is $\\index G {\\map {N_G} x}$."}721{"_id": "812", "title": "Conjugacy Class of Element of Center is Singleton", "text": "Let $G$ be a group. Let $\\map Z G$ denote the center of $G$. The elements of $\\map Z G$ form singleton conjugacy classes, and the elements of $G \\setminus \\map Z G$ belong to multi-element conjugacy classes."}722{"_id": "813", "title": "Conjugacy Class Equation", "text": "Let $G$ be a group. Let $\\order G$ denote the order of $G$. Let $\\map Z G$ denote the center of $G$. Let $x \\in G$. Let $\\map {N_G} x$ denote the normalizer of $x$ in $G$. Let $\\index G {\\map {N_G} x}$ denote the index of $\\map {N_G} x$ in $G$. Let $m$ be the number of non-singleton conjugacy classes of $G$. Let $x_j: j \\in \\set {1, 2, \\ldots, m}$ be arbitrary elements of those conjugacy classes. Then: :$\\displaystyle \\order G = \\order {\\map Z G} + \\sum_{j \\mathop = 1}^m \\index G {\\map {N_G} {x_j} }$"}723{"_id": "814", "title": "Group of Order Prime Squared is Abelian", "text": "A group whose order is the square of a prime is abelian."}724{"_id": "815", "title": "Center of Group of Prime Power Order is Non-Trivial", "text": "Let $G$ be a group whose order is the power of a prime. Then the center of $G$ is non-trivial: :$\\forall G: \\order G = p^r: p \\in \\mathbb P, r \\in \\N_{>0}: \\map Z G \\ne \\set e$"}725{"_id": "816", "title": "Center of Group of Order Prime Cubed", "text": "Let $G$ be a group of order $p^3$, where $p$ is a prime. Let $\\map Z G$ be the center of $G$. Then $\\order {\\map Z G} \\ne p^2$."}726{"_id": "817", "title": "Prime Power Group has Non-Trivial Proper Normal Subgroup", "text": "Let $G$ be a group, whose identity is $e$, such that $\\order G = p^n: n > 1, p \\in \\mathbb P$. Then $G$ has a proper normal subgroup which is non-trivial."}727{"_id": "818", "title": "Composition Series of Group of Prime Power Order", "text": "Let $G$ be a group whose identity is $e$, and whose order is a prime power: :$\\order G = p^n, p \\in \\mathbb P, n \\ge 1$ Then $G$ has a composition series: :$\\set e = G_0 \\subset G_1 \\subset \\ldots \\subset G_n = G$ such that $\\order {G_k} = p^k$, $G_k \\lhd G_{k + 1}$ and $G_{k + 1} / G_k$ is cyclic and of order $p$."}728{"_id": "819", "title": "Sum Rule for Counting", "text": "Let there be: : $r_1$ different objects in the set $S_1$ : $r_2$ different objects in the set $S_2$ : $\\ldots$ : $r_m$ different objects in the set $S_m$. Let $\\displaystyle \\bigcap_{i \\mathop = 1}^m S_i = \\varnothing$. Then the number of ways to select an object from one of the $m$ sets is $\\displaystyle \\sum_{i \\mathop = 1}^m r_i$."}729{"_id": "820", "title": "Product Rule for Counting", "text": "Let it be possible to choose an element $\\alpha$ from a given set $S$ in $m$ different ways. Let it be possible to choose an element $\\beta$ from a given set $T$ in $n$ different ways. Then the ordered pair $\\tuple {\\alpha, \\beta}$ can be chosen from the cartesian product $S \\times T$ in $m n$ different ways."}730{"_id": "822", "title": "External Direct Product of Abelian Groups is Abelian Group", "text": "Let $\\struct {G, \\circ_1}$ and $\\struct {H, \\circ_2}$ be groups. Then the group direct product $\\struct {G \\times H, \\circ}$ is abelian {{iff}} both $\\struct {G, \\circ_1}$ and $\\struct {H, \\circ_2}$ are abelian."}731{"_id": "824", "title": "Group Direct Product of Cyclic Groups", "text": "Let $G$ and $H$ both be finite cyclic groups with orders $n = \\order G$ and $m = \\order H$ respectively. Then their group direct product $G \\times H$ is cyclic {{iff}} $g$ and $h$ are coprime, that is, $g \\perp h$."}732{"_id": "825", "title": "Group Direct Product of Infinite Cyclic Groups", "text": "The group direct product of two infinite cyclic groups is not cyclic."}733{"_id": "826", "title": "Order of Group Element in Group Direct Product", "text": "Let $G$ and $H$ be finite groups. Let $g \\in G: \\order g = m, h \\in H: \\order h = n$. Then $\\order {\\tuple {g, h} }$ in $G \\times H$ is $\\lcm \\set {m, n}$."}734{"_id": "827", "title": "Subgroup Product is Internal Group Direct Product iff Surjective", "text": "Let $G$ be a group. Let $\\sequence {H_n}$ be a sequence of subgroups of $G$. Let $\\displaystyle \\phi: \\prod_{k \\mathop = 1}^n H_k \\to G$ be a mapping defined by: :$\\displaystyle \\map \\phi {h_1, h_2, \\ldots, h_n} = \\prod_{k \\mathop = 1}^n h_k$ Then $\\phi$ is surjective {{iff}}: : $\\displaystyle G = \\prod_{k \\mathop = 1}^n H_k$ That is, {{iff}} $G$ is the internal group direct product of $H_1, H_2, \\ldots, H_n$."}735{"_id": "828", "title": "Internal Group Direct Product is Injective", "text": "Let $G$ be a group whose identity is $e$. Let $H_1, H_2$ be subgroups of $G$. Let $\\phi: H_1 \\times H_2 \\to G$ be a mapping defined by: :$\\map \\phi {h_1, h_2} = h_1 h_2$ Then $\\phi$ is injective {{iff}}: :$H_1 \\cap H_2 = \\set e$"}736{"_id": "829", "title": "Internal Group Direct Product Isomorphism", "text": "Let $G$ be a group. Let $H_1, H_2$ be subgroups of $G$. Let $\\phi: H_1 \\times H_2 \\to G$ be the mapping defined by $\\map \\phi {h_1, h_2} := h_1 h_2$. If $\\phi$ is a (group) isomorphism, then both $H_1$ and $H_2$ are normal subgroups of $G$."}737{"_id": "830", "title": "Internal Group Direct Product of Normal Subgroups", "text": "Let $G$ be a group whose identity is $e$. Let $H_1, H_2$ be subgroups of $G$. Let $\\phi: H_1 \\times H_2 \\to G$ be a mapping defined by $\\map \\phi {h_1, h_2} = h_1 h_2$. Let $H_1$ and $H_2$ be normal subgroups of $G$, and let $H_1 \\cap H_2 = \\set e$. Then $\\phi$ is a (group) homomorphism."}738{"_id": "831", "title": "Internal Direct Product Theorem", "text": "Let $\\struct {G, \\circ}$ be a group whose identity is $e$. Let $H_1, H_2 \\le G$. Then $G$ is the internal group direct product of $H_1$ and $H_2$ {{iff}}: :$(1): \\quad G = H_1 \\circ H_2$ :$(2): \\quad H_1 \\cap H_2 = \\set e$ :$(3): \\quad H_1, H_2 \\lhd G$ where $H_1 \\lhd G$ denotes that $H_1$ is a normal subgroup of $G$."}739{"_id": "832", "title": "Inclusion Mapping is Surjection iff Identity", "text": "Let $T$ be a set. Let $S\\subseteq T$ be a subset. Let $i_S: S \\to T$ be the inclusion mapping. Then: :$i_S: S \\to T$ is surjective {{iff}} $i_S: S \\to T = I_S: S \\to S$ where $I_S: S \\to S$ denotes the identity mapping on $S$. Alternatively, this theorem can be worded as: :$i_S: S \\to S = I_S: S \\to S$ It follows directly that from Surjection by Restriction of Codomain, the surjective restriction of $i_S: S \\to T$ to $i_S: S \\to \\Img {i_S}$ is itself the identity mapping."}740{"_id": "834", "title": "Internal Direct Product Generated by Subgroups", "text": "Let $G$ be a group whose identity is $e$. Let $\\sequence {H_n}$ be a sequence of subgroups of $G$. Then: :the subgroup generated by $\\displaystyle \\bigcup_{k \\mathop = 1}^n H_k$ is the internal group direct product of $\\sequence {H_n}$ {{iff}}: :$\\sequence {H_n}$ is an independent sequence of subgroups such that every element of $H_i$ commutes with every element of $H_j$ whenever $1 \\le i < j \\le n$."}741{"_id": "835", "title": "Internal Group Direct Product Commutativity", "text": "Let $\\struct {G, \\circ}$ be a group whose identity is $e$. Let $H_1, H_2 \\le G$. Let $\\struct {G, \\circ}$ be the internal group direct product of $H_1$ and $H_2$. Then: :$\\forall h_1 \\in H_1, h_2 \\in H_2: h_1 \\circ h_2 = h_2 \\circ h_1$"}742{"_id": "836", "title": "Internal and External Group Direct Products are Isomorphic", "text": "Let $G$ be a group whose identity is $e$. Then $G$ is the (external) group direct product of $G_1, G_2, \\ldots, G_n$ {{iff}} $G$ is the internal group direct product of $N_1, N_2, \\ldots, N_n$ such that: :$\\forall i \\in \\N_n: N_i \\cong G_i$ where: :$\\cong$ denotes (group) isomorphism :$\\N_n$ denotes $\\set {1, 2, \\ldots, n}$"}743{"_id": "838", "title": "Pullback of Quotient Group Isomorphism is Subgroup", "text": "Let $\\struct {G, \\circ}$ be a group whose identity element is $e_G$. Let $\\struct {H, *}$ be a group whose identity element is $e_H$. Let $N \\lhd G, K \\lhd H$ be normal subgroups of $G$ and $H$ respectively. Let: :$G / N \\cong H / K$ where: :$G / N$ denotes the quotient of $G$ by $N$ :$\\cong$ denotes group isomorphism. Let $\\theta: G / N \\to H / K$ be such a group isomorphism. Let $G \\times^\\theta H$ be the pullback of $G$ and $H$ via $\\theta$. Then $G \\times^\\theta H$ is a subgroup of $G \\times H$."}744{"_id": "839", "title": "Group/Examples/x+y over 1+xy", "text": "Let $G = \\set {x \\in \\R: -1 < x < 1}$ be the set of all real numbers whose absolute value is less than $1$. Let $\\circ: G \\times G \\to \\R$ be the binary operation defined as: :$\\forall x, y \\in G: x \\circ y = \\dfrac {x + y} {1 + x y}$ The algebraic structure $\\struct {G, \\circ}$ is a group."}745{"_id": "840", "title": "Group/Examples/inv x = 1 - x", "text": "Let $S = \\set {x \\in \\R: 0 < x < 1}$. Then an operation $\\circ$ can be found such that $\\struct {S, \\circ}$ is a group such that the inverse of $x \\in S$ is $1 - x$."}746{"_id": "841", "title": "Group/Examples/Self-Inverse and Cancellable Elements", "text": "Let $S$ be a set with an operation which assigns to each $\\tuple {a, b} \\in S \\times S$ an element $a \\ast b \\in S$ such that: : $(1): \\quad \\exists e \\in S: a \\ast b = e \\iff a = b$ : $(2): \\quad \\forall a, b, c \\in S: \\paren {a \\ast c} \\ast \\paren {b \\ast c} = a \\ast b$ Then $\\struct {S, \\circ}$ is a group, where $\\circ$ is defined as $a \\circ b = a \\ast \\paren {e \\ast b}$."}747{"_id": "842", "title": "Complex Numbers under Addition form Abelian Group", "text": "Let $\\C$ be the set of complex numbers. The structure $\\struct {\\C, +}$ is an infinite abelian group."}748{"_id": "843", "title": "Non-Zero Complex Numbers under Multiplication form Abelian Group", "text": "Let $\\C_{\\ne 0}$ be the set of complex numbers without zero, that is: :$\\C_{\\ne 0} = \\C \\setminus \\set 0$ The structure $\\struct {\\C_{\\ne 0}, \\times}$ is an infinite abelian group."}749{"_id": "844", "title": "Real Numbers under Addition form Abelian Group", "text": "Let $\\R$ be the set of real numbers. The structure $\\struct {\\R, +}$ is an infinite abelian group."}750{"_id": "845", "title": "Non-Zero Real Numbers under Multiplication form Abelian Group", "text": "Let $\\R_{\\ne 0}$ be the set of real numbers without zero: :$\\R_{\\ne 0} = \\R \\setminus \\set 0$ The structure $\\struct {\\R_{\\ne 0}, \\times}$ is an uncountable abelian group."}751{"_id": "846", "title": "Rational Numbers under Addition form Abelian Group", "text": "Let $\\Q$ be the set of rational numbers. The structure $\\struct {\\Q, +}$ is a countably infinite abelian group."}752{"_id": "847", "title": "Non-Zero Rational Numbers under Multiplication form Abelian Group", "text": "Let $\\Q_{\\ne 0}$ be the set of non-zero rational numbers: :$\\Q_{\\ne 0} = \\Q \\setminus \\set 0$ The structure $\\struct {\\Q_{\\ne 0}, \\times}$ is a countably infinite abelian group."}753{"_id": "848", "title": "Integers under Multiplication form Countably Infinite Commutative Monoid", "text": "The set of integers under multiplication $\\struct {\\Z, \\times}$ is a countably infinite commutative monoid."}754{"_id": "849", "title": "Additive Group of Rationals is Subgroup of Reals", "text": "Let $\\struct {\\Q, +}$ be the additive group of rational numbers. Let $\\struct {\\R, +}$ be the additive group of real numbers. Then $\\struct {\\Q, +}$ is a normal subgroup of $\\struct {\\R, +}$."}755{"_id": "850", "title": "Additive Group of Integers is Subgroup of Rationals", "text": "Let $\\struct {\\Z, +}$ be the additive group of integers. Let $\\struct {\\Q, +}$ be the additive group of rational numbers. Then $\\struct {\\Z, +}$ is a subgroup of $\\struct {\\Q, +}$."}756{"_id": "852", "title": "Multiplicative Group of Reals is Subgroup of Complex", "text": "Let $\\struct {\\R_{\\ne 0}, \\times}$ be the multiplicative group of real numbers. Let $\\struct {\\C_{\\ne 0}, \\times}$ be the multiplicative group of complex numbers. Then $\\struct {\\R_{\\ne 0}, \\times}$ is a normal subgroup of $\\struct {\\C_{\\ne 0}, \\times}$."}757{"_id": "853", "title": "Multiplicative Group of Rationals is Subgroup of Reals", "text": "Let $\\struct {\\Q_{\\ne 0}, \\times}$ be the multiplicative group of rational numbers. Let $\\struct {\\R_{\\ne 0}, \\times}$ be the multiplicative group of real numbers. Then $\\struct {\\Q_{\\ne 0}, \\times}$ is a normal subgroup of $\\left({\\R_{\\ne 0}, \\times}\\right)$."}758{"_id": "854", "title": "Circle Group is Infinite Abelian Group", "text": "The circle group $\\struct {K, \\times}$ is an uncountably infinite abelian group under the operation of complex multiplication."}759{"_id": "855", "title": "Homomorphism from Reals to Circle Group", "text": "Let $\\struct {\\R, +}$ be the additive group of real numbers. Let $\\struct {K, \\times}$ be the circle group. Let $\\phi: \\struct {\\R, +} \\to \\struct {K, \\times}$ be the mapping defined as: :$\\forall x \\in \\R: \\map \\phi x = e^{i x}$ Then $\\phi$ is a (group) homomorphism."}760{"_id": "856", "title": "Integers under Addition form Infinite Cyclic Group", "text": "The additive group of integers $\\struct {\\Z, +}$ is an infinite cyclic group which is generated by the element $1 \\in \\Z$."}761{"_id": "858", "title": "Inverse of Generator of Cyclic Group is Generator", "text": "Let $\\gen g = G$ be a cyclic group. Then: :$G = \\gen {g^{-1} }$ where $g^{-1}$ denotes the inverse of $g$. Thus, in general, a generator of a cyclic group is not unique."}762{"_id": "859", "title": "Generators of Infinite Cyclic Group", "text": "Let $\\gen g = G$ be an infinite cyclic group. Then the only other generator of $G$ is $g^{-1}$. Thus an infinite cyclic group has exactly $2$ generators."}763{"_id": "860", "title": "Subgroup of Integers is Ideal", "text": "Let $\\struct {\\Z, +}$ be the additive group of integers. Every subgroup of $\\struct {\\Z, +}$ is an ideal of the ring $\\struct {\\Z, +, \\times}$."}764{"_id": "861", "title": "Additive Group of Integers is Subgroup of Reals", "text": "Let $\\struct {\\Z, +}$ be the additive group of integers. Let $\\struct {\\R, +}$ be the additive group of real numbers. Then $\\struct {\\Z, +}$ is a subgroup of $\\struct {\\R, +}$."}765{"_id": "862", "title": "Quotient Group of Reals by Integers is Circle Group", "text": "Let $\\struct {\\Z, +}$ be the additive group of integers. Let $\\struct {\\R, +}$ be the additive group of real numbers. Let $K$ be the circle group. Then the quotient group of $\\struct {\\R, +}$ by $\\struct {\\Z, +}$ is isomorphic to $K$."}766{"_id": "863", "title": "Integers Modulo m under Addition form Cyclic Group", "text": "Let $\\Z_m$ be the set of integers modulo $m$. Let $+_m$ be the operation of addition modulo $m$. Let $\\struct {\\Z_m, +_m}$ denote the additive group of integers modulo $m$. Then $\\struct {\\Z_m, +_m}$ is a cyclic group of order $m$, generated by the element $\\eqclass 1 m \\in \\Z_m$."}767{"_id": "864", "title": "Integers Modulo m under Multiplication form Commutative Monoid", "text": "The structure: :$\\struct {\\Z_m, \\times}$ (where $\\Z_m$ is the set of integers modulo $m$) is a commutative monoid."}768{"_id": "865", "title": "Multiplicative Inverse in Ring of Integers Modulo m", "text": "Let $\\struct {\\Z_m, +_m, \\times_m}$ be the ring of integers modulo $m$. Then $\\eqclass k m \\in \\Z_m$ has an inverse in $\\struct {\\Z_m, \\times_m}$ {{iff}} $k \\perp m$."}769{"_id": "866", "title": "Reduced Residue System is Subset of Set of All Residue Classes", "text": "Let $\\Z_m$ be the set of set of residue classes modulo $m$. Let $\\Z'_m$ be the reduced residue system modulo $m$. Then: :$\\forall m \\in \\Z_{> 1}: \\O \\subset \\Z'_m \\subset \\Z_m$"}770{"_id": "867", "title": "Reduced Residue System under Multiplication forms Abelian Group", "text": "Let $\\Z_m$ be the set of set of residue classes modulo $m$. Let $\\struct {\\Z'_m, \\times}$ denote the multiplicative group of reduced residues modulo $m$. Then $\\struct {\\Z'_m, \\times}$ is an abelian group, precisely equal to the group of units of $\\Z_m$."}771{"_id": "868", "title": "Ring of Integers Modulo Prime is Field", "text": "Let $m \\in \\Z: m \\ge 2$. Let $\\struct {\\Z_m, +, \\times}$ be the ring of integers modulo $m$. Then: :$m$ is prime {{iff}}: :$\\struct {\\Z_m, +, \\times}$ is a field."}772{"_id": "869", "title": "Subgroups of Additive Group of Integers", "text": "Let $\\struct {\\Z, +}$ be the additive group of integers. Let $n \\Z$ be the additive group of integer multiples of $n$. Every non-trivial subgroup of $\\struct {\\Z, +}$ has the form $n \\Z$."}773{"_id": "870", "title": "Integer Multiples under Addition form Infinite Cyclic Group", "text": "Let $n \\Z$ be the set of integer multiples of $n$. Then $\\struct {n \\Z, +}$ is a countably infinite cyclic group. It is generated by $n$ and $-n$: :$n \\Z = \\gen n$ :$n \\Z = \\gen {-n}$ Hence $\\struct {n \\Z, +}$ can be justifiably referred to as the additive group of integer multiples."}774{"_id": "871", "title": "Quotient Group of Integers by Multiples", "text": "Let $\\struct {\\Z, +}$ be the additive group of integers. Let $\\struct {m \\Z, +}$ be the additive group of integer multiples of $m$. Let $\\struct {\\Z_m, +_m}$ be the additive group of integers modulo $m$. Then the quotient group of $\\struct {\\Z, +}$ by $\\struct {m \\Z, +}$ is $\\struct {\\Z_m, +_m}$. Thus: :$\\index \\Z {m \\Z} = m$"}775{"_id": "872", "title": "Euler's Theorem", "text": "Let $a, m \\in \\Z$ be coprime integers: $a \\perp m$. Let $\\map \\phi m$ be the Euler $\\phi$ function of $m$. Then: :$a^{\\map \\phi m} \\equiv 1 \\pmod m$"}776{"_id": "873", "title": "Symmetry Group is Group", "text": "Let $P$ be a geometric figure. Let $S_P$ be the set of all symmetries of $P$. Let $\\circ$ denote composition of mappings. The symmetry group $\\struct {S_P, \\circ}$ is indeed a group."}777{"_id": "875", "title": "Area of Square", "text": "A square has an area of $L^2$ where $L$ is the length of a side of the square. Thus we have that the area is a function of the length of the side: :$\\forall L \\in \\R_{\\ge 0}: \\map \\Area L = L^2$ where it is noted that the domain of $L$ is the set of non-negative real numbers."}778{"_id": "876", "title": "Order of Symmetric Group", "text": "Let $S$ be a finite set of cardinality $n$. Let $\\struct {\\map \\Gamma S, \\circ}$ be the symmetric group on $S$. Then $\\struct {\\map \\Gamma S, \\circ}$ has $n!$ elements (see factorial)."}779{"_id": "877", "title": "Powers of Permutation Element", "text": "Let $S_n$ denote the symmetric group on $n$ letters. Let $\\pi \\in S_n$, and let $i \\in \\N^*_n$. Let $k \\in \\Z: k > 0$ be the smallest such that: : $\\pi^k \\left({i}\\right) \\in \\left\\{{i, \\pi \\left({i}\\right), \\pi^2 \\left({i}\\right), \\ldots, \\pi^{k-1} \\left({i}\\right)}\\right\\}$ Then $\\pi^k \\left({i}\\right) = i$."}780{"_id": "879", "title": "Equality of Cycles", "text": "Let $S_n$ denote the symmetric group on $n$ letters, realised as the permutations of $\\left\\{{1, \\ldots, n}\\right\\}$. Let $\\rho = \\begin{bmatrix} a_0 & \\cdots & a_{k-1} \\end{bmatrix}$, $\\sigma = \\begin{bmatrix} b_0 & \\cdots & b_{k-1} \\end{bmatrix} \\in S_n$ be $k$-cycles of $S_n$. For any $d \\in \\Z$, by Integer is Congruent to Integer less than Modulus we can associate to $d$ a unique integer $\\tilde d \\in \\left\\{{0, \\ldots, k-1}\\right\\}$ such that $d \\equiv \\tilde d \\pmod k$. Define $a_d$ and $b_d$ for any $d \\in \\Z$ by $a_d = a_{\\tilde d}$ and $b_d = b_{\\tilde d}$ Choose $i, j \\in \\left\\{{1, \\ldots, k}\\right\\}$ such that: :$\\displaystyle a_i = \\min\\left\\{ {a_0, \\ldots, a_{k-1} }\\right\\}$ :$\\displaystyle b_j = \\min\\left\\{ {b_0, \\ldots, b_{k-1} }\\right\\}$ Then $\\rho = \\sigma$ {{iff}} for all $d \\in \\Z$, $a_{i + d} = b_{j + d}$. That is, $\\rho = \\sigma$ {{iff}} they are identical when written with the lowest element first."}781{"_id": "880", "title": "Identity Permutation is Disjoint from All", "text": "Let $S_n$ denote the symmetric group on $n$ letters. Let $e \\in S_n$ be the identity permutation on $S_n$. Then $e$ is disjoint from every permutation $\\pi$ on $S_n$ (including itself)."}782{"_id": "881", "title": "Disjoint Permutations Commute", "text": "Let $S_n$ denote the symmetric group on $n$ letters. Let $\\rho, \\sigma \\in S_n$ such that $\\rho$ and $\\sigma$ are disjoint. Then $\\rho \\sigma = \\sigma \\rho$."}783{"_id": "882", "title": "Permutation Induces Equivalence Relation", "text": "Let $S_n$ denote the symmetric group on $n$ letters. Let $\\pi \\in S_n$. Let $\\mathcal R_\\pi$ be the relation defined by: :$i \\mathrel {\\mathcal R_\\pi} j \\iff \\exists k \\in \\Z: \\map {\\pi^k} i = j$ Then $\\mathcal R_\\pi$ is an equivalence relation."}784{"_id": "883", "title": "Existence and Uniqueness of Cycle Decomposition", "text": "Let $S_n$ denote the symmetric group on $n$ letters. Every element of $S_n$ may be uniquely expressed as a cycle decomposition, up to the order of factors."}785{"_id": "885", "title": "Order of Product of Disjoint Permutations", "text": "Let $S_n$ denote the symmetric group on $n$ letters. Let $\\pi$ be a product of disjoint permutations of orders $k_1, k_2, \\ldots, k_r$. Then: :$\\order \\pi = \\lcm \\set {k_1, k_2, \\ldots, k_r}$ where: :$\\order \\pi$ denotes the order of $\\pi$ in $S_n$ :$\\lcm$ denotes lowest common multiple."}786{"_id": "886", "title": "Group Action defines Permutation Representation", "text": "Let $\\map \\Gamma X$ be the set of permutations on a set $X$. Let $G$ be a group. Let $\\phi: G \\times X \\to X$ be a group action. For $g \\in G$, let $\\phi_g: X \\to X$ be the mapping defined as: :$\\map {\\phi_g} x = \\map \\phi {g, x}$ Let $\\tilde \\phi: G \\to \\map \\Gamma X$ be the mapping associated to $\\phi$, defined by: :$\\map {\\tilde \\phi} g := \\phi_g$ Then $\\tilde \\phi$ is a group homomorphism."}787{"_id": "887", "title": "Group Action determines Bijection", "text": "Let $*$ be a group action of $G$ on $X$. Then each $g \\in G$ determines a bijection $\\phi_g: X \\to X$ given by: :$\\map {\\phi_g} x = g * x$ Its inverse is: :$\\phi_{g^{-1} }: X \\to X$. These bijection are sometimes called '''transformations''' of $X$."}788{"_id": "888", "title": "Group Action Induces Equivalence Relation", "text": "Let $G$ be a group whose identity is $e$. Let $X$ be a set. Let $*: G \\times S \\to S$ be a group action. Let $\\mathcal R_G$ be the relation induced by $G$, that is: :$x \\mathrel {\\mathcal R_G} y \\iff y \\in \\Orb x$ where: :$\\Orb x$ denotes the orbit of $x \\in X$. Then: :$\\mathcal R_G$ is an equivalence relation. :The equivalence class of an element is its orbit."}789{"_id": "889", "title": "Partition Equation", "text": "Let group $G$ act on a finite set $X$. Let the distinct orbits of $X$ under the action of $G$ be: :$\\Orb {x_1}, \\Orb {x_2}, \\ldots, \\Orb {x_s}$ Then: :$\\card X = \\card {\\Orb {x_1} } + \\card {\\Orb {x_2} } + \\cdots + \\card {\\Orb {x_s} }$"}790{"_id": "890", "title": "Definition:Stabilizer", "text": "Let $G$ be a group. Let $X$ be a set. Let $*: G \\times X \\to X$ be a group action. For each $x \\in X$, the '''stabilizer of $x$ by $G$''' is defined as: :$\\Stab x := \\set {g \\in G: g * x = x}$ where $*$ denotes the group action."}791{"_id": "891", "title": "Stabilizer is Subgroup", "text": "Let $\\struct {G, \\circ}$ be a group which acts on a set $X$. Let $\\Stab x$ be the stabilizer of $x$ by $G$. Then for each $x \\in X$, $\\Stab x$ is a subgroup of $G$."}792{"_id": "892", "title": "Orbit-Stabilizer Theorem", "text": "Let $G$ be a group which acts on a finite set $X$. Let $x \\in X$. Let $\\Orb x$ denote  the orbit of $x$. Let $\\Stab x$ denote the stabilizer of $x$ by $G$. Let $\\index G {\\Stab x}$ denote the index of $\\Stab x$ in $G$. Then: :$\\order {\\Orb x} = \\index G {\\Stab x} = \\dfrac {\\order G} {\\order {\\Stab x} }$"}793{"_id": "894", "title": "Action of Group on Coset Space is Group Action", "text": "Let $G$ be a group whose identity is $e$. Let $H$ be a subgroup of $G$. Let $*: G \\times G / H \\to G / H$ be the action on the (left) coset space: :$\\forall g \\in G, \\forall g' H \\in G / H: g * \\paren {g' H} := \\paren {g g'} H$ Then $G$ is a group action."}794{"_id": "895", "title": "Group Action on Sets with k Elements", "text": "Let $\\struct {G, \\circ}$ be a finite group whose identity is $e$. Let $\\Bbb S = \\set {S \\subseteq G: \\card S = k}$, that is, the set of all of subsets of $G$ which have exactly $k$ elements. Let $G$ act on $\\Bbb S$ by the rule: :$\\forall S \\in \\Bbb S: g * S = g S = \\set {x \\in G: x = g s: s \\in S}$ This is a group action, and: : $\\forall S \\in \\Bbb S: \\order {\\Stab S} \\divides \\card S$ where $\\Stab S$ denotes the stabilizer of $S$ by $G$."}795{"_id": "898", "title": "Quotient of Transformation Group acts Effectively", "text": "Let $G$ be a transformation group (which may or may not be effective) acting on $X$. Then the quotient group $G / G_0$, where $G_0$ is the kernel, ''does'' act effectively on $X$."}796{"_id": "899", "title": "Condition for Group to Act Effectively on Left Coset Space", "text": "Let $G$ be a group whose identity is $e$. Let $H$ be a subgroup of $G$. Then $G$ acts effectively on the left coset space $G / H$ {{iff}}: :$\\ds \\bigcap_{a \\mathop \\in G} H^a = \\set e$ where $H^a$ denotes the conjugate of $H$ by $a$."}797{"_id": "900", "title": "Set of Permutations is Largest Effective Transformation Group", "text": "The set of permutations of a set $X$ forms the largest effective transformation group of $X$."}798{"_id": "901", "title": "Conjugacy Action is Group Action", "text": "Let $\\left({G, \\circ}\\right)$ be a group whose identity is $e$."}799{"_id": "902", "title": "Conjugacy Action on Identity", "text": "Let $G$ be a group whose identity is $e$. For the conjugacy action: :$\\order {\\Orb e} = 1$ and thus: :$\\Stab e = G$"}800{"_id": "903", "title": "Cauchy's Lemma (Group Theory)", "text": "Let $\\struct {G, \\circ}$ be a group of finite order whose identity is $e$. Let $p$ be a prime number which divides the order of $G$. Then $\\struct {G, \\circ}$ has an element of order $p$."}801{"_id": "904", "title": "Transposition is Self-Inverse", "text": "All transpositions are self-inverse."}802{"_id": "905", "title": "Conjugates of Transpositions", "text": "Let $k_1, k_2, k_3 \\in \\left\\{{1, 2, \\ldots, n}\\right\\}$. Then: : $(1): \\quad \\begin{bmatrix} k_1 & k_2 \\end{bmatrix} = \\begin{bmatrix} k_3 & k_2 \\end{bmatrix} \\begin{bmatrix} k_1 & k_3 \\end{bmatrix} \\begin{bmatrix} k_3 & k_2 \\end{bmatrix}$ : $(2): \\quad \\begin{bmatrix} k_1 & k_2 \\end{bmatrix} = \\begin{bmatrix} k_3 & k_1 \\end{bmatrix} \\begin{bmatrix} k_3 & k_2 \\end{bmatrix} \\begin{bmatrix} k_3 & k_1 \\end{bmatrix}$"}803{"_id": "906", "title": "K-Cycle can be Factored into Transpositions", "text": "Every $k$-cycle can be factorised into the product of $k - 1$ transpositions. This factorisation is not unique."}804{"_id": "907", "title": "Sign of Permutation is Plus or Minus Unity", "text": "Let $n \\in \\N$ be a natural number. Let $\\N_n$ denote the set of natural numbers $\\set {1, 2, \\ldots, n}$. Let $S_n$ denote the symmetric group on $n$ letters. Let $\\sequence {x_k}_{k \\mathop \\in \\N_n}$ be a finite sequence in $\\R$. Let $\\pi \\in S_n$. Let $\\map {\\Delta_n} {x_1, x_2, \\ldots, x_n}$ be the product of differences of $\\tuple {x_1, x_2, \\ldots, x_n}$. Let $\\map \\sgn \\pi$ be the sign of $\\pi$. Let $\\pi \\cdot \\map {\\Delta_n} {x_1, x_2, \\ldots, x_n}$ be defined as: :$\\pi \\cdot \\map {\\Delta_n} {x_1, x_2, \\ldots, x_n} := \\map {\\Delta_n} {x_{\\map \\pi 1}, x_{\\map \\pi 2}, \\ldots, x_{\\map \\pi n} }$ Then either: :$\\pi \\cdot \\Delta_n = \\Delta_n$ or: :$\\pi \\cdot \\Delta_n = -\\Delta_n$ That is: :$\\map \\sgn \\pi = \\begin{cases} 1 & :\\pi \\cdot \\Delta_n = \\Delta_n \\\\ -1 & : \\pi \\cdot \\Delta_n = -\\Delta_n \\end{cases}$ Thus: :$\\pi \\cdot \\Delta_n = \\map \\sgn \\pi \\Delta_n$"}805{"_id": "908", "title": "Parity Function is Homomorphism", "text": "Let $S_n$ denote the symmetric group on $n$ letters. Let $\\pi \\in S_n$. Let $\\map \\sgn \\pi$ be the sign of $\\pi$. Let the parity function of $\\pi$ be defined as: :Parity of $\\pi = \\begin{cases} \\mathrm {Even} & : \\map \\sgn \\pi = 1 \\\\ \\mathrm {Odd} & : \\map \\sgn \\pi = -1 \\end{cases}$ The mapping $\\sgn: S_n \\to C_2$, where $C_2$ is the cyclic group of order 2, is a homomorphism."}806{"_id": "909", "title": "Permutation on Polynomial is Group Action", "text": "Let $n \\in \\Z: n > 0$. Let $F_n$ be the set of all polynomials in $n$ variables $x_1, x_2, \\ldots, x_n$: :$F = \\set {\\map f {x_1, x_2, \\ldots, x_n}: f \\text{ is a polynomial in $n$ variables} }$ Let $S_n$ denote the symmetric group on $n$ letters. Let $*: S_n \\times F \\to F$ be the mapping defined as: :$\\forall \\pi \\in S_n, f \\in F: \\pi * \\map f {x_1, x_2, \\ldots, x_n} = \\map f {x_{\\map \\pi 1}, x_{\\map \\pi 2}, \\ldots, x_{\\map \\pi n} }$ Then $*$ is a group action."}807{"_id": "910", "title": "Parity of Inverse of Permutation", "text": "Let $S_n$ denote the symmetric group on $n$ letters. Then: :$\\forall \\pi \\in S_n: \\map \\sgn \\pi = \\map \\sgn {\\pi^{-1} }$"}808{"_id": "911", "title": "Parity of Conjugate of Permutation", "text": "Let $S_n$ denote the symmetric group on $n$ letters. Then: :$\\forall \\pi, \\rho \\in S_n: \\map \\sgn {\\pi \\rho \\pi^{-1} } = \\map \\sgn \\rho$ where $\\map \\sgn \\pi$ is the sign of $\\pi$."}809{"_id": "912", "title": "Parity of K-Cycle", "text": "Let $\\pi$ be a $k$-cycle. Then: :$\\map \\sgn \\pi = \\begin{cases} 1 & : k \\ \\text {odd} \\\\ -1 & : k \\ \\text {even} \\end{cases}$ Thus: :$\\map \\sgn \\pi = \\paren {-1}^{k - 1}$ or equivalently: :$\\map \\sgn \\pi = \\paren {-1}^{k + 1}$"}810{"_id": "913", "title": "Alternating Group is Normal Subgroup of Symmetric Group", "text": "Let $n \\ge 2$ be a natural number. Let $S_n$ denote the symmetric group on $n$ letters. Let $A_n$ be the alternating group on $n$ letters. Then $A_n$ is a normal subgroup of $S_n$ whose index is $2$."}811{"_id": "914", "title": "Group of Permutations either All or Half Even", "text": "Let $G$ be a group of permutations. Then either ''exactly half'' of the permutations in $G$ are even, or they are ''all'' even."}812{"_id": "915", "title": "Cycle Decomposition of Conjugate", "text": "Let $S_n$ denote the symmetric group on $n$ letters. Let $\\pi, \\rho \\in S_n$. The cycle decomposition of the permutation $\\pi \\rho \\pi^{-1}$ can be obtained from that of $\\rho$ by replacing each $i$ in the cycle decomposition of $\\rho$ with $\\map \\pi i$."}813{"_id": "916", "title": "Conjugate Permutations have Same Cycle Type", "text": "Let $n \\ge 1$ be a natural number. Let $G$ be a subgroup of the symmetric group on $n$ letters $S_n$. Let $\\sigma, \\rho \\in G$. Then $\\sigma$ and $\\rho$ are conjugate {{iff}} they have the same cycle type."}814{"_id": "917", "title": "Transpositions of Adjacent Elements generate Symmetric Group", "text": "Let $n \\in \\Z: n > 1$. Let $S_n$ denote the symmetric group on $n$ letters. Then the transpositions $a_k = \\begin{pmatrix} k & k + 1 \\end{pmatrix}$ for $1 \\le k < n$ are a set of generators for $S_n$. They satisfy the relations: {{begin-eqn}} {{eqn | l = a_k^2       | r = e       | c = (for $1 \\le k < n$) }} {{eqn | l = \\paren {a_k a_{k + 1} }^3       | r = e       | c = (for $1 \\le k < n - 1$) }} {{eqn | l = \\paren {a_i a_j}^2       | r = e       | c = (for $1 \\le i, j < n, \\size {i - j} > 1$) }} {{end-eqn}}"}815{"_id": "919", "title": "Permutation of Cosets", "text": "Let $G$ be a group and let $H \\le G$. Let $\\mathbb S$ be the set of all distinct left cosets of $H$ in $G$. Then: :$(1): \\quad$ For any $g \\in G$, the mapping $\\theta_g: \\mathbb S \\to \\mathbb S$ defined by $\\map {\\theta_g} {x H} = g x H$ is a permutation of $\\mathbb S$. :$(2): \\quad$ The mapping $\\theta$ defined by $\\map \\theta g = \\theta_g$ is a homomorphism from $G$ into the symmetric group on $\\mathbb S$. :$(3): \\quad$ The kernel of $\\theta$ is the subgroup $\\displaystyle \\bigcap_{x \\mathop \\in G} x H x^{-1}$."}816{"_id": "921", "title": "Basic Results about Modules", "text": "Let $\\struct {G, +_G}$ be an abelian group whose identity is $e$. Let $\\struct {R, +_R, \\times_R}$ be a ring whose zero is $0_R$. Let $\\struct {G, +_G, \\circ}_R$ be an $R$-module. Let $x \\in G, \\lambda \\in R, n \\in \\Z$. Let $\\sequence {x_m}$ be a sequence of elements of $G$. Let $\\sequence {\\lambda_m}$ be a sequence of elements of $R$ that is, scalars. Then:"}817{"_id": "922", "title": "Basic Results about Unitary Modules", "text": "Let $\\struct {G, +_G}$ be an abelian group whose identity is $e$. Let $\\struct {R, +_R, \\times_R}$ be a ring with unity whose zero is $0_R$ and whose unity is $1_R$. Let $\\struct {G, +_G, \\circ}_R$ be an unitary $R$-module. Let $x \\in G, n \\in \\Z$. Then:"}818{"_id": "923", "title": "Epimorphism preserves Modules", "text": "Let $\\left({G, +_G, \\circ}\\right)_R$ be an $R$-module. Let $\\left({H, +_H, \\circ}\\right)_R$ be an $R$-algebraic structure. Let $\\phi: G \\to H$ be an epimorphism. Then $H$ is an $R$-module."}819{"_id": "924", "title": "Condition for Linear Transformation", "text": "Let $G$ be a unitary $R$-module, and let $H$ be an $R$-module. Let $\\phi: G \\to H$ be a mapping. Then $\\phi$ is a linear transformation {{iff}}: :$\\forall x, y \\in G: \\forall \\lambda, \\mu \\in R: \\map \\phi {\\lambda x + \\mu y} = \\lambda \\map \\phi x + \\mu \\map \\phi y$"}820{"_id": "925", "title": "Module of All Mappings is Module", "text": "Let $\\struct {R, +_R, \\times_R}$ be a ring. Let $\\struct {G, +_G, \\circ}_R$ be an $R$-module. Let $S$ be a set. Let $\\struct {G^S, +_G', \\circ}_R$ be the module of all mappings from $S$ to $G$. Then $\\struct {G^S, +_G', \\circ}_R$ is an $R$-module."}821{"_id": "926", "title": "Kuratowski's Theorem", "text": "The following conditions on a graph $\\Gamma$ are equivalent: :$(1): \\quad \\Gamma$ is planar :$(2): \\quad \\Gamma$ contains no subdivision of either the complete graph $K_5$ or the complete bipartite graph $K_{3, 3}$."}822{"_id": "927", "title": "Fundamental Theorem of Finite Abelian Groups", "text": "Every finite abelian group is an internal group direct product of cyclic groups whose orders are prime powers. The number of terms in the product and the orders of the cyclic groups are uniquely determined by the group."}823{"_id": "928", "title": "Finite Direct Product of Modules is Module", "text": "Let $\\struct {R, +_R, \\times_R}$ be a ring. Let $\\struct {G_1, +_1, \\circ_1}_R, \\struct {G_2, +_2, \\circ_2}_R, \\ldots, \\struct {G_n, +_n, \\circ_n}_R$ be $R$-modules. Let: :$\\ds G = \\prod_{k \\mathop = 1}^n G_k$ be their direct product. Then $G$ is a module."}824{"_id": "929", "title": "Subring Module", "text": "Let $\\struct {R, +, \\times}$ be a ring. Let $\\struct {S, +_S, \\times_S}$ be a subring of $R$. Let $\\struct {G, +_G, \\circ}_R$ be an $R$-module. Let $\\circ_S$ be the restriction of $\\circ$ to $S \\times G$. Then $\\struct {G, +_G, \\circ_S}_S$ is an $S$-module. The module $\\struct {G, +_G, \\circ_S}_S$ is called the '''$S$-module obtained from $\\struct {G, +_G, \\circ}_R$ by restricting scalar multiplication'''. {{refactor|Extract the below into its own page}} If $\\struct {G, +_G, \\circ}_R$ is a unitary $R$-module and $1_R \\in S$, then $\\struct{G, +_G, \\circ_S}_S$ is also unitary."}825{"_id": "931", "title": "Projection on Cartesian Product of Modules", "text": "Let $G$ be the cartesian product of a sequence $\\sequence {G_n}$ of $R$-modules. Then for each $j \\in \\closedint 1 n$, the projection $\\pr_j$ on the $j$th co-ordinate is an epimorphism from $G$ onto $G_j$."}826{"_id": "933", "title": "Submodule Test", "text": "Let $\\left({G, +, \\circ}\\right)_R$ be a unitary $R$-module. Let $H$ be a non-empty subset of $G$. Then $\\left({H, +, \\circ}\\right)_R$ is a submodule of $G$ iff: :$\\forall x, y \\in H: \\forall \\lambda \\in R: x + y \\in H, \\lambda \\circ x \\in H$"}827{"_id": "934", "title": "Module is Submodule of Itself", "text": "Let $\\left({G, +_G, \\circ}\\right)_R$ be an $R$-module. Then $\\left({G, +_G, \\circ}\\right)_R$ is a submodule of itself."}828{"_id": "935", "title": "Null Module Submodule of All", "text": "Let $\\left({G, +_G, \\circ}\\right)_R$ be an $R$-module. Then the null module: :$\\left({\\left\\{{e_G}\\right\\}, +_G, \\circ}\\right)_R$ is a submodule of $\\left({G, +_G, \\circ}\\right)_R$."}829{"_id": "937", "title": "First Sylow Theorem", "text": "Let $p$ be a prime number. Let $G$ be a group such that: :$\\order G = k p^n$ where: :$\\order G$ denotes the order of $G$ :$p$ is not a divisor of $k$. Then $G$ has at least one Sylow $p$-subgroup."}830{"_id": "938", "title": "H-Cobordism Theorem", "text": "Let $X^n, Y^n$ be two simply connected manifolds. Let $n \\in \\N: n \\ge 5$ and $\\exists W$ such that $W$ is an h-cobordism between $X$ and $Y$. Then $\\exists \\psi: W \\to X \\times \\closedint 0 1$ such that $\\psi$ is a diffeomorphism. In particular, $X$ and $Y$ are diffeomorphic."}831{"_id": "939", "title": "Group has Subgroups of All Prime Power Factors", "text": "Let $p$ be a prime. Let $G$ be a finite group of order $n$. If $p^k \\divides n$ then $G$ has at least one subgroup of order $p^k$."}832{"_id": "940", "title": "Normalizer of Sylow p-Subgroup", "text": "Let $P$ be a Sylow $p$-subgroup of a finite group $G$. Let $\\map {N_G} P$ be the normalizer of $P$. Then any $p$-subgroup of $\\map {N_G} P$ is contained in $P$. In particular, $P$ is the unique Sylow $p$-subgroup of $\\map {N_G} P$."}833{"_id": "941", "title": "Second Sylow Theorem", "text": "Let $P$ be a Sylow $p$-subgroup of the finite group $G$. Let $Q$ be any $p$-subgroup of $G$. Then $Q$ is a subset of a conjugate of $P$."}834{"_id": "942", "title": "Third Sylow Theorem", "text": "All the Sylow $p$-subgroups of a finite group are conjugate."}835{"_id": "943", "title": "Fourth Sylow Theorem", "text": "The number of Sylow $p$-subgroups of a finite group is congruent to $1 \\pmod p$."}836{"_id": "944", "title": "Sylow p-Subgroup is Unique iff Normal", "text": "A group $G$ has exactly one Sylow $p$-subgroup $P$ {{iff}} $P$ is normal."}837{"_id": "945", "title": "Intersection of Normal Subgroup with Sylow P-Subgroup", "text": "Let $P$ be a Sylow $p$-subgroup of a finite group $G$. Let $N$ be a normal subgroup of $G$. Then $P \\cap N$ is a Sylow $p$-subgroup of $N$."}838{"_id": "946", "title": "Quotient of Sylow P-Subgroup", "text": "Let $P$ be a Sylow $p$-subgroup of a finite group $G$. Let $N$ be a normal subgroup of $G$. Then $P N / N$ is a Sylow $p$-subgroup of $G / N$."}839{"_id": "947", "title": "Fifth Sylow Theorem", "text": "The number of Sylow $p$-subgroups of a finite group is a divisor of their common index."}840{"_id": "948", "title": "Finite Submodule of Function Space", "text": "Let $\\left({G, +}\\right)$ be a group whose identity is $e$. Let $R$ be a ring. Let $\\left({G, +, \\circ}\\right)_R$ be an $R$-module. Let $S$ be a set. Let $G^S$ the set of all mappings $f: S \\to G$. Let $G^{\\left({S}\\right)}$ be the set of all mappings $f: S \\to G$ such that $f \\left({x}\\right) = e$ for all but finitely many elements $x$ of $S$. Then: : $\\left({G^{\\left({S}\\right)}, +', \\circ}\\right)_R$ is a submodule of $\\left({G^S, +, \\circ}\\right)_R$ where $+'$ is the operation induced on $G^{\\left({S}\\right)}$ by $+$."}841{"_id": "949", "title": "Polynomial Functions form Submodule of All Functions", "text": "Let $K$ be a commutative ring with unity. Let $K^K$ be the $K$-module mappings $f: K \\to K$. Let $P \\left({K}\\right) \\subseteq K^K$ be the set of all polynomial functions on $K$. Then $P \\left({K}\\right)$ is a $K$-submodule of $K^K$."}842{"_id": "950", "title": "Intersection and Sum of Submodules", "text": "Let $\\left({G, +, \\circ}\\right)_R$ be an $R$-module. Let $H$ and $K$ be submodules of $G$. Then $H + K$ and $H \\cap K$ are also submodules of $G$. The intersection of any set of submodules of $G$ is a submodule. Thus if $S \\subseteq G$, the intersection of all submodules of $G$ containing $S$ is the smallest submodule of $G$ containing $S$."}843{"_id": "951", "title": "Linear Transformation of Submodule", "text": "Let $G$ and $H$ be $R$-modules. Let $\\phi: G \\to H$ be a linear transformation. Then: :$(1): \\quad$ If $M$ is a submodule of $G$, $\\phi \\sqbrk M$ is a submodule of $H$ :$(2): \\quad$ If $N$ is a submodule of $H$, $\\phi^{-1} \\sqbrk N$ is a submodule of $G$ :$(3): \\quad$ The codomain of $\\phi$ is a submodule of $H$ :$(4): \\quad$ The kernel of $\\phi$ is a submodule of $G$."}844{"_id": "953", "title": "Poincaré Conjecture", "text": "Let $\\Sigma^m$ be a smooth $m$-manifold. Let $\\Sigma^m$ satisfy: :$H_0 \\struct {\\Sigma; \\Z} = 0$ and: :$H_m \\struct {\\Sigma; \\Z} = \\Z$ {{explain|Definition of the notation $H_0 \\struct {\\Sigma; \\Z}$, nature of $H_0$ and $H_m$}} Then $\\Sigma^m$ is homeomorphic to the $m$-sphere $\\Bbb S^m$."}845{"_id": "954", "title": "Linear Transformation of Generated Module", "text": "Let $G$ and $H$ be $R$-modules. Let $\\phi$ and $\\psi$ be linear transformations $G$ into $H$.  Let $S$ be a generator for $G$. Suppose that $\\forall x \\in S: \\map \\phi x = \\map \\psi x$. Then $\\phi = \\psi$."}846{"_id": "955", "title": "Generated Submodule is Linear Combinations", "text": "Let $G$ be a unitary $R$-module. Let $S \\subseteq G$. Then the submodule $H$ generated by $S$ is the set of all linear combinations of $S$."}847{"_id": "956", "title": "Empty Set is Linearly Independent", "text": "The empty set is a linearly independent set."}848{"_id": "957", "title": "Subset of Module Containing Identity is Linearly Dependent", "text": "Let $G$ be a group whose identity is $e$. Let $\\struct {R, +, \\circ}$ be a ring whose zero is $0_R$. Let $\\struct {G, +_G, \\circ}_R$ be an $R$-module. Let $H \\subseteq G$ such that $e \\in H$. Then $H$ is a linearly dependent set."}849{"_id": "958", "title": "Subset of Linearly Independent Set is Linearly Independent", "text": "A subset of a linearly independent set is also linearly independent."}850{"_id": "959", "title": "Superset of Linearly Dependent Set", "text": "Any set containing a linearly dependent set is also linearly dependent."}851{"_id": "960", "title": "Number of Ordered Bases from a Basis", "text": "Each basis of $n$ elements determines $n!$ ordered bases."}852{"_id": "961", "title": "Classification of Compact Two-Manifolds", "text": "Any smooth, compact, path-connected manifold of dimension $2$ is diffeomorphic to the sphere $\\mathbb S^2$, a connected sum of tori $\\mathbb T^2$, or a connected sum of projective spaces $\\mathbb{RP}^2$. Any such $2$-manifold with boundary is diffeomorphic to the sphere $\\mathbb S^2$, a connected sum of tori $\\mathbb T^2$, or a connected sum of projective spaces $\\mathbb{RP}^2$, with a number of open disks removed. The Euler characteristic, orientability, and number of boundary curves suffice to describe a surface."}853{"_id": "962", "title": "Standard Ordered Basis is Basis", "text": "Let $\\struct {R, +, \\circ}$ be a ring with unity whose zero is $0_R$ and whose unity is $1_R$. Let $n$ be a positive integer. For each $j \\in \\closedint 1 n$, let $e_j$ be the ordered $n$-tuple of elements of $R$ whose $j$th entry is $1_R$ and all of whose other entries is $0_R$. Then $\\sequence {e_n}$ is an ordered basis of the $R$-module $R^n$. This ordered basis is called the '''standard ordered basis of $R^n$'''. The corresponding set $\\set {e_1, e_2, \\ldots, e_n}$ is called the '''standard basis of $R^n$'''."}854{"_id": "964", "title": "Unique Representation by Ordered Basis", "text": "Let $G$ be a unitary $R$-module. Then $\\sequence {a_n}$ is an ordered basis of $G$ {{Iff}}: :For every $x \\in G$ there exists one and only one sequence $\\sequence {\\lambda_n}$ of scalars such that $\\displaystyle x = \\sum_{k \\mathop = 1}^n \\lambda_k a_k$."}855{"_id": "965", "title": "Isomorphism from R^n via n-Term Sequence", "text": "Let $G$ be a unitary $R$-module. Let $\\sequence {a_k}_{1 \\mathop \\le k \\mathop \\le n}$ be an ordered basis of $G$. Let $R^n$ be the $R$-module $R^n$. Let $\\psi: R^n \\to G$ be defined as: :$\\displaystyle \\map \\psi {\\sequence {\\lambda_k}_{1 \\mathop \\le k \\mathop \\le n} } = \\sum_{k \\mathop = 1}^n \\lambda_k a_k$ Then $\\psi$ is an isomorphism."}856{"_id": "966", "title": "Unitary R-Modules with n-Element Bases Isomorphic", "text": "Any two unitary $R$-modules having bases of $n$ elements are isomorphic."}857{"_id": "967", "title": "R-Module R^n is n-Dimensional", "text": "The $R$-module $R^n$ is $n$-dimensional."}858{"_id": "968", "title": "Unique Linear Transformation Between Modules", "text": "Let $G$ and $H$ be unitary $R$-modules. Let $\\left \\langle {a_n} \\right \\rangle$ be an ordered basis of $G$. Let $\\left \\langle {b_n} \\right \\rangle$ be a sequence of elements of $H$. Then there is a unique linear transformation $\\phi: G \\to H$ satisfying $\\forall k \\in \\left[{1 \\,.\\,.\\, n}\\right]: \\phi \\left({a_k}\\right) = b_k$"}859{"_id": "969", "title": "Linear Transformation from Ordered Basis less Kernel", "text": "Let $G$ and $H$ be unitary $R$-modules. Let $\\phi: G \\to H$ be a non-zero linear transformation. Let $G$ be $n$-dimensional. Let $\\left \\langle {a_n} \\right \\rangle$ be any ordered basis of $G$ such that $\\left\\{{a_k: r + 1 \\le k \\le n}\\right\\}$ is the basis of the kernel of $\\phi$. Then $\\left \\langle {\\phi \\left({a_r}\\right)} \\right \\rangle$ is an ordered basis of the image of $\\phi$."}860{"_id": "970", "title": "Addition of Linear Transformations", "text": "Let $\\left({G, +_G, \\circ}\\right)_R$ and $\\left({H, +_H, \\circ}\\right)_R$ be $R$-modules. Let $\\phi: G \\to H$ and $\\psi: G \\to H$ be linear transformations. Let $\\phi +_H \\psi$ be the operation on $H^G$ induced by $+_H$ as defined in Induced Structure. Then $\\phi +_H \\psi: G \\to H$ is a linear transformation."}861{"_id": "971", "title": "Negative Linear Transformation", "text": "Let $\\left({G, +_G, \\circ}\\right)_R$ and $\\left({H, +_H, \\circ}\\right)_R$ be $R$-modules. Let $\\phi: G \\to H$ be a linear transformation. Let $- \\phi$ be the negative of $\\phi$ as defined in Induced Structure Inverse. Then $- \\phi: G \\to H$ is also a linear transformation."}862{"_id": "973", "title": "Morse-Sard Theorem", "text": "Let $f: X \\to Y$ be any smooth map of manifolds. Then almost every point in $Y$ is a regular value of $f$."}863{"_id": "974", "title": "Linear Transformation from Center of Scalar Ring", "text": "Let $\\struct {G, +_G, \\circ}_R$ and $\\struct {H, +_H, \\circ}_R$ be $R$-modules. Let $\\phi: G \\to H$ be a linear transformation. Let $\\map Z R$ be the center of the scalar ring $R$. Let $\\lambda \\in \\map Z R$. Then $\\lambda \\circ \\phi$ is a linear transformation."}864{"_id": "976", "title": "Product of Linear Transformations", "text": "Let $R$ be a commutative ring with unity whose zero is $0_R$ and whose unity is $1_R$. Let $\\struct {G, +_G, \\circ}_R$ be a unitary $R$-module such that $\\map \\dim G = n$. Let $\\struct {H, +_H, \\circ}_R$ be a unitary $R$-module such that $\\map \\dim H = m$. Let $\\map {\\LL_R} {G, H}$ be the set of all linear transformations from $G$ to $H$. Then: :$\\map \\dim {\\map {\\LL_R} {G, H} } = n m$ Let $\\sequence {a_n}$ be an ordered basis for $G$. Let $\\sequence {b_m}$ be an ordered basis for $H$. Let $\\phi_{i j}: G \\to H$ be the unique linear transformation defined for each $i \\in \\closedint 1 n, j \\in \\closedint 1 m$ which satisfies: :$\\forall k \\in \\closedint 1 n: \\map {\\phi_{i j} } {a_k} = \\delta_{i k} b_j$ where $\\delta$ is the Kronecker delta. Then: :$\\set {\\phi_{i j}: i \\in \\closedint 1 n, j \\in \\closedint 1 m}$ is a basis for $\\map \\dim {\\map {\\LL_R} {G, H} }$."}865{"_id": "978", "title": "Dimension of Algebraic Dual", "text": "Let $G$ be an $n$-dimensional $R$-module. Let $G^*$ be the algebraic dual of $G$. Let $G^{**}$ be the algebraic dual of $G^*$. Then $G^*$ and $G^{**}$ are also $n$-dimensional."}866{"_id": "979", "title": "Annihilator is Submodule of Algebraic Dual", "text": "Let $R$ be a commutative ring. Let $G$ be a module over $R$. Let $M$ be a submodule of $G$. Let $G^*$ be the algebraic dual of $G$. Then the annihilator $M^\\circ$ of $M$ is a submodule of $G^*$. Similarly, let $N$ be a submodule of $G^*$. Let $G^{**}$ be the algebraic dual of $G^*$. Then the annihilator $N^\\circ$ of $N$ is a submodule of $G^{**}$."}867{"_id": "981", "title": "Properties of Evaluation Linear Transformation", "text": "Let $R$ be a commutative ring. Let $G$ be an $R$-module. Let $G^*$ be the algebraic dual of $G$. Let $\\left \\langle {x, t'} \\right \\rangle$ be the evaluation linear transformation from $G$ to $G^{**}$. Then the mapping $\\phi: G \\times G^* \\to R$ defined as $\\forall \\left({x, t'}\\right) \\in G \\times G^*: \\phi \\left({x, t'}\\right) = \\left \\langle {x, t'} \\right \\rangle$ satisfies the following properties: : $(1): \\quad \\forall x, y \\in G: \\forall t' \\in G^*: \\left \\langle {x + y, t'} \\right \\rangle = \\left \\langle {x, t'} \\right \\rangle + \\left \\langle {y, t'} \\right \\rangle$ : $(2): \\quad \\forall x \\in G: \\forall s', t' \\in G^*: \\left \\langle {x, s' + t'} \\right \\rangle = \\left \\langle {x, s'} \\right \\rangle + \\left \\langle {x, t'} \\right \\rangle$ : $(3): \\quad \\forall x \\in G: \\forall s', t' \\in G^*: \\forall \\lambda \\in R: \\left \\langle {\\lambda x, t'} \\right \\rangle = \\lambda \\left \\langle {x, t'} \\right \\rangle = \\left \\langle {x, \\lambda t'} \\right \\rangle$"}868{"_id": "982", "title": "Zero Vector Space Product iff Factor is Zero", "text": "Let $F$ be a field whose zero is $0_F$ and whose unity is $1_F$. Let $\\struct {\\mathbf V, +, \\circ}_F$ be a vector space over $F$, as defined by the vector space axioms. Let $\\mathbf v \\in \\mathbf V, \\lambda \\in F$. Then: :$\\lambda \\circ \\mathbf v = \\bszero \\iff \\paren {\\lambda = 0_F \\lor x = \\bszero}$"}869{"_id": "983", "title": "Homomorphic Image of Vector Space", "text": "Let $\\struct {K, +_K, \\times_K}$ be a division ring. Let $\\struct {V, +_V, \\circ_V}_K$ be a $K$-vector space. Let $\\struct {W, +_W, \\circ_W}_K$ be a $K$-algebraic structure. Let $\\phi: V \\to W$ be a homomorphism, i.e. a linear transformation. Then the homomorphic image of $\\phi$ is a $K$-vector space."}870{"_id": "984", "title": "Direct Product of Vector Spaces is Vector Space", "text": "Let $K$ be a field. Let $V_1, V_2, \\ldots, V_n$ be $K$-vector spaces. Let $\\struct {V, + , \\circ}_K$ be their direct product. Then $\\struct {V, + , \\circ}_K$ is a $K$-vector space."}871{"_id": "985", "title": "Definition:Vector Space on Cartesian Product", "text": "Let $\\struct {K, +, \\circ}$ be a division ring. Let $n \\in \\N_{>0}$. Let $+: K^n \\times K^n \\to K^n$ be defined as: :$\\tuple {\\alpha_1, \\ldots, \\alpha_n} + \\tuple {\\beta_1, \\ldots, \\beta_n} = \\tuple {\\alpha_1 +_R \\beta_1, \\ldots, \\alpha_n +_R \\beta_n}$ Let $\\times: K \\times K^n \\to K^n$ be defined as: :$\\lambda \\times \\tuple {\\alpha_1, \\ldots, \\alpha_n} = \\tuple {\\lambda \\times_R \\alpha_1, \\ldots, \\lambda \\times_R \\alpha_n}$ Then $\\struct {K^n, +, \\times}_K$ is '''the $K$-vector space $K^n$'''."}872{"_id": "986", "title": "Definition:Vector Space over Division Subring", "text": "Let $\\struct {R, +, \\circ}$ be a ring with unity whose unity is $1_R$. Let $S$ be a division subring of $R$, such that $1_R \\in S$. Then $\\struct {R, +, \\circ_S}_S$, where $\\circ_S$ is the restriction of $\\circ$ to $S \\times R$, is the '''vector space on $R$ over the division subring $S$'''."}873{"_id": "988", "title": "Division Ring is Vector Space over Prime Subfield", "text": "Let $\\struct {K, +, \\times}$ be a division ring. Let $\\struct {S, +, \\times}$ be the prime subfield of $K$ Then $\\struct {K, +, \\times_S}_S$ is an $S$-vector space, where $\\times_S$ is the restriction of $\\times$ to $S \\times K$."}874{"_id": "989", "title": "Vector Space over Subring", "text": "Let $K$ be a division subring of the division ring $\\struct {L, +_L, \\times_L}$. Let $\\struct {G, +_G, \\circ}_L$ be a $L$-vector space. Then $\\struct {G, +_G, \\circ_K}_K$ is a $K$-vector space, where $\\circ_K$ is the restriction of $\\circ$ to $K \\times G$. The $K$-vector space $\\struct {G, +_G, \\circ_K}_K$ is called the '''$K$-vector space obtained from $\\struct {L, +_L, \\times_L}$ by restricting scalar multiplication'''."}875{"_id": "990", "title": "Subspaces of Dimension 2 Real Vector Space", "text": "Take the $\\R$-vector space $\\left({\\R^2, +, \\times}\\right)_\\R$. Let $S$ be a subspace of $\\left({\\R^2, +, \\times}\\right)_\\R$. Then $S$ is one of: : $(1): \\quad \\left({\\R^2, +, \\times}\\right)_\\R$ : $(2): \\quad \\left\\{{0}\\right\\}$ : $(3): \\quad$ A line through the origin."}876{"_id": "991", "title": "Subspace of Real Continuous Functions", "text": "Let $\\mathbb J = \\set {x \\in \\R: a \\le x \\le b}$ be a closed interval of the real number line $\\R$. Let $\\map \\CC {\\mathbb J}$ be the set of all continuous real functions on $\\mathbb J$. Then $\\struct {\\map \\CC {\\mathbb J}, +, \\times}_\\R$ is a subspace of the $\\R$-vector space $\\struct {\\R^{\\mathbb J}, +, \\times}_\\R$."}877{"_id": "992", "title": "Singleton is Linearly Independent", "text": "Let $K$ be a division ring. Let $\\struct {G, +_G}$ be a group whose identity is $e$. Let $\\struct {G, +_G, \\circ}_K$ be a $K$-vector space whose zero is $0_K$. Let $x \\in G: x \\ne e$. Then $\\set x$ is a linearly independent subset of $G$."}878{"_id": "993", "title": "Linearly Dependent Sequence of Vector Space", "text": "Let $\\struct {G, +}$ be a group whose identity is $\\mathbf 0$. Let $\\struct {G, +, \\circ}_K$ be a $K$-vector space. Let $\\sequence {a_k}_{1 \\mathop \\le k \\mathop \\le n}$ be a sequence of distinct non-zero vectors of $G$. Then $\\sequence {a_k}_{1 \\mathop \\le k \\mathop \\le n}$ is linearly dependent {{iff}}: :$\\exists p \\in \\closedint 2 n: a_p$ is a linear combination of $\\sequence {a_k}_{1 \\mathop \\le k \\mathop \\le p - 1}$"}879{"_id": "994", "title": "Vector Space has Basis Between Linearly Independent Set and Finite Spanning Set", "text": "Let $K$ be a division ring. Let $G$ be a finitely generated $K$-vector space. Let $H$ be a linearly independent subset of $G$. Let $F$ be a finite generator for $G$ such that $H \\subseteq F$. Then there is a basis $B$ for $G$ such that $H \\subseteq B \\subseteq F$."}880{"_id": "997", "title": "Bases of Finitely Generated Vector Space have Equal Cardinality", "text": "Let $K$ be a division ring. Let $G$ be a finitely generated $K$-vector space. Then any two bases of $G$ are finite and equivalent. {{improve|I still think there's a simpler way of saying \"they have the same number of elements\" than bringing in all that top-heavy technical set theoretic language of \"set equivalence\" and \"cardinality\" and so on. Since we are talking about a finite set, the complexities which arise with regard to transfinites do not arise, so there is no direct need to go into such depth.}}"}881{"_id": "998", "title": "Linearly Independent Subset also Independent in Generated Subspace", "text": "Let $G$ be a finitely generated $K$-vector space. Let $S$ be a linearly independent subset of $G$. Let $M$ be the subspace of $G$ generated by $S$. If $M \\ne G$, then $\\forall b \\in G: b \\notin M$, the set $S \\cup \\set b$ is linearly independent."}882{"_id": "999", "title": "Sufficient Conditions for Basis of Finite Dimensional Vector Space", "text": "Let $K$ be a division ring. Let $n \\ge 0$ be a natural number. Let $E$ be an $n$-dimensional vector space over $K$. Let $B \\subseteq E$ be a subset such that $\\card B = n$. {{TFAE}} : $(1): \\quad$ $B$ is a basis of $E$. : $(2): \\quad$ $B$ is linearly independent. : $(3): \\quad$ $B$ is a generator for $E$."}883{"_id": "1000", "title": "Dimension of Proper Subspace is Less Than its Superspace", "text": "Let $G$ be a vector space whose dimension is $n$. Let $H$ be a subspace of $G$. Then $H$ is finite dimensional, and $\\map \\dim H \\le \\map \\dim G$. If $H$ is a proper subspace of $G$, then $\\map \\dim H < \\map \\dim G$."}884{"_id": "1001", "title": "Results concerning Generators and Bases of Vector Spaces", "text": "Let $E$ be a vector space of $n$ dimensions. Let $G$ be a generator for $E$. Then $G$ has the following properties:"}885{"_id": "1002", "title": "Grassmann's Identity", "text": "Let $K$ be a division ring. Let $\\struct {G, +_G, \\circ}_K$ be a $K$-vector space. Let $M$ and $N$ be finite-dimensional subspaces of $G$. Then the sum $M + N$ and intersection $M \\cap N$ are finite-dimensional, and: :$\\map \\dim {M + N} + \\map \\dim {M \\cap N} = \\map \\dim M + \\map \\dim N$"}886{"_id": "1003", "title": "Rank Plus Nullity Theorem", "text": "Let $G$ be an $n$-dimensional vector space. Let $H$ be a vector space. Let $\\phi: G \\to H$ be a linear transformation. Let $\\map \\rho \\phi$ and $\\map \\nu \\phi$ be the rank and nullity respectively of $\\phi$. Then the image of $\\phi$ is finite-dimensional, and: :$\\map \\rho \\phi + \\map \\nu \\phi = n$ By definition of rank and nullity, it can be seen that this is equivalent to the alternative way of stating this result: :$\\map \\dim {\\Img \\phi} + \\map \\dim {\\map \\ker \\phi} = \\map \\dim G$ {{wtd|and the theorem is applicable to matrices}}"}887{"_id": "1004", "title": "Linear Transformation of Vector Space Monomorphism", "text": "Let $G$ and $H$ be a $K$-vector space. Let $\\phi: G \\to H$ be a linear transformation. Then $\\phi$ is a monomorphism {{iff}} for every linearly independent sequence $\\sequence {a_n}$ of vectors of $G$, $\\sequence {\\map \\phi {a_n} }$ is a linearly independent sequence of vectors of $H$."}888{"_id": "1005", "title": "Linear Transformation of Vector Space Equivalent Statements", "text": "Let $G$ and $H$ be $n$-dimensional vector spaces. Let $\\phi: G \\to H$ be a linear transformation. Then these statements are equivalent: : $(1): \\quad \\phi$ is an isomorphism. : $(2): \\quad \\phi$ is a monomorphism. : $(3): \\quad \\phi$ is an epimorphism. : $(4): \\quad \\phi \\left({B}\\right)$ is a basis of $H$ for every basis $B$ of $G$. : $(5): \\quad \\phi \\left({B}\\right)$ is a basis of $H$ for some basis $B$ of $G$."}889{"_id": "1006", "title": "Results Concerning Annihilator of Vector Subspace", "text": "Let $G$ be an $n$-dimensional vector space over a field. Let $J: G \\to G^{**}$ be the evaluation isomorphism. Let $G^*$ be the algebraic dual of $G$. Let $G^{**}$ be the algebraic dual of $G^*$. Let $M$ be an $m$-dimensional subspace of $G$. Let $N$ be a $p$-dimensional subspace of $G^*$. Let $M^\\circ$ be the annihilator of $M$. Then: :$(1): \\quad M^\\circ$ is an $\\paren {n - m}$-dimensional subspace of $G^*$, and $M^{\\circ \\circ} = \\map J M$ :$(2): \\map {\\quad J^{-1} } {N^\\circ}$ is an $\\paren {n - p}$-dimensional subspace of $G$ :$(3): \\quad$ The mapping $M \\to M^\\circ$ is a bijection from the set of all $m$-dimensional subspaces of $G$ onto the set of all $\\paren {n - m}$-dimensional subspaces of $G^*$ :$(4): \\quad$ Its inverse is the bijection $N \\to \\map {J^{-1} } {N^\\circ}$."}890{"_id": "1007", "title": "Vector Subspace Dimension One Less", "text": "Let $K$ be a field. Let $M$ be a subspace of the $n$-dimensional vector space $K^n$. The following statements are equivalent: :$(1): \\quad \\map \\dim M = n - 1$ :$(2): \\quad M$ is the kernel of a nonzero linear form :$(3): \\quad$ There exists a sequence $\\sequence {\\alpha_n} $ of scalars, not all of which are zero, such that: :::$M = \\set {\\tuple {\\lambda_1, \\ldots, \\lambda_n} \\in K^n: \\alpha_1 \\lambda_1 + \\cdots + \\alpha_n \\lambda_n = 0}$ Also, suppose the above hold. Let $\\sequence {\\beta_n}$ be a sequence of scalars such that: :$M = \\set {\\tuple {\\lambda_1, \\ldots, \\lambda_n} \\in K^n: \\beta_1 \\lambda_1 + \\cdots + \\beta_n \\lambda_n = 0}$ Then there is a non-zero scalar $\\gamma$ such that: :$\\forall k \\in \\closedint 1 n: \\beta_k = \\gamma \\alpha_k$"}891{"_id": "1008", "title": "Rank and Nullity of Transpose", "text": "Let $G$ and $H$ be $n$-dimensional vector spaces over a field. Let $\\mathcal L \\left({G, H}\\right)$ be the set of all linear transformations from $G$ to $H$. Let $u \\in \\mathcal L \\left({G, H}\\right)$. Let $u^t$ be the transpose of $u$. Then: : $(1): \\quad$ $u$ and $u^t$ have the same rank and nullity : $(2): \\quad$ $\\ker \\left({u^t}\\right)$ is the annihilator of the image of $u$ : $(3): \\quad$ The image of $u^t$ is the annihilator of $\\ker \\left({u}\\right)$."}892{"_id": "1009", "title": "Linear Operator on the Plane", "text": "Let $\\phi$ be a linear operator on the real vector space of two dimensions $\\R^2$. Then $\\phi$ is completely determined by an ordered tuple of $4$ real numbers."}893{"_id": "1010", "title": "Similarity Mapping of Plane is Linear Operator", "text": "Let $G$ be a vector space over a field $K$. Let $\\beta \\in K$. Then the mapping: :$s_\\beta: G \\to G$ defined by $\\map {s_\\beta} {\\mathbf x} = \\beta \\mathbf x$ is a linear operator on $G$. If $\\beta \\ne 0$ then $s_\\beta$ is an automorphism of $G$, and $\\paren {s_\\beta}^{-1} = s_{\\beta^{-1} }$ The linear operators $s_\\beta$, where $\\beta \\ne 0$, are called '''similarities of $G$'''."}894{"_id": "1011", "title": "Cantor-Dedekind Hypothesis", "text": "The points on an infinite straight line are in one-to-one correspondence with the set $\\R$ of real numbers. Hence the set of all points on an infinite straight line and $\\R$ are equinumerous."}895{"_id": "1013", "title": "Full Angle measures 2 Pi Radians", "text": "One full angle is equal to $2 \\pi$ radians. :$2 \\pi \\approx 6 \\cdotp 28318 \\, 53071 \\, 79586 \\, 4769 \\ldots$ {{OEIS|A019692}}"}896{"_id": "1014", "title": "Rotation of Plane about Origin is Linear Operator", "text": "Let $r_\\alpha$ be the plane rotation of the plane about the origin through an angle of $\\alpha$. That is, let $r_\\alpha: \\R^2 \\to \\R^2$ be the mapping defined as: :$\\forall x \\in \\R^2: \\map {r_\\alpha} x = \\text { the point into which a rotation of } \\alpha \\text{ carries } x$ Then $r_\\alpha$ is a linear operator determined by the ordered sequence: : $\\tuple {\\cos \\alpha  -\\sin \\alpha, \\sin \\alpha + \\cos \\alpha}$"}897{"_id": "1015", "title": "Stretching and Contraction Mappings of Plane are Linear Operators", "text": "Let $s_\\beta: \\R^2 \\to \\R^2$ be a similarity of $\\R^2$. Then $s_{-1}$ is the same as the rotation $r_{\\pi}$ of the plane about the origin one half turn. If $\\beta \\ge 1$, then $s_\\beta$ is called a '''stretching''', and if $0 < \\beta \\le 1$, $s_\\beta$ is called a '''contraction'''. If $\\beta < 0$, then $s_\\beta$ is a stretching or contraction followed by a rotation one half turn. It is also the same as a rotation one half turn followed by a stretching or contraction."}898{"_id": "1016", "title": "Reflection of Plane in Line through Origin is Linear Operator", "text": "Let $M$ be a straight line in the plane passing through the origin. Then the '''reflection''' $s_M$ of $\\R^2$ in $M$ is the rotation of the plane in space through one half turn about $M$ as an axis. :$s_M \\circ s_M = I_{\\R^2}$ and hence: :$s_M = s_M^{-1}$ If $M$ is the $x$-axis then $\\map {s_M} {\\lambda_1, \\lambda_2} = \\tuple {\\lambda_1, -\\lambda_2}$. If $M$ is the $y$-axis then $\\map {s_M} {\\lambda_1, \\lambda_2} = \\tuple {-\\lambda_1, \\lambda_2}$. In general, $s_M$ is a linear operator for every straight line $M$ through the origin."}899{"_id": "1017", "title": "Projection of Straight Line on Another in Plane", "text": "Let $M$ and $N$ be distinct straight lines through the plane through the origin. Let $\\operatorname{pr}_{M, N}$ be the projection on $M$ along $N$. $M$ and $N$ are respectively the codomain and kernel of $\\operatorname{pr}_{M, N}$. {{explain|As the kernel is a concept defined in relation to a homomorphism, it needs to be clarified what homomorphism is being considered.}} :$\\operatorname{pr}_{M, N} \\left({x}\\right) = x \\iff x \\in M$ If $M$ is the $x$-axis and $N$ is the $y$-axis, then $\\operatorname{pr}_{M, N} \\left({\\lambda_1, \\lambda_2}\\right) = \\left({\\lambda_1, 0}\\right)$. If $M$ is the $y$-axis and $N$ is the $x$-axis, then $\\operatorname{pr}_{M, N} \\left({\\lambda_1, \\lambda_2}\\right) = \\left({0, \\lambda_2}\\right)$. Any such projection is a linear operator."}900{"_id": "1019", "title": "Condition for Straight Lines in Plane to be Parallel", "text": "Let $L: \\alpha_1 x + \\alpha_2 y = \\beta$ be a straight line in $\\R^2$. Then the straight line $L'$ is parallel to $L$ iff there is a $\\beta' \\in \\R^2$ such that: :$L' = \\set {\\tuple {x, y} \\in \\R^2: \\alpha_1 x + \\alpha_2 y = \\beta'}$"}901{"_id": "1020", "title": "Equation of Plane", "text": "A plane $P$ is the set of all $\\tuple {x, y, z} \\in \\R^3$, where: :$\\alpha_1 x + \\alpha_2 y + \\alpha_3 z = \\gamma$ where $\\alpha_1, \\alpha_2, \\alpha_3, \\gamma \\in \\R$ are given, and not all of $\\alpha_1, \\alpha_2, \\alpha_3$ are zero."}902{"_id": "1024", "title": "Planes are Subspaces of Space", "text": "The two-dimensional subspaces of $\\R^3$ are precisely the homogeneous planes of solid analytic geometry."}903{"_id": "1025", "title": "Matrix Space Semigroup under Hadamard Product", "text": "Let $\\map {\\MM_S} {m, n}$ be the matrix space over a semigroup $\\struct {S, \\cdot}$. Then the algebraic structure $\\struct {\\map {\\MM_S} {m, n}, \\circ}$, where $\\circ$ is the Hadamard product, is also a semigroup. If $\\struct {S, \\cdot}$ is a commutative semigroup then so is $\\struct {\\map {\\MM_S} {m, n}, \\circ}$. If $\\struct {S, \\cdot}$ is a monoid then so is $\\struct {\\map {\\MM_S} {m, n}, \\circ}$."}904{"_id": "1026", "title": "Hadamard Product over Group forms Group", "text": "Let $\\struct {G, \\cdot}$ be a group whose identity is $e$. Let $\\map {\\MM_G} {m, n}$ be a $m \\times n$ matrix space over $\\struct {G, \\cdot}$. Then $\\struct {\\map {\\MM_G} {m, n}, \\circ}$, where $\\circ$ is Hadamard product, is also a group."}905{"_id": "1027", "title": "Matrix Multiplication is Associative", "text": "Let $R$ be a ring. Matrix multiplication (conventional) is associative."}906{"_id": "1028", "title": "Linear Transformations Isomorphic to Matrix Space", "text": "Let $R$ be a ring with unity. Let $F$, $G$ and $H$ be free $R$-modules of finite dimension $p,n,m>0$ respectively. Let $\\left \\langle {a_p} \\right \\rangle$, $\\left \\langle {b_n} \\right \\rangle$ and $\\left \\langle {c_m} \\right \\rangle$ be ordered bases  Let $\\operatorname{Hom} \\left({G, H}\\right)$ be the set of all linear transformations from $G$ to $H$. Let $\\mathcal M_R \\left({m, n}\\right)$ be the $m \\times n$ matrix space over $R$. Let $\\left[{u; \\left \\langle {c_m} \\right \\rangle, \\left \\langle {b_n} \\right \\rangle}\\right]$ be the matrix of $u$ relative to $\\left \\langle {b_n} \\right \\rangle$ and $\\left \\langle {c_m} \\right \\rangle$. Let $M: \\operatorname{Hom} \\left({G, H}\\right) \\to \\mathcal M_R \\left({m, n}\\right)$ be defined as: :$\\forall u \\in \\operatorname{Hom} \\left({G, H}\\right): M \\left({u}\\right) = \\left[{u; \\left \\langle {c_m} \\right \\rangle, \\left \\langle {b_n} \\right \\rangle}\\right]$ Then $M$ is an isomorphism of modules, and: :$\\forall u \\in \\operatorname{Hom} \\left({F, G}\\right), v \\in \\operatorname{Hom} \\left({G, H}\\right): \\left[{v \\circ u; \\left \\langle {c_m} \\right \\rangle, \\left \\langle {a_p} \\right \\rangle}\\right] = \\left[{v; \\left \\langle {c_m} \\right \\rangle, \\left \\langle {b_n} \\right \\rangle}\\right] \\left[{u; \\left \\langle {b_n} \\right \\rangle, \\left \\langle {a_p} \\right \\rangle}\\right]$"}907{"_id": "1029", "title": "Matrix Multiplication Distributes over Matrix Addition", "text": "Matrix multiplication (conventional) is distributive over matrix entrywise addition."}908{"_id": "1030", "title": "Unit Matrix is Unity of Ring of Square Matrices", "text": "Let $R$ be a ring with unity whose zero is $0_R$ and whose unity is $1_R$. Let $n \\in \\Z_{>0}$ be a (strictly) positive integer. Let $\\struct {\\map {\\MM_R} n, +, \\times}$ denote the ring of square matrices of order $n$ over $R$. The unit matrix over $R$: :$\\mathbf I_n = \\begin {pmatrix} 1_R & 0_R & 0_R & \\cdots & 0_R \\\\ 0_R & 1_R & 0_R & \\cdots & 0_R \\\\ 0_R & 0_R & 1_R & \\cdots & 0_R \\\\ \\vdots & \\vdots & \\vdots & \\ddots & \\vdots \\\\ 0_R & 0_R & 0_R & \\cdots & 1_R \\end {pmatrix}$ is the identity element of $\\struct {\\map {\\MM_R} n, +, \\times}$."}909{"_id": "1031", "title": "Matrix Multiplication is Closed", "text": "Let $\\struct {R, +, \\circ}$ be a ring. Let $\\map {\\MM_R} n$ be a $n \\times n$ matrix space over $R$. Then matrix multiplication (conventional) over $\\map {\\MM_R} n$ is closed."}910{"_id": "1032", "title": "Square Matrices over Real Numbers under Multiplication form Monoid", "text": "Let $\\map {\\mathcal M_\\R} n$ be a $n \\times n$ matrix space over the set of real numbers $\\R$. Then the set of all $n \\times n$ real matrices $\\map {\\mathcal M_\\R} n$ under matrix multiplication (conventional) forms a monoid."}911{"_id": "1033", "title": "Ring of Square Matrices over Commutative Ring with Unity", "text": "Let $R$ be a commutative ring with unity. Let $n \\in \\Z_{>0}$ be a (strictly) positive integer. Let $\\struct {\\map {\\MM_R} n, +, \\times}$ denote the ring of square matrices of order $n$ over $R$. Then $\\struct {\\map {\\MM_R} n, +, \\times}$ is a ring with unity. However, for $n \\ge 2$, $\\struct {\\map {\\MM_R} n, +, \\times}$ is not a commutative ring."}912{"_id": "1034", "title": "Invertible Matrix corresponds to Automorphism", "text": "Let $R$ be a ring with unity. Let $n \\in \\Z_{>0}$ be a (strictly) positive integer. Let $G$ be an $n$-dimensional $R$-module. Let $\\map {\\mathcal M_R} n$ be the $n \\times n$ matrix space over $R$. Let $\\map {\\mathcal L_R} G$ be the set of all linear operators on $G$. Then the invertible elements of the ring of square matrices $\\struct {\\map {\\mathcal M_R} n, +, \\times}$ correspond directly to automorphisms of $\\map {\\mathcal L_R} G$."}913{"_id": "1035", "title": "Change of Basis is Invertible", "text": "Let $R$ be a ring with unity. Let $M$ be a free $R$-module of finite dimension $n>0$. Let $\\mathcal A$ and $\\mathcal B$ be ordered bases of $M$. Let $\\mathbf P$ be the change of basis matrix from $\\mathcal A$ to $\\mathcal B$. Then $\\mathbf P$ is invertible, and its inverse $\\mathbf P^{-1}$ is the change of basis matrix from $\\mathcal B$ to $\\mathcal A$."}914{"_id": "1037", "title": "Matrix Corresponding to Change of Basis under Linear Transformation", "text": "Let $R$ be a ring with unity. Let $G$ and $H$ be free $R$-modules of finite dimensions $n,m>0$ respectively. Let $\\left \\langle {a_n} \\right \\rangle$ and $\\left \\langle {{a_n}'} \\right \\rangle$ be ordered bases of $G$. Let $\\left \\langle {b_m} \\right \\rangle$ and $\\left \\langle {{b_m}'} \\right \\rangle$ be ordered bases of $H$. Let $u: G \\to H$ be a linear transformation, and let $\\left[{u; \\left \\langle {b_m} \\right \\rangle, \\left \\langle {a_n} \\right \\rangle}\\right]$ be the matrix of $u$ relative to $\\left \\langle {a_n} \\right \\rangle$ and $\\left \\langle {b_m} \\right \\rangle$. Let: : $\\mathbf A = \\left[{u; \\left \\langle {b_m} \\right \\rangle, \\left \\langle {a_n} \\right \\rangle}\\right]$ : $\\mathbf B = \\left[{u; \\left \\langle {{b_m}'} \\right \\rangle, \\left \\langle {{a_n}'} \\right \\rangle}\\right]$ Then: :$\\mathbf B = \\mathbf Q^{-1} \\mathbf A \\mathbf P$ where: : $\\mathbf P$ is the matrix corresponding to the change of basis from $\\left \\langle {a_n} \\right \\rangle$ to $\\left \\langle {{a_n}'} \\right \\rangle$ : $\\mathbf Q$ is the matrix corresponding to the change of basis from $\\left \\langle {b_m} \\right \\rangle$ to $\\left \\langle {{b_m}'} \\right \\rangle$."}915{"_id": "1038", "title": "Matrix Equivalence is Equivalence Relation", "text": "Matrix equivalence is an equivalence relation."}916{"_id": "1041", "title": "Equivalent Matrices have Equal Rank", "text": "Let $\\mathbf A$ and $\\mathbf B$ be $m \\times n$ matrices over a field $K$. Let $\\map \\phi {\\mathbf A}$ denote the rank of $\\mathbf A$. Let $\\mathbf A \\equiv \\mathbf B$ denote that $\\mathbf A$ and $\\mathbf B$ are matrix equivalent. Then: :$\\mathbf A \\equiv \\mathbf B$ {{iff}}: :$\\map \\phi {\\mathbf A} = \\map \\phi {\\mathbf B}$"}917{"_id": "1043", "title": "Transpose of Matrix Product", "text": "Let $\\mathbf A$ and $\\mathbf B$ be matrices over a commutative ring such that $\\mathbf A \\mathbf B$ is defined. Then $\\mathbf B^\\intercal \\mathbf A^\\intercal$ is defined, and: :$\\paren {\\mathbf A \\mathbf B}^\\intercal = \\mathbf B^\\intercal \\mathbf A^\\intercal$ where $\\mathbf X^\\intercal$ is the transpose of $\\mathbf X$."}918{"_id": "1045", "title": "General Linear Group is Group", "text": "Let $K$ be a field. Let $\\GL {n, K}$ be the general linear group of order $n$ over $K$. Then $\\GL {n, K}$ is a group."}919{"_id": "1046", "title": "Transpose of Row Matrix is Column Matrix", "text": "Let $\\mathbf x = \\sqbrk x_{1 n} = \\begin {bmatrix} x_1 & x_2 & \\cdots & x_n \\end {bmatrix}$ be a row matrix. Then $\\mathbf x^\\intercal$, the transpose of $\\mathbf x$, is a column matrix: :$\\begin {bmatrix} x_1 & x_2 & \\cdots & x_n \\end{bmatrix}^\\intercal = \\begin {bmatrix} x_1 \\\\ x_2 \\\\ \\vdots \\\\ x_n \\end {bmatrix}$"}920{"_id": "1047", "title": "Transpose of Transpose of Matrix", "text": "Let $\\mathbf A$ be a matrix. Let $\\mathbf A^\\intercal$ be the transpose of $\\mathbf A$. Then: :$\\paren {\\mathbf A^\\intercal}^\\intercal = \\mathbf A$"}921{"_id": "1048", "title": "Solution to Simultaneous Linear Equations", "text": "Let $\\displaystyle \\forall i \\in \\closedint 1 m: \\sum _{j \\mathop = 1}^n {\\alpha_{i j} x_j} = \\beta_i$ be a system of simultaneous linear equations. where all of $\\alpha_1, \\ldots, a_n, x_1, \\ldots x_n, \\beta_i, \\ldots, \\beta_m$ are elements of a field $K$. Then $x = \\tuple {x_1, x_2, \\ldots, x_n}$ is a solution of this system {{iff}}: :$\\sqbrk \\alpha_{m n} \\sqbrk x_{n 1} = \\sqbrk \\beta_{m 1}$ where $\\sqbrk a_{m n}$ is an $m \\times n$ matrix."}922{"_id": "1049", "title": "Infinite Cyclic Group is Isomorphic to Integers", "text": "Let $G$ be an infinite cyclic group. Then $G$ is isomorphic to the additive group of integers: $G \\cong \\struct {\\Z, +}$."}923{"_id": "1050", "title": "Subgroup of Infinite Cyclic Group is Infinite Cyclic Group", "text": "Let $G = \\gen a$ be an infinite cyclic group generated by $a$, whose identity is $e$. Let $g \\in G, g \\ne e: \\exists k \\in \\Z, k \\ne 0: g = a^k$. Let $H = \\gen g$. Then $H \\le G$ and $H \\cong G$. Thus, all non-trivial subgroups of an infinite cyclic group are themselves infinite cyclic groups. A subgroup of $G = \\gen a$ is denoted as follows: :$n G := \\gen {a^n}$ This notation is usually used in the context of $\\struct {\\Z, +}$, where $n \\Z$ is (informally) understood as '''the set of integer multiples of $n$'''."}924{"_id": "1051", "title": "Quotient Group of Infinite Cyclic Group by Subgroup", "text": "Let $C_n$ be the cyclic group of order $n$. Then: :$C_n \\cong \\dfrac {\\struct {\\Z, +} } {\\struct {n \\Z, +} } = \\dfrac \\Z {n \\Z}$ where: :$\\Z$ is the additive group of integers :$n \\Z$ is the additive group of integer multiples :$\\Z / n \\Z$ is the quotient group of $\\Z$ by $n \\Z$. Thus, every cyclic group is isomorphic to one of: :$\\Z, \\dfrac \\Z \\Z, \\dfrac \\Z {2 \\Z}, \\dfrac \\Z {3 \\Z}, \\dfrac \\Z {4 \\Z}, \\ldots$"}925{"_id": "1052", "title": "Ring Operations on Coset Space of Ideal", "text": "Let $\\struct {R, +, \\circ}$ be a ring. Let $\\powerset R$ be the power set of $R$. Let $J$ be an ideal of $R$. Let $X$ and $Y$ be cosets of $J$. Let $X +_\\mathcal P Y$ be the sum of $X$ and $Y$, where $+_\\mathcal P$ is the operation induced on $\\powerset R$ by $+$. Similarly, let $X \\circ_\\mathcal P Y$ be the product of $X$ and $Y$, where $\\circ_\\mathcal P$ is the operation induced on $\\powerset R$ by $\\circ$. Then: * The sum $X +_\\mathcal P Y$ in $\\powerset R$ is also their sum in the quotient ring $R / J$. * The product $X \\circ_\\mathcal P Y$ in $\\powerset R$ may be a proper subset of their product in $R / J$."}926{"_id": "1053", "title": "Property of Being an Ideal is not Transitive", "text": "Let $J_1$ be an ideal of a ring $R$. Let $J_2$ be an ideal of $J_1$. Then $J_2$ need not necessarily be an ideal of $R$."}927{"_id": "1054", "title": "Ideals Containing Ideal Form Lattice", "text": "Let $J$ be an ideal of a ring $R$. Let $\\mathbb L_J$ be the set of all ideal of $R$ which contain $J$. Then the ordered set $\\struct {\\mathbb L_J, \\subseteq}$ is a lattice."}928{"_id": "1055", "title": "Ideals Containing Ideal Isomorphic to Quotient Ring", "text": "Let $J$ be an ideal of a ring $R$. Let $\\mathbb L_J$ be the set of all ideals of $R$ which contain $J$. Let the ordered set $\\left({\\mathbb L \\left({R / J}\\right), \\subseteq}\\right)$ be the set of all ideals of $R / J$. Let the mapping $\\Phi_J: \\left({\\mathbb L_J, \\subseteq}\\right) \\to \\left({\\mathbb L \\left({R / J}\\right), \\subseteq}\\right)$ be defined as: :$\\forall a \\in \\mathbb L_J: \\Phi_J \\left({a}\\right) = q_J \\left({a}\\right)$ where $q_J: a \\to a / J$ is the quotient epimorphism from $a$ to $a / J$ from the definition of quotient ring. Then $\\Phi_J$ is an isomorphism."}929{"_id": "1056", "title": "Ring of Integers is Principal Ideal Domain", "text": "The integers $\\Z$ form a principal ideal domain."}930{"_id": "1057", "title": "Principal Ideals of Integers", "text": "Let $J$ be a non-zero ideal of $\\Z$. Then $J = \\ideal b$ where $b$ is the smallest strictly positive integer belonging to $J$."}931{"_id": "1058", "title": "Natural Numbers Set Equivalent to Ideals of Integers", "text": "Let the mapping $\\psi: \\N \\to$ the set of all ideals of $\\Z$ be defined as: :$\\forall b \\in \\N: \\psi \\left({b}\\right) = \\left({b}\\right)$ where $\\left({b}\\right)$ is the principal ideal of $\\Z$ generated by $b$. Then $\\psi$ is a bijection."}932{"_id": "1059", "title": "Canonical Epimorphism from Integers by Principal Ideal", "text": "Let $m$ be a strictly positive integer. Let $\\left({m}\\right)$ be the principal ideal of $\\Z$ generated by $m$. The restriction to $\\N_m$ of the canonical epimorphism $q_m$ from the ring $\\left({\\Z, +, \\times}\\right)$ onto $\\left({\\Z, +, \\times}\\right) / \\left({m}\\right)$ is an isomorphism from the ring $\\left({\\N_m, +_m, \\times_m}\\right)$ of integers modulo $m$ onto the quotient ring $\\left({\\Z, +, \\times}\\right) / \\left({m}\\right)$. In particular, $\\left({\\Z, +, \\times}\\right) / \\left({m}\\right)$ has $m$ elements."}933{"_id": "1060", "title": "Integer Divisor is Equivalent to Subset of Ideal", "text": "Let $\\Z$ be the set of all integers. Let $\\Z_{>0}$ be the set of strictly positive integers. Let $m \\in \\Z_{>0}$ and let $n \\in \\Z$. Let $\\ideal m$ be the principal ideal of $\\Z$ generated by $m$. Then: :$m \\divides n \\iff \\ideal n \\subseteq \\ideal m$"}934{"_id": "1061", "title": "Principal Ideals in Integral Domain", "text": "Let $\\struct {D, +, \\circ}$ be an integral domain. Let $U_D$ be the group of units of $D$. Let $\\ideal x$ be the principal ideal of $D$ generated by $x$. Let $x, y \\in \\struct {D, +, \\circ}$. Then:"}935{"_id": "1062", "title": "Principal Ideal Domain is Unique Factorization Domain", "text": "Every principal ideal domain is a unique factorization domain."}936{"_id": "1063", "title": "Maximal Ideal iff Quotient Ring is Field", "text": "Let $\\struct {R, +, \\circ}$ be a commutative ring with unity whose zero is $0_R$ and whose unity is $1_R$. Let $J$ be an ideal of $R$. The following are equivalent: :$(1): \\quad$ $J$ is a maximal ideal. :$(2): \\quad$ The quotient ring $R / J$ is a field."}937{"_id": "1064", "title": "Prime Number iff Generates Principal Maximal Ideal", "text": "Let $\\Z_{>0}$ be the set of strictly positive integers. Let $p \\in \\Z_{>0}$. Let $\\ideal p$ be the principal ideal of $\\Z$ generated by $p$.  Then $p$ is prime {{iff}} $\\ideal p$ is a maximal ideal of $\\Z$."}938{"_id": "1065", "title": "Integral Domain of Prime Order is Field", "text": "Let $\\left({\\Z_p, +_p, \\times_p}\\right)$ be the ring of integers modulo $p$. The following statements are equivalent: : $(1): \\quad p$ is a prime. : $(2): \\quad \\left({\\Z_p, +_p, \\times_p}\\right)$ is an integral domain. : $(3): \\quad \\left({\\Z_p, +_p, \\times_p}\\right)$ is a field."}939{"_id": "1066", "title": "Quotient Ring of Integers and Zero", "text": "Let $\\struct {\\Z, +, \\times}$ be the integral domain of integers. Let $\\ideal 0$ be the principal ideal of $\\struct {\\Z, +, \\times}$ generated by $0$. The quotient ring $\\struct {\\Z / \\ideal 0, +, \\times}$ is isomorphic to $\\struct {\\Z, +, \\times}$."}940{"_id": "1067", "title": "Quotient Ring of Integers and Principal Ideal from Unity", "text": "Let $\\left({\\Z, +, \\times}\\right)$ be the integral domain of integers. Let $\\left({1}\\right)$ be the principal ideal of $\\left({\\Z, +, \\times}\\right)$ generated by $1$. The quotient ring $\\left({\\Z, +, \\times}\\right) / \\left({1}\\right)$ is isomorphic to the null ring."}941{"_id": "1068", "title": "Principal Ideal of Principal Ideal Domain is of Irreducible Element iff Maximal", "text": "Let $\\struct {D, +, \\circ}$ be a principal ideal domain. Let $\\ideal p$ be the principal ideal of $D$ generated by $p$. Then $p$ is irreducible {{iff}} $\\ideal p$ is a maximal ideal of $D$."}942{"_id": "1069", "title": "Subring Generated by Unity of Ring with Unity", "text": "Let $\\left({R, +, \\circ}\\right)$ be a ring with unity whose zero is $0_R$ and whose unity is $1_R$. Let the mapping $g: \\Z \\to R$ be defined as $\\forall n \\in \\Z: g \\left({n}\\right) = n 1_R$, where $n 1_R$ the $n$th power of $1_R$. Let $\\left({x}\\right)$ be the principal ideal of $\\left({R, +, \\circ}\\right)$ generated by $x$. Then $g$ is an epimorphism from $\\Z$ onto the subring $S$ of $R$ generated by $1_R$. If $R$ has no proper zero divisors, then $g$ is the only nonzero homomorphism from $\\Z$ into $R$. The kernel of $g$ is either: : $(1): \\quad \\left({0_R}\\right)$, in which case $g$ is an isomorphism from $\\Z$ onto $S$ or: : $(2): \\quad \\left({p}\\right)$ for some prime $p$, in which case $S$ is isomorphic to the field $\\Z_p$."}943{"_id": "1070", "title": "Null Ring iff Characteristic is One", "text": "The only ring whose characteristic is $1$ is the null ring."}944{"_id": "1071", "title": "Characteristic of Finite Ring with No Zero Divisors", "text": "Let $\\struct {R, +, \\circ}$ be a finite ring with unity with no proper zero divisors whose zero is $0_R$ and whose unity is $1_R$. Let $n \\ne 0$ be the characteristic of $R$. Then: :$(1): \\quad n$ must be a prime number :$(2): \\quad n$ is the order of all non-zero elements in $\\struct {R, +}$. It follows that $\\struct {R, +} \\cong C_n$, where $C_n$ is the cyclic group of order $n$."}945{"_id": "1072", "title": "Integral Domain with Characteristic Zero", "text": "In an integral domain with characteristic zero, every non-zero element has infinite order under ring addition."}946{"_id": "1073", "title": "Characteristic of Field is Zero or Prime", "text": "Let $F$ be a field. Then the characteristic of $F$ is either zero or a prime number."}947{"_id": "1074", "title": "Field of Characteristic Zero has Unique Prime Subfield", "text": "Let $F$ be a field, whose zero is $0_F$ and whose unity is $1_F$, with characteristic zero. Then there exists a unique $P \\subseteq F$ such that: :$(1): \\quad P$ is a subfield of $F$ :$(2): \\quad P$ is isomorphic to the field of rational numbers $\\struct {\\Q, +, \\times}$. That is, $P \\cong \\Q$ is a unique minimal subfield of $F$, and all other subfields of $F$ contain $P$. This field $P$ is called the prime subfield of $F$."}948{"_id": "1075", "title": "Field of Prime Characteristic has Unique Prime Subfield", "text": "Let $F$ be a field whose characteristic is $p$. Then there exists a unique $P \\subseteq F$ such that: :$(1): \\quad P$ is a subfield of $F$ :$(2): \\quad P \\cong \\Z_p$. That is, $P \\cong \\Z_p$ is a unique minimal subfield of $F$, and all other subfields of $F$ contain $P$. This field $P$ is called the prime subfield of $F$."}949{"_id": "1076", "title": "Intersection of All Division Subrings is Prime Subfield", "text": "Let $\\struct {K, +, \\circ}$ be a division ring. Let $P$ be the intersection of the set of all division subrings of $K$. Then $P$ is the prime subfield of $K$."}950{"_id": "1077", "title": "Characteristic of Ordered Integral Domain is Zero", "text": "Let $\\left({D, +, \\circ}\\right)$ be an ordered integral domain whose zero is $0_D$ and whose unity is $1_D$. Let $\\operatorname{Char} \\left({D}\\right)$ be the characteristic of $D$. Then $\\operatorname{Char} \\left({D}\\right) = 0$. Let $g: \\Z \\to D$ be the mapping defined as: :$\\forall n \\in \\Z: g \\left({n}\\right) = n \\cdot 1_D$ where $n \\cdot 1_D$ is defined as the $n$th power of $1_D$. Then $g$ is the only monomorphism from the ordered ring $\\Z$ onto the ordered ring $D$."}951{"_id": "1078", "title": "Monomorphism from Rational Numbers to Totally Ordered Field", "text": "Let $\\struct {F, +, \\circ, \\le}$ be a totally ordered field. There is one and only one (ring) monomorphism from the totally ordered field $\\Q$ onto $F$. Its image is the prime subfield of $F$."}952{"_id": "1079", "title": "Set of Polynomials over Integral Domain is Subring", "text": "Let $\\struct {R, +, \\circ}$ be a commutative ring. Let $\\struct {D, +, \\circ}$ be an integral subdomain of $R$. Then $\\forall x \\in R$, the set $D \\sqbrk x$ of polynomials in $x$ over $D$ is a subring of $R$."}953{"_id": "1081", "title": "Polynomials Closed under Ring Product", "text": "Let $\\left({R, +, \\circ}\\right)$ be a commutative ring. Let $\\displaystyle f = \\sum_{k \\in Z} a_k \\mathbf X^k$, $\\displaystyle g =  \\sum_{k \\in Z} b_k \\mathbf X^k$ be polynomials in the indeterminates $\\left\\{{X_j: j \\in J}\\right\\}$ over $R$, where $Z$ is the set of all multiindices indexed by $\\left\\{{X_j: j \\in J}\\right\\}$. Define the product :$\\displaystyle f \\otimes g = \\sum_{k \\in Z} c_k \\mathbf X^k$ where :$\\displaystyle c_k = \\sum_{\\substack{p + q = k \\\\ p, q \\in Z}} a_p b_q$ Then $f \\otimes g$ is a polynomial."}954{"_id": "1083", "title": "Ring of Polynomial Forms is Integral Domain", "text": "Let $\\struct {R, +, \\circ}$ be a commutative ring with unity. Let $\\struct {D, +, \\circ}$ be an integral subdomain of $R$. Let $X \\in R$ be transcendental over $D$. Let $D \\sqbrk X$ be the ring of polynomials in $X$ over $D$. Then $D \\sqbrk X$ is an integral domain."}955{"_id": "1085", "title": "Injection is Bijection iff Inverse is Injection", "text": "Let $\\phi: S \\to T$ be an injection. Then $\\phi$ is a bijection {{iff}} its inverse $\\phi^{-1}$ is also an injection."}956{"_id": "1086", "title": "Epimorphism from Polynomial Forms to Polynomial Functions", "text": "Let $D$ be an integral domain. Let $D \\sqbrk X$ be the ring of polynomial forms in $X$ over $D$. Let $\\map P D$ be the ring of polynomial functions over $D$. The mapping $\\kappa: D \\sqbrk X \\to \\map P D$ given by: :$\\displaystyle \\map \\kappa {\\sum_{k \\mathop = 0}^n {a_k \\circ X^k} } = f$ where $\\displaystyle f = \\sum_{k \\mathop = 0}^n {a_k \\circ x^k}, x \\in D$ is a ring epimorphism."}957{"_id": "1087", "title": "Division Theorem for Polynomial Forms over Field", "text": "Let $\\struct {F, +, \\circ}$ be a field whose zero is $0_F$ and whose unity is $1_F$. Let $X$ be transcendental over $F$. Let $F \\sqbrk X$ be the ring of polynomials in $X$ over $F$. Let $d$ be an element of $F \\sqbrk X$ of degree $n \\ge 1$. Then $\\forall f \\in F \\sqbrk X: \\exists q, r \\in F \\sqbrk X: f = q \\circ d + r$ such that either: :$(1): \\quad r = 0_F$ or: :$(2): \\quad r \\ne 0_F$ and $r$ has degree that is less than $n$."}958{"_id": "1088", "title": "Polynomial Forms over Field form Principal Ideal Domain", "text": "Let $\\struct {F, +, \\circ}$ be a field whose zero is $0_F$ and whose unity is $1_F$. Let $X$ be transcendental over $F$. Let $F \\sqbrk X$ be the ring of polynomials in $X$ over $F$. Then $F \\sqbrk X$ is a principal ideal domain."}959{"_id": "1089", "title": "Equal Consecutive Prime Number Gaps are Multiples of Six", "text": "If you list the gaps between consecutive primes greater than $5$: :$2, 4, 2, 4, 2, 4, 6, 2, 6, 4, 2, 4, \\ldots$ you will notice that consecutive gaps that are equal are of the form $6 x$. This is ''always'' the case. {{OEIS|A001223}}"}960{"_id": "1090", "title": "Standard Discrete Metric is Metric", "text": "The standard discrete metric is a metric."}961{"_id": "1091", "title": "Derivative of Constant", "text": "Let $\\map {f_c} x$ be the constant function on $\\R$, where $c \\in \\R$. Then: :$\\map {f_c'} x = 0$"}962{"_id": "1092", "title": "Derivative of Identity Function", "text": "Let $X$ be either set of either the real numbers $\\R$ or the complex numbers $\\C$. Let $I_X: X \\to X$ be the identity function. Then: :$\\map {I_X'} x = 1$ where $\\map {I_X'} x$ denotes the derivative of $I_X$ {{WRT|Differentiation}} $x$. This can be presented for each of $\\R$ and $\\C$:"}963{"_id": "1093", "title": "Product Rule for Derivatives", "text": "Let $\\map f x, \\map j x, \\map k x$ be real functions defined on the open interval $I$. Let $\\xi \\in I$ be a point in $I$ at which both $j$ and $k$ are differentiable. Let $\\map f x = \\map j x \\map k x$. Then: :$\\map {f'} \\xi = \\map j \\xi \\map {k'} \\xi + \\map {j'} \\xi \\map k \\xi$ It follows from the definition of derivative that if $j$ and $k$ are both differentiable on the interval $I$, then: :$\\forall x \\in I: \\map {f'} x = \\map j x \\map {k'} x + \\map {j'} x \\map k x$ Using Leibniz's notation for derivatives, this can be written as: :$\\map {\\dfrac \\d {\\d x} } {y \\, z} = y \\dfrac {\\d z} {\\d x} + \\dfrac {\\d y} {\\d x} z$ where $y$ and $z$ represent functions of $x$."}964{"_id": "1094", "title": "Derivative of Composite Function", "text": "Let $f, g, h$ be continuous real functions such that: :$\\forall x \\in \\R: \\map h x = \\map {f \\circ g} x = \\map f {\\map g x}$ Then: :$\\map {h'} x = \\map {f'} {\\map g x} \\map {g'} x$ where $h'$ denotes the derivative of $h$. Using the $D_x$ notation: :$\\map {D_x} {\\map f {\\map g x} } = \\map {D_{\\map g x} } {\\map f {\\map g x} } \\, \\map {D_x} {\\map g x}$ This is often informally referred to as the '''chain rule (for differentiation)'''."}965{"_id": "1095", "title": "Derivative of Inverse Function", "text": "Let $I = \\closedint a b$ and $J = \\closedint c d$ be closed real intervals. Let $I^o = \\openint a b$ and $J^o = \\openint c d$ be the corresponding open real intervals. Let $f: I \\to J$ be a real function which is continuous on $I$ and differentiable on $I^o$ such that $J = f \\sqbrk I$. Let either: :$\\forall x \\in I^o: D \\map f x > 0$ or: :$\\forall x \\in I^o: D \\map f x < 0$ Then: :$f^{-1}: J \\to I$ exists and is continuous on $J$ :$f^{-1}$ is differentiable on $J^o$ :$\\forall y \\in J^o: D \\map {f^{-1} } y = \\dfrac 1 {D \\map f x}$"}966{"_id": "1096", "title": "Upper Sum Never Smaller than Lower Sum", "text": "Let $\\closedint a b$ be a closed interval of the set $\\R$ of real numbers. Let $P = \\set {x_0, x_1, x_2, \\ldots, x_{n - 1}, x_n}$ be a finite subdivision of $\\closedint a b$. Let $f: \\R \\to \\R$ be a real function. Let $f$ be bounded on $\\closedint a b$. Let $\\map L P$ be the lower sum of $\\map f x$ on $\\closedint a b$ belonging to the subdivision $P$. Let $\\map U P$ be the upper sum of $\\map f x$ on $\\closedint a b$ belonging to the subdivision $P$. Then $\\map L P \\le \\map U P$."}967{"_id": "1097", "title": "Dedekind's Theorem", "text": "Let $\\tuple {L, R}$ be a Dedekind cut of the set of real numbers $\\R$. Then there exists a unique real number which is a producer of $\\tuple {L, R}$."}968{"_id": "1098", "title": "Complex Addition is Closed", "text": "The set of complex numbers $\\C$ is closed under addition: :$\\forall z, w \\in \\C: z + w \\in \\C$"}969{"_id": "1099", "title": "Complex Addition is Associative", "text": "The operation of addition on the set of complex numbers $\\C$ is associative: :$\\forall z_1, z_2, z_3 \\in \\C: z_1 + \\paren {z_2 + z_3} = \\paren {z_1 + z_2} + z_3$"}970{"_id": "1100", "title": "Complex Addition is Commutative", "text": "The operation of addition on the set of complex numbers is commutative: :$\\forall z, w \\in \\C: z + w = w + z$"}971{"_id": "1101", "title": "Integers are Countably Infinite", "text": "The set $\\Z$ of integers is countably infinite."}972{"_id": "1102", "title": "Permutation of Determinant Indices", "text": "Let $\\mathbf A = \\sqbrk a_n$ be a square matrix of order $n$ over a field. Let $\\lambda: \\N_{> 0} \\to \\N_{> 0}$ be any fixed permutation on $\\N_{> 0}$. Let $\\map \\det {\\mathbf A}$ be the determinant of $\\mathbf A$. Let $\\struct {S_n, \\circ}$ be the symmetric group of $n$ letters.  Then: :$\\displaystyle \\map \\det {\\mathbf A} = \\sum_{\\mu \\mathop \\in S_n} \\paren {\\map \\sgn \\mu \\map \\sgn \\lambda \\prod_{k \\mathop = 1}^n a_{\\map \\lambda k, \\map \\mu k} }$ :$\\displaystyle \\map \\det {\\mathbf A} = \\sum_{\\mu \\mathop \\in S_n} \\paren {\\map \\sgn \\mu \\map \\sgn \\lambda \\prod_{k \\mathop = 1}^n a_{\\map \\mu k, \\map \\lambda k} }$ where: :the summation $\\displaystyle \\sum_{\\mu \\mathop \\in S_n}$ goes over all the $n!$ permutations of $\\set {1, 2, \\ldots, n}$ :$\\map \\sgn \\mu$ is the sign of the permutation $\\mu$."}973{"_id": "1103", "title": "Determinant of Transpose", "text": "Let $\\mathbf A = \\sqbrk a_n$ be a square matrix of order $n$. Let $\\map \\det {\\mathbf A}$ be the determinant of $\\mathbf A$. Let $\\mathbf A^\\intercal$ be the transpose of $\\mathbf A$. Then: :$\\map \\det {\\mathbf A} = \\map \\det {\\mathbf A^\\intercal}$"}974{"_id": "1104", "title": "Determinant with Rows Transposed", "text": "If two rows of a matrix with determinant $D$ are transposed, its determinant becomes $-D$."}975{"_id": "1105", "title": "Square Matrix with Duplicate Rows has Zero Determinant", "text": "If two rows of a square matrix over a commutative ring $\\struct {R, +, \\circ}$ are the same, then its determinant is zero."}976{"_id": "1106", "title": "Determinant with Row Multiplied by Constant", "text": "Let $\\mathbf A = \\sqbrk a_n$ be a square matrix of order $n$. Let $\\map \\det {\\mathbf A}$ be the determinant of $\\mathbf A$. Let $\\mathbf B$ be the matrix resulting from one row of $\\mathbf A$ having been multiplied by a constant $c$. Then: :$\\map \\det {\\mathbf B} = c \\map \\det {\\mathbf A}$ That is, multiplying one row of a square matrix by a constant multiplies its determinant by that constant."}977{"_id": "1107", "title": "Determinant as Sum of Determinants", "text": "Let $\\begin{vmatrix}   a_{11} & \\cdots & a_{1s} & \\cdots & a_{1n} \\\\ \\vdots & \\ddots & \\vdots & \\ddots & \\vdots \\\\   a_{r1} & \\cdots & a_{rs} & \\cdots & a_{rn} \\\\ \\vdots & \\ddots & \\vdots & \\ddots & \\vdots \\\\   a_{n1} & \\cdots & a_{ns} & \\cdots & a_{nn} \\end{vmatrix}$ be a determinant. Then $\\begin{vmatrix}   a_{11} & \\cdots & a_{1s} & \\cdots & a_{1n} \\\\ \\vdots & \\ddots & \\vdots & \\ddots & \\vdots \\\\   a_{r1} + a'_{r1} & \\cdots & a_{rs} + a'_{rs} & \\cdots & a_{rn} + a'_{rn} \\\\ \\vdots & \\ddots & \\vdots & \\ddots & \\vdots \\\\   a_{n1} & \\cdots & a_{ns} & \\cdots & a_{nn} \\end{vmatrix} = \\begin{vmatrix}   a_{11} & \\cdots & a_{1s} & \\cdots & a_{1n} \\\\ \\vdots & \\ddots & \\vdots & \\ddots & \\vdots \\\\   a_{r1} & \\cdots & a_{rs} & \\cdots & a_{rn} \\\\ \\vdots & \\ddots & \\vdots & \\ddots & \\vdots \\\\   a_{n1} & \\cdots & a_{ns} & \\cdots & a_{nn} \\end{vmatrix} + \\begin{vmatrix}   a_{11} & \\cdots & a_{1s} & \\cdots & a_{1n} \\\\ \\vdots & \\ddots & \\vdots & \\ddots & \\vdots \\\\   a'_{r1} & \\cdots & a'_{rs} & \\cdots & a'_{rn} \\\\ \\vdots & \\ddots & \\vdots & \\ddots & \\vdots \\\\   a_{n1} & \\cdots & a_{ns} & \\cdots & a_{nn} \\end{vmatrix}$. Similarly: Then $\\begin{vmatrix}   a_{11} & \\cdots & a_{1s} + a'_{1s} & \\cdots & a_{1n} \\\\ \\vdots & \\ddots & \\vdots & \\ddots & \\vdots \\\\   a_{r1} & \\cdots & a_{rs} + a'_{rs} & \\cdots & a_{rn} \\\\ \\vdots & \\ddots & \\vdots & \\ddots & \\vdots \\\\   a_{n1} & \\cdots & a_{ns} + a'_{ns} & \\cdots & a_{nn} \\end{vmatrix} = \\begin{vmatrix}   a_{11} & \\cdots & a_{1s} & \\cdots & a_{1n} \\\\ \\vdots & \\ddots & \\vdots & \\ddots & \\vdots \\\\   a_{r1} & \\cdots & a_{rs} & \\cdots & a_{rn} \\\\ \\vdots & \\ddots & \\vdots & \\ddots & \\vdots \\\\   a_{n1} & \\cdots & a_{ns} & \\cdots & a_{nn} \\end{vmatrix} + \\begin{vmatrix}   a_{11} & \\cdots & a'_{1s} & \\cdots & a_{1n} \\\\ \\vdots & \\ddots & \\vdots & \\ddots & \\vdots \\\\   a_{r1} & \\cdots & a'_{rs} & \\cdots & a_{rn} \\\\ \\vdots & \\ddots & \\vdots & \\ddots & \\vdots \\\\   a_{n1} & \\cdots & a'_{ns} & \\cdots & a_{nn} \\end{vmatrix}$."}978{"_id": "1108", "title": "Multiple of Row Added to Row of Determinant", "text": "Let $\\mathbf A = \\begin {bmatrix} a_{1 1} & a_{1 2} & \\cdots & a_{1 n} \\\\  \\vdots &  \\vdots & \\ddots &  \\vdots \\\\ a_{r 1} & a_{r 2} & \\cdots & a_{r n} \\\\  \\vdots &  \\vdots & \\ddots &  \\vdots \\\\ a_{s 1} & a_{s 2} & \\cdots & a_{s n} \\\\  \\vdots &  \\vdots & \\ddots &  \\vdots \\\\ a_{n 1} & a_{n 2} & \\cdots & a_{n n} \\\\ \\end {bmatrix}$ be a square matrix of order $n$. Let $\\map \\det {\\mathbf A}$ denote the determinant of $\\mathbf A$. Let $\\mathbf B = \\begin{bmatrix} a_{1 1}             & a_{1 2}             & \\cdots &             a_{1 n} \\\\  \\vdots             & \\vdots              & \\ddots &              \\vdots \\\\ a_{r 1} + k a_{s 1} & a_{r 2} + k a_{s 2} & \\cdots & a_{r n} + k a_{s n} \\\\  \\vdots             & \\vdots              & \\ddots &              \\vdots \\\\ a_{s 1}             & a_{s 2}             & \\cdots &             a_{s n} \\\\  \\vdots             & \\vdots              & \\ddots &              \\vdots \\\\ a_{n 1}             & a_{n 2}             & \\cdots &             a_{n n} \\\\ \\end{bmatrix}$. Then $\\map \\det {\\mathbf B} = \\map \\det {\\mathbf A}$. That is, the value of a determinant remains unchanged if a constant multiple of any row is added to any other row."}979{"_id": "1109", "title": "Determinant of Matrix Product", "text": "Let $\\mathbf A = \\sqbrk a_n$ and $\\mathbf B = \\sqbrk b_n$ be a square matrices of order $n$. Let $\\map \\det {\\mathbf A}$ be the determinant of $\\mathbf A$. Let $\\mathbf A \\mathbf B$ be the (conventional) matrix product of $\\mathbf A$ and $\\mathbf B$. Then: :$\\map \\det {\\mathbf A \\mathbf B} = \\map \\det {\\mathbf A} \\map \\det {\\mathbf B}$ That is, the determinant of the product is equal to the product of the determinants."}980{"_id": "1110", "title": "Expansion Theorem for Determinants", "text": "Let $\\mathbf A = \\sqbrk a_n$ be a square matrix of order $n$. Let $D = \\map \\det {\\mathbf A}$ be the determinant of $\\mathbf A$: :$\\displaystyle \\map \\det {\\mathbf A} := \\sum_{\\lambda} \\paren {\\map \\sgn \\lambda \\prod_{k \\mathop = 1}^n a_{k \\map \\lambda k} } = \\sum_\\lambda \\map \\sgn \\lambda a_{1 \\map \\lambda 1} a_{2 \\map \\lambda 2} \\cdots a_{n \\map \\lambda n}$ where: :the summation $\\displaystyle \\sum_\\lambda$ goes over all the $n!$ permutations of $\\set {1, 2, \\ldots, n}$ :$\\map \\sgn \\lambda$ is the sign of the permutation $\\lambda$. Let $a_{p q}$ be an element of $\\mathbf A$. Let $A_{p q}$ be the cofactor of $a_{p q}$ in $D$. Then: :$(1): \\quad \\displaystyle \\forall r \\in \\closedint 1 n: D = \\sum_{k \\mathop = 1}^n a_{r k} A_{r k}$ :$(2): \\quad \\displaystyle \\forall r \\in \\closedint 1 n: D = \\sum_{k \\mathop = 1}^n a_{k r} A_{k r}$ Thus the value of a determinant can be found either by: :multiplying all the elements in a row by their cofactors and adding up the products or: :multiplying all the elements in a column by their cofactors and adding up the products. The identity: :$\\displaystyle D = \\sum_{k \\mathop = 1}^n a_{r k} A_{r k}$ is known as the '''expansion of $D$ in terms of row $r$''', while: :$\\displaystyle D = \\sum_{k \\mathop = 1}^n a_{k r} A_{k r}$ is known as the '''expansion of $D$ in terms of column $r$'''."}981{"_id": "1111", "title": "Determinant with Unit Element in Otherwise Zero Row", "text": "Let $D$ be the determinant: :$D = \\begin {vmatrix}       1 &       0 & \\cdots &       0 \\\\ b_{2 1} & b_{2 2} & \\cdots & b_{2 n} \\\\  \\vdots &  \\vdots & \\ddots &  \\vdots \\\\ b_{n 1} & b_{n 2} & \\cdots & b_{n n} \\end {vmatrix}$ Then: :$D = \\begin {vmatrix} b_{2 2} & \\cdots & b_{2 n} \\\\  \\vdots & \\ddots &  \\vdots \\\\ b_{n 2} & \\cdots & b_{n n} \\end {vmatrix}$"}982{"_id": "1112", "title": "Vandermonde Determinant", "text": "The '''Vandermonde determinant of order $n$''' is the determinant defined as follows: :$V_n = \\begin {vmatrix} 1 & x_1 & x_1^2 & \\cdots & x_1^{n - 2} & x_1^{n - 1} \\\\ 1 & x_2 & x_2^2 & \\cdots & x_2^{n - 2} & x_2^{n - 1} \\\\ \\vdots & \\vdots & \\vdots & \\ddots & \\vdots & \\vdots \\\\ 1 & x_n & x_n^2 & \\cdots & x_n^{n - 2} & x_n^{n - 1} \\end {vmatrix}$ Its value is given by: :$\\displaystyle V_n = \\prod_{1 \\mathop \\le i \\mathop < j \\mathop \\le n} \\paren {x_j - x_i}$"}983{"_id": "1114", "title": "Equality of Polynomials", "text": "$f$ and $g$ are equal as polynomials {{iff}} $f$ and $g$ are equal as functions. Thus we can say $f = g$ without ambiguity as to what it means. {{explain|In the exposition, the term was \"equal as forms\", but it has now morphed into \"equal as polynomials\". Needs to be resolved.}}"}984{"_id": "1115", "title": "Value of Adjugate of Determinant", "text": "Let $D$ be the determinant of order $n$. Let $D^*$ be the adjugate of $D$. Then $D^* = D^{n-1}$."}985{"_id": "1116", "title": "Determinant of Diagonal Matrix", "text": "Let $\\mathbf A = \\begin{bmatrix} a_{11} & 0 & \\cdots & 0 \\\\ 0 & a_{22} & \\cdots & 0 \\\\ \\vdots & \\vdots & \\ddots & \\vdots \\\\ 0 & 0 & \\cdots & a_{nn} \\\\ \\end{bmatrix}$ be a diagonal matrix. Then the determinant of $\\mathbf A$ is the product of the elements of $\\mathbf A$. That is: :$\\ds \\map \\det {\\mathbf A} = \\prod_{i \\mathop = 1}^n a_{ii}$"}986{"_id": "1119", "title": "Matrix is Row Equivalent to Reduced Echelon Matrix", "text": "Let $\\mathbf A = \\sqbrk a_{m n}$ be a matrix of order $m \\times n$ over a field $F$. Then $A$ is row equivalent to a reduced echelon matrix of order $m \\times n$."}987{"_id": "1120", "title": "Square Matrix is Row Equivalent to Triangular Matrix", "text": "Let $\\mathbf A = \\left[{a}\\right]_n$ be a square matrix of order $n$ over a commutative ring $R$. Then $\\mathbf A$ can be converted to an upper or lower triangular matrix by elementary row operations."}988{"_id": "1121", "title": "Product of Triangular Matrices", "text": "Let $\\mathbf A = \\sqbrk a_n, \\mathbf B = \\sqbrk b_n$ be upper triangular matrices of order $n$. Let $\\mathbf C = \\mathbf A \\mathbf B$. Then :$(1): \\quad$ the diagonal elements of $\\mathbf C$ are given by: ::::$\\forall j \\in \\closedint 1 n: c_{j j} = a_{j j} b_{j j}$ :::That is, the diagonal elements of $\\mathbf C$ are those of the factor matrices multiplied together. :$(2): \\quad$ The matrix $\\mathbf C$ is itself upper triangular. The same applies if both $\\mathbf A$ and $\\mathbf B$ are lower triangular matrices."}989{"_id": "1122", "title": "Effect of Elementary Row Operations on Determinant", "text": "Let $\\mathbf A = \\sqbrk a_n$ be a square matrix of order $n$. Let $\\map \\det {\\mathbf A}$ denote the determinant of $\\mathbf A$. Take the elementary row operations: {{begin-axiom}} {{axiom | n = \\text {ERO} 1         | t = For some $\\lambda$, multiply row $i$ by $\\lambda$         | m = r_i \\to \\lambda r_i }} {{axiom | n = \\text {ERO} 2         | t = For some $\\lambda$, add $\\lambda$ times row $j$ to row $i$         | m = r_i \\to r_i + \\lambda r_j }} {{axiom | n = \\text {ERO} 3         | t = Exchange rows $i$ and $j$         | m = r_i \\leftrightarrow r_j }} {{end-axiom}} Applying $\\text {ERO} 1$ has the effect of multiplying $\\map \\det {\\mathbf A}$ by $\\lambda$. Applying $\\text {ERO} 2$ has no effect on $\\map \\det {\\mathbf A}$. Applying $\\text {ERO} 3$ has the effect of multiplying $\\map \\det {\\mathbf A}$ by $-1$."}990{"_id": "1123", "title": "Modulus in Terms of Conjugate", "text": "Let $z = a + i b$ be a complex number. Let $\\cmod z$ be the modulus of $z$. Let $\\overline z$ be the conjugate of $z$. Then: :$\\cmod z^2 = z \\overline z$"}991{"_id": "1124", "title": "Existence and Uniqueness of Positive Root of Positive Real Number", "text": "Let $x \\in \\R$ be a real number such that $x \\ge 0$. Let $n \\in \\Z$ be an integer such that $n \\ne 0$. Then there always exists a unique $y \\in \\R: \\paren {y \\ge 0} \\land \\paren {y^n = x}$. Hence the justification for the terminology '''the positive $n$th root of $x$''' and the notation $x^{1/n}$."}992{"_id": "1125", "title": "Triangle Inequality", "text": "=== Geometry === {{:Triangle Inequality/Geometry}} === Real Numbers === {{:Triangle Inequality/Real Numbers}} === Complex Numbers === {{:Triangle Inequality/Complex Numbers}}"}993{"_id": "1126", "title": "Even Power is Non-Negative", "text": "Let $x \\in \\R$ be a real number. Let $n \\in \\Z$ be an even integer. Then $x^n \\ge 0$. That is, all even powers are positive."}994{"_id": "1127", "title": "Sign of Odd Power", "text": "Let $x \\in \\R$ be a real number. Let $n \\in \\Z$ be an odd integer. Then: :$x^n = 0 \\iff x = 0$ :$x^n > 0 \\iff x > 0$ :$x^n < 0 \\iff x < 0$ That is, the sign of an odd power matches the number it is a power of."}995{"_id": "1128", "title": "Product of Absolute Values on Ordered Integral Domain", "text": "Let $\\struct {D, +, \\times, \\le}$ be an ordered integral domain whose zero is denoted by $0_D$. For all $a \\in D$, let $\\size a$ denote the absolute value of $a$. Then: :$\\size a \\times \\size b = \\size {a \\times b}$"}996{"_id": "1129", "title": "Negative of Absolute Value", "text": "Let $x \\in \\R$ be a real number. Let $\\size x$ denote the absolute value of $x$. Then: :$-\\size x \\le x \\le \\size x$"}997{"_id": "1130", "title": "Order of Squares in Ordered Ring", "text": "Let $\\struct {R, +, \\circ, \\le}$ be an ordered ring whose zero is $0_R$ and whose unity is $1_R$. Let $x, y \\in \\struct {R, +, \\circ, \\le}$ such that $0_R \\le x, y$. Then: :$x \\le y \\implies x \\circ x \\le y \\circ y$ When $R$ is one of the standard sets of numbers, that is $\\Z, \\Q, \\R$, then this translates into: :If $x, y$ are positive then $x \\le y \\implies x^2 \\le y^2$."}998{"_id": "1131", "title": "Continuum Property", "text": "Let $S \\subset \\R$ be a non-empty subset of the set of real numbers such that $S$ is bounded above. Then $S$ admits a supremum in $\\R$. This is known as the '''least upper bound property''' of the real numbers. Similarly, let $S \\subset \\R$ be a non-empty subset of the set of real numbers such that $S$ is bounded below. Then $S$ admits an infimum in $\\R$. This is sometimes called the '''greatest lower bound property''' of the real numbers. The two properties taken together are called the '''continuum property of $\\R$'''. This can also be stated as: :The set $\\R$ of real numbers is Dedekind complete."}999{"_id": "1132", "title": "Complex Multiplication is Closed", "text": "The set of complex numbers $\\C$ is closed under multiplication: :$\\forall z, w \\in \\C: z \\times w \\in \\C$"}1000{"_id": "1133", "title": "Complex Multiplication is Associative", "text": "The operation of multiplication on the set of complex numbers $\\C$ is associative: :$\\forall z_1, z_2, z_3 \\in \\C: z_1 \\paren {z_2 z_3} = \\paren {z_1 z_2} z_3$"}1001{"_id": "1134", "title": "Arithmetic Mean is Never Less than Harmonic Mean", "text": "Let $x_1, x_2, \\ldots, x_n \\in \\R_{> 0}$ be strictly positive real numbers. Let $A_n $ be the arithmetic mean of $x_1, x_2, \\ldots, x_n$. Let $H_n$ be the harmonic mean of $x_1, x_2, \\ldots, x_n$. Then $A_n \\ge H_n$."}1002{"_id": "1135", "title": "Multiple of Supremum", "text": "Let $S \\subseteq \\R: S \\ne \\varnothing$ be a non-empty subset of the set of real numbers $\\R$. Let $S$ be bounded above. Let $z \\in \\R: z > 0$ be a positive real number. Then: :$\\displaystyle \\map {\\sup_{x \\mathop \\in S} } {z x} = z \\map {\\sup_{x \\mathop \\in S} } x$"}1003{"_id": "1136", "title": "Cauchy's Mean Theorem", "text": "Let $x_1, x_2, \\ldots, x_n \\in \\R$ be real numbers which are all positive. Let $A_n$ be the arithmetic mean of $x_1, x_2, \\ldots, x_n$. Let $G_n$ be the geometric mean of $x_1, x_2, \\ldots, x_n$. Then: :$A_n \\ge G_n$ with equality holding {{iff}}: :$\\forall i, j \\in \\set {1, 2, \\ldots, n}: x_i = x_j$ That is, {{iff}} all terms are equal."}1004{"_id": "1137", "title": "Distance on Real Numbers is Metric", "text": "Let $x, y \\in \\R$ be real numbers. Let $\\map d {x, y}$ be the distance between $x$ and $y$: :$\\map d {x, y} = \\size {x - y}$ Then $\\map d {x, y}$ is a metric on $\\R$. Thus it follows that $\\tuple {\\R, d}$ is a metric space."}1005{"_id": "1138", "title": "Combination Theorem for Sequences/Real", "text": "Let $\\sequence {x_n}$ and $\\sequence {y_n}$ be sequences in $\\R$. Let $\\sequence {x_n}$ and $\\sequence {y_n}$ be convergent to the following limits: :$\\displaystyle \\lim_{n \\mathop \\to \\infty} x_n = l$ :$\\displaystyle \\lim_{n \\mathop \\to \\infty} y_n = m$ Let $\\lambda, \\mu \\in \\R$. Then the following results hold: === Sum Rule === {{:Combination Theorem for Sequences/Real/Sum Rule}} === Difference Rule === {{:Combination Theorem for Sequences/Real/Difference Rule}} === Multiple Rule === {{:Combination Theorem for Sequences/Real/Multiple Rule}} === Combined Sum Rule === {{:Combination Theorem for Sequences/Real/Combined Sum Rule}} === Product Rule === {{:Combination Theorem for Sequences/Real/Product Rule}} === Quotient Rule === {{:Combination Theorem for Sequences/Real/Quotient Rule}}"}1006{"_id": "1139", "title": "Convergent Sequence in Metric Space is Bounded", "text": "Let $M = \\struct {A, d}$ be a metric space. Let $\\sequence {x_n}$ be a sequence in $M$ which is convergent, and so $x_n \\to l$ as $n \\to \\infty$. Then $\\sequence {x_n}$ is bounded."}1007{"_id": "1140", "title": "Convergent Sequence Minus Limit", "text": "Let $X$ be  one of the standard number fields $\\Q, \\R, \\C$. Let $\\left \\langle {x_n} \\right \\rangle$ be a sequence in $X$ which converges to $l$. That is: : $\\displaystyle \\lim_{n \\mathop \\to \\infty} x_n = l$ Then: : $\\displaystyle \\lim_{n \\mathop \\to \\infty} \\left|{x_n - l}\\right| = 0$"}1008{"_id": "1141", "title": "Monotone Convergence Theorem (Real Analysis)", "text": "Every bounded monotone sequence is convergent."}1009{"_id": "1142", "title": "One Plus Reciprocal to the Nth", "text": "Let $\\sequence {x_n}$ be the sequence in $\\R$ defined as $x_n = \\paren {1 + \\dfrac 1 n}^n$. Then $\\sequence {x_n}$ converges to a limit as $n$ increases without bound."}1010{"_id": "1143", "title": "Between two Real Numbers exists Rational Number", "text": "Let $a, b \\in \\R$ be real numbers such that $a < b$. Then: : $\\exists r \\in \\Q: a < r < b$"}1011{"_id": "1144", "title": "Power over Factorial", "text": "Let $x \\in \\R: x > 0$ be a positive real number. Let $\\sequence {x_n}$ be the sequence in $\\R$ defined as $x_n = \\dfrac {x^n} {n!}$. Then $\\sequence {x_n}$ converges to zero."}1012{"_id": "1145", "title": "Sequence of Powers of Reciprocals is Null Sequence", "text": "Let $r \\in \\Q_{>0}$ be a strictly positive rational number. Let $\\sequence {x_n}$ be the sequence in $\\R$ defined as: : $x_n = \\dfrac 1 {n^r}$ Then $\\sequence {x_n}$ is a null sequence."}1013{"_id": "1146", "title": "Euler's Number: Limit of Sequence implies Limit of Series", "text": "Let Euler's number $e$ be defined as: :$\\displaystyle e := \\lim_{n \\to \\infty} \\left({1 + \\frac 1 n}\\right)^n$ Then: :$\\displaystyle e = \\sum_{k \\mathop \\ge 0} \\frac 1 {k!}$ That is: :$\\displaystyle e = \\frac 1 {0!} + \\frac 1 {1!} + \\frac 1 {2!} + \\frac 1 {3!} + \\frac 1 {4!} \\cdots$"}1014{"_id": "1147", "title": "Degree of Field Extensions is Multiplicative", "text": "Let $E, K$ and $F$ be fields. Let $E / K$ and $K / F$ be finite field extensions. Then $E / F$ is a finite field extension, and: :$\\index E F = \\index E K \\index K F$ where $\\index E F$ denotes the degree of $E / F$"}1015{"_id": "1148", "title": "Convergent Sequence in Metric Space has Unique Limit", "text": "Let $\\left({X, d}\\right)$ be a metric space. Let $\\left \\langle {x_n} \\right \\rangle$ be a sequence in $\\left({X, d}\\right)$. Then $\\left \\langle {x_n} \\right \\rangle$ can have at most one limit."}1016{"_id": "1149", "title": "Lower and Upper Bounds for Sequences", "text": "Let $\\sequence {x_n}$ be a sequence in $\\R$. Let $x_n \\to l$ as $n \\to \\infty$. Then: :$(1): \\quad \\forall n \\in \\N: x_n \\ge a \\implies l \\ge a$ :$(2): \\quad \\forall n \\in \\N: x_n \\le b \\implies l \\le b$"}1017{"_id": "1151", "title": "Ring Homomorphism from Division Ring is Monomorphism or Zero Homomorphism", "text": "Let $\\struct {R, +_R, \\circ}$ and $\\struct {S, +_S, *}$ be rings whose zeros are $0_R$ and $0_S$ respectively. Let $\\phi: R \\to S$ be a ring homomorphism. If $R$ is a division ring, then either: : $(1): \\quad \\phi$ is a monomorphism (that is, $\\phi$ is injective) : $(2): \\quad \\phi$ is the zero homomorphism (that is, $\\forall a \\in R: \\map \\phi a = 0_S$)."}1018{"_id": "1152", "title": "Kernel of Ring Epimorphism is Ideal", "text": "Let $\\phi: \\struct {R_1, +_1, \\circ_1} \\to \\struct {R_2, +_2, \\circ_2}$ be a ring epimorphism. Then: :The kernel of $\\phi$ is an ideal of $R_1$. :There is a unique ring isomorphism $g: R_1 / K \\to R_2$ such that: ::$g \\circ q_K = \\phi$ :$\\phi$ is a ring isomorphism {{iff}} $K = \\set {0_{R_1} }$."}1019{"_id": "1153", "title": "Unbounded Monotone Sequence Diverges to Infinity", "text": "Let $\\sequence {x_n}$ be a sequence in $\\R$. Let $\\sequence {x_n}$ be monotone, that is either increasing or decreasing."}1020{"_id": "1154", "title": "Reciprocal of Null Sequence", "text": "Let $\\sequence {x_n}$ be a sequence in $\\R$. Let $\\forall n \\in \\N: x_n > 0$. Then: :$x_n \\to 0$ as $n \\to \\infty$ {{iff}} $\\size {\\dfrac 1 {x_n} } \\to \\infty$ as $n \\to \\infty$"}1021{"_id": "1155", "title": "Modulus of Limit", "text": "Let $X$ be one of the standard number fields $\\Q, \\R, \\C$. Let $\\sequence {x_n}$ be a sequence in $X$. Let $\\sequence {x_n}$ be convergent to the limit $l$. That is, let $\\displaystyle \\lim_{n \\mathop \\to \\infty} x_n = l$. Then :$\\displaystyle \\lim_{n \\mathop \\to \\infty} \\cmod {x_n} = \\cmod l$ where $\\cmod {x_n}$ is the modulus of $x_n$."}1022{"_id": "1157", "title": "Strictly Increasing Sequence of Natural Numbers", "text": "Let $\\N_{>0}$ be the set of natural numbers without zero: : $\\N_{>0} = \\left\\{{1, 2, 3, \\ldots}\\right\\}$ Let $\\left \\langle {n_r} \\right \\rangle$ be strictly increasing sequence in $\\N_{>0}$. Then: : $\\forall r \\in \\N_{>0}: n_r \\ge r$"}1023{"_id": "1158", "title": "Limit of Subsequence equals Limit of Sequence", "text": "Let $T = \\struct {S, \\tau}$ be a topological space. Let $\\sequence {x_n}$ be a sequence in $T$. Let $l \\in S$ such that: :$\\displaystyle \\lim_{n \\mathop \\to \\infty} x_n = l$ Let $\\sequence {x_{n_r} }$ be a subsequence of $\\sequence {x_n}$. Then: :$\\displaystyle \\lim_{r \\mathop \\to \\infty} x_{n_r} = l$ That is, the limit of a convergent sequence in a topological space equals the limit of any subsequence of it."}1024{"_id": "1159", "title": "Root of Number Greater than One", "text": "Let $x \\in \\R$ be a real number. Let $n \\in \\N^*$ be a natural number such that $n > 0$. Then $x \\ge 1 \\implies x^{1/n} \\ge 1$ where $x^{1/n}$ is the $n$th root of $x$."}1025{"_id": "1160", "title": "Limit of Root of Positive Real Number", "text": "Let $x \\in \\R: x > 0$ be a real number. Let $\\sequence {x_n}$ be the sequence in $\\R$ defined as: :$x_n = x^{1 / n}$ Then $x_n \\to 1$ as $n \\to \\infty$."}1026{"_id": "1161", "title": "Hero's Method", "text": "Let $a \\in \\R$ be a real number such that $a > 0$. Let $x_1 \\in \\R$ be a real number such that $x_1 > 0$. Let $\\sequence {x_n}$ be the sequence in $\\R$ defined recursively by: :$\\forall n \\in \\N_{>0}: x_{n + 1} = \\dfrac {x_n + \\dfrac a {x_n} } 2$ Then $x_n \\to \\sqrt a$ as $n \\to \\infty$."}1027{"_id": "1162", "title": "Limit of Integer to Reciprocal Power", "text": "Let $\\sequence {x_n}$ be the real sequence defined as $x_n = n^{1/n}$, using exponentiation. Then $\\sequence {x_n}$ converges with a limit of $1$."}1028{"_id": "1163", "title": "Difference Between Adjacent Square Roots Converges", "text": "Let $\\sequence {x_n}$ be the sequence in $\\R$ defined as $x_n = \\sqrt {n + 1} - \\sqrt n$. Then $\\sequence {x_n}$ converges to a zero limit."}1029{"_id": "1164", "title": "Peak Point Lemma", "text": "Let $\\sequence {x_n}$ be a sequence in $\\R$ which is infinite. Then $\\sequence {x_n}$ has an infinite subsequence which is monotone."}1030{"_id": "1165", "title": "Bolzano-Weierstrass Theorem", "text": "Every bounded sequence of real numbers has a convergent subsequence."}1031{"_id": "1166", "title": "Existence of Maximum and Minimum of Bounded Sequence", "text": "Let $\\sequence {x_n}$ be a bounded sequence in $\\R$ (which may or may not be convergent). Let $L$ be the set of all real numbers which are the limit of some subsequence of $\\sequence {x_n}$. Then $L$ has both a maximum and a minimum."}1032{"_id": "1170", "title": "Interval Defined by Absolute Value", "text": "Let $\\xi, \\delta \\in \\R$ be real numbers. Let $\\delta > 0$. Then:"}1033{"_id": "1171", "title": "Difference of Two Powers", "text": "Let $\\mathbb F$ denote one of the standard number systems, that is $\\Z$, $\\Q$, $\\R$ and $\\C$. Let $n \\in \\N$ such that $n \\ge 2$. Then for all $a, b \\in \\mathbb F$: {{begin-eqn}} {{eqn | l = a^n - b^n       | r = \\paren {a - b} \\sum_{j \\mathop = 0}^{n - 1} a^{n - j - 1} b^j       | c =  }} {{eqn | r = \\paren {a - b} \\paren {a^{n - 1} + a^{n - 2} b + a^{n - 3} b^2 + \\dotsb + a b^{n - 2} + b^{n - 1} }       | c =  }} {{end-eqn}}"}1034{"_id": "1172", "title": "Polynomial Factor Theorem", "text": "Let $P \\paren x$ be a polynomial in $x$ over a field $K$ of degree $n$. Then: :$\\xi \\in K: P \\paren \\xi = 0 \\iff \\map P x = \\paren {x - \\xi} \\map Q x$ where $Q$ is a polynomial of degree $n - 1$. Hence, if $\\xi_1, \\xi_2, \\ldots, \\xi_n \\in K$ such that all are different, and $\\map P {\\xi_1} = \\map P {\\xi_2} = \\dotsb = \\map P {\\xi_n} = 0$, then: :$\\displaystyle \\map P x = k \\prod_{j \\mathop = 1}^n \\paren {x - \\xi_j}$ where $k \\in K$."}1035{"_id": "1173", "title": "Telescoping Series/Example 1", "text": "Let $\\left \\langle {b_n} \\right \\rangle$ be a sequence in $\\R$. Let $\\left \\langle {a_n} \\right \\rangle$ be a sequence whose terms are defined as: :$a_k = b_k - b_{k + 1}$ Then: :$\\displaystyle \\sum_{k \\mathop = 1}^n a_k = b_1 - b_{n + 1}$"}1036{"_id": "1174", "title": "Terms in Convergent Series Converge to Zero", "text": "Let $\\sequence {a_n}$ be a sequence in any of the standard number fields $\\Q$, $\\R$, or $\\C$. Suppose that the series $\\displaystyle \\sum_{n \\mathop = 1}^\\infty a_n$ converges in any of the standard number fields $\\Q$, $\\R$, or $\\C$. Then: :$\\displaystyle \\lim_{n \\mathop \\to \\infty} a_n = 0$ {{expand|Expand (on a different page) to Banach spaces}}"}1037{"_id": "1175", "title": "Linear Combination of Convergent Series", "text": "Let $\\sequence {a_n}_{n \\mathop \\ge 1}$ and $\\sequence {b_n}_{n \\mathop \\ge 1}$ be sequences of real numbers. Let the two series $\\displaystyle \\sum_{n \\mathop = 1}^\\infty a_n$ and $\\displaystyle \\sum_{n \\mathop = 1}^\\infty b_n$ converge to $\\alpha$ and $\\beta$ respectively. Let $\\lambda, \\mu \\in \\R$ be real numbers. Then the series $\\displaystyle \\sum_{n \\mathop = 1}^\\infty \\paren {\\lambda a_n + \\mu b_n}$ converges to $\\lambda \\alpha + \\mu \\beta$."}1038{"_id": "1176", "title": "Abel's Theorem", "text": "Let $\\displaystyle \\sum_{k \\mathop = 0}^\\infty a_k$ be a convergent series in $\\R$. Then: :$\\displaystyle \\lim_{x \\mathop \\to 1^-} \\paren {\\sum_{k \\mathop = 0}^\\infty a_k x^k} = \\sum_{k \\mathop = 0}^\\infty a_k$ where $\\displaystyle \\lim_{x \\mathop \\to 1^-}$ denotes the limit from the left."}1039{"_id": "1177", "title": "Alternating Series Test", "text": "Let $\\sequence {a_n}_{N \\mathop \\ge 0}$ be a decreasing sequence of positive terms in $\\R$ which converges with a limit of zero. That is, let $\\forall n \\in \\N: a_n \\ge 0, a_{n + 1} \\le a_n, a_n \\to 0$ as $n \\to \\infty$ Then the series: :$\\displaystyle \\sum_{n \\mathop = 1}^\\infty \\paren {-1}^{n - 1} a_n = a_1 - a_2 + a_3 - a_4 + \\dotsb$  converges."}1040{"_id": "1178", "title": "Euler's Identity", "text": ":$e^{i \\pi} + 1 = 0$"}1041{"_id": "1179", "title": "Intermediate Value Theorem", "text": "Let $f: S \\to \\R$ be a real function on some subset $S$ of $\\R$. Let $I \\subseteq S$ be a real interval. Let $f: I \\to \\R$ be continuous on $I$. Let $a, b \\in I$. Let $k \\in \\R$ lie between $\\map f a$ and $\\map f b$. That is, either: :$\\map f a < k < \\map f b$ or: :$\\map f b < k < \\map f a$ Then $\\exists c \\in \\openint a b$ such that $\\map f c = k$."}1042{"_id": "1180", "title": "Tail of Convergent Series tends to Zero", "text": "Let $\\sequence {a_n}_{n \\mathop \\ge 1}$ be a sequence of real numbers. Let $\\displaystyle \\sum_{n \\mathop = 1}^\\infty a_n$ be a convergent series. Let $N \\in \\N_{\\ge 1}$ be a natural number. Let $\\displaystyle \\sum_{n \\mathop = N}^\\infty a_n$ be the tail of the series $\\displaystyle \\sum_{n \\mathop = 1}^\\infty a_n$. Then: :$\\displaystyle \\sum_{n \\mathop = N}^\\infty a_n$ is convergent :$\\displaystyle \\sum_{n \\mathop = N}^\\infty a_n \\to 0$ as $N \\to \\infty$. That is, the tail of a convergent series tends to zero."}1043{"_id": "1181", "title": "Comparison Test", "text": "Let $\\displaystyle \\sum_{n \\mathop = 1}^\\infty b_n$ be a convergent series of positive real numbers. Let $\\sequence {a_n}$ be a sequence $\\R$ or sequence in $\\C$. Let $\\forall n \\in \\N_{>0}: \\cmod {a_n} \\le b_n$. Then the series $\\displaystyle \\sum_{n \\mathop = 1}^\\infty a_n$ converges absolutely."}1044{"_id": "1182", "title": "Ratio Test", "text": "Let $\\displaystyle \\sum_{n \\mathop = 1}^\\infty a_n$ be a series of real numbers in $\\R$, or a series of complex numbers in $\\C$. Let the sequence $\\sequence {a_n}$ satisfy: :$\\displaystyle \\lim_{n \\mathop \\to \\infty} \\size {\\frac {a_{n + 1} } {a_n} } = l$ :If $l > 1 $, then $\\displaystyle \\sum_{n \\mathop = 1}^\\infty a_n$ diverges. :If $l < 1 $, then $\\displaystyle \\sum_{n \\mathop = 1}^\\infty a_n$ converges absolutely."}1045{"_id": "1183", "title": "Limit of Subsequence of Bounded Sequence", "text": "Let $\\sequence {x_n}$ be a sequence in $\\R$. Let $\\sequence {x_n}$ be bounded. Let $b \\in \\R$ be a real number. Suppose that $\\forall N: \\exists n > N: x_n \\ge b$. Then $\\sequence {x_n}$ has a subsequence which converges to a limit $l \\ge b$."}1046{"_id": "1184", "title": "Terms of Bounded Sequence Within Bounds", "text": "Let $\\sequence {x_n}$ be a sequence in $\\R$. Let $\\sequence {x_n}$ be bounded. Let the limit superior of $\\sequence {x_n}$ be $\\overline l$. Let the limit inferior of $\\sequence {x_n}$ be $\\underline l$. Then: :$\\forall \\epsilon > 0: \\exists N: \\forall n > N: x_n < \\overline l + \\epsilon$ :$\\forall \\epsilon > 0: \\exists N: \\forall n > N: x_n > \\underline l - \\epsilon$"}1047{"_id": "1185", "title": "Convergence of Limsup and Liminf", "text": "Let $\\sequence {x_n}$ be a sequence in $\\R$. Let the limit superior of $\\sequence {x_n}$ be $\\overline l$. Let the limit inferior of $\\sequence {x_n}$ be $\\underline l$. Then $\\left \\langle {x_n} \\right \\rangle$ converges to a limit $l$ {{iff}} $\\overline l = \\underline l = l$. Hence a bounded real sequence converges {{iff}} all its convergent subsequences have the same limit."}1048{"_id": "1186", "title": "Limsup and Liminf are Limits of Bounds", "text": "Let $\\sequence {x_n}$ be a sequence in $\\R$. Let $\\sequence {x_n}$ be bounded. Let $\\displaystyle \\overline l = \\limsup_{n \\mathop \\to \\infty} x_n$ be the limit superior and $\\displaystyle \\liminf_{n \\mathop \\to \\infty} x_n$ the limit inferior of $\\sequence {x_n}$. Then: :$\\displaystyle \\overline l = \\limsup_{n \\mathop \\to \\infty} x_n = \\map {\\lim_{n \\mathop \\to \\infty} } {\\sup_{k \\mathop \\ge n} x_k}$ :$\\displaystyle \\underline l = \\liminf_{n \\mathop \\to \\infty} x_n = \\map {\\lim_{n \\mathop \\to \\infty} } {\\inf_{k \\mathop \\ge n} x_k}$"}1049{"_id": "1187", "title": "Nth Root Test", "text": "Let $\\displaystyle \\sum_{n \\mathop = 1}^\\infty a_n$ be a series of real numbers $\\R$ or complex numbers $\\C$. Let the sequence $\\sequence {a_n}$ be such that the limit superior $\\displaystyle \\limsup_{n \\mathop \\to \\infty} \\size {a_n}^{1/n} = l$. Then: :If $l > 1$, the series $\\displaystyle \\sum_{n \\mathop = 1}^\\infty a_n$ diverges. :If $l < 1$, the series $\\displaystyle \\sum_{n \\mathop = 1}^\\infty a_n$ converges absolutely."}1050{"_id": "1188", "title": "Series of Power over Factorial Converges", "text": "The series $\\displaystyle \\sum_{n \\mathop = 0}^\\infty \\frac {x^n} {n!}$ converges for all real values of $x$."}1051{"_id": "1189", "title": "Area of Triangle in Terms of Side and Altitude", "text": "The area of a triangle $\\triangle ABC$ is given by: :$\\dfrac {c \\cdot h_c} 2 = \\dfrac {b \\cdot h_b} 2 = \\dfrac {a \\cdot h_a} 2$ where: :$a, b, c$ are the sides :$h_a, h_b, h_c$ are the altitudes from $A$, $B$ and $C$ respectively."}1052{"_id": "1190", "title": "Absolutely Convergent Series is Convergent", "text": "Let $V$ be a Banach space with norm $\\norm {\\, \\cdot \\,}$. Let $\\displaystyle \\sum_{n \\mathop = 1}^\\infty a_n$ be an absolutely convergent series in $V$. Then $\\displaystyle \\sum_{n \\mathop = 1}^\\infty a_n$ is convergent."}1053{"_id": "1191", "title": "Limit Comparison Test", "text": "Let $\\left \\langle {a_n} \\right \\rangle$ and $\\left \\langle {b_n} \\right \\rangle$ be sequences in $\\R$. Let $\\displaystyle \\frac {a_n}{b_n} \\to l$ as $n \\to \\infty$ where $l \\in \\R_{>0}$. Then the series $\\displaystyle \\sum_{n \\mathop = 1}^\\infty a_n$ and $\\displaystyle \\sum_{n \\mathop = 1}^\\infty b_n$ are either both convergent or both divergent."}1054{"_id": "1192", "title": "Area of Parallelogram", "text": "The area of a parallelogram equals the product of one of its bases and the associated altitude."}1055{"_id": "1193", "title": "Area of Triangle in Terms of Inradius and Exradii", "text": "The area of a $\\triangle ABC$ is given by the formula: :$(ABC) = \\rho_a \\left({s - a}\\right) = \\rho_b \\left({s - b}\\right) = \\rho_c \\left({s - c}\\right) = \\rho s = \\sqrt {\\rho_a \\rho_b \\rho_c \\rho}$ where: :$s$ is the semiperimeter :$I$ is the incenter :$\\rho$ is the inradius :$I_a, I_b, I_c$ are the excenters :$\\rho_a, \\rho_b, \\rho_c$ are the exradii from $I_a, I_b, I_c$, respectively."}1056{"_id": "1194", "title": "Area of Triangle in Terms of Circumradius", "text": "Let $\\triangle ABC$ be a triangle whose sides are of lengths $a, b, c$. Then the area $\\AA$ of $\\triangle ABC$ is given by: :$\\AA = \\dfrac {a b c} {4 R}$ where $R$ is the circumradius of $\\triangle ABC$."}1057{"_id": "1195", "title": "Stewart's Theorem", "text": "Let $\\triangle ABC$ be a triangle with sides $a, b, c$. Let $CP$ be a cevian from $C$ to $P$. :400px Then: :$a^2 \\cdot AP + b^2 \\cdot PB = c \\paren {CP^2 + AP \\cdot PB}$"}1058{"_id": "1196", "title": "Length of Median of Triangle", "text": "Let $\\triangle ABC$ be a triangle. Let $CD$ be the median of $\\triangle ABC$ which bisects $AB$. :400px The length $m_c$ of $CD$ is given by: :${m_c}^2 = \\dfrac {a^2 + b^2} 2 - \\dfrac {c^2} 4$"}1059{"_id": "1198", "title": "Length of Angle Bisector", "text": "Let $\\triangle ABC$ be a triangle. Let $AD$ be the angle bisector of $\\angle BAC$ in $\\triangle ABC$. :300px Let $d$ be the length of $AD$. Then $d$ is given by: :$d^2 = \\dfrac {b c} {\\paren {b + c}^2} \\paren {\\paren {b + c}^2 - a^2}$ where $a$, $b$, and $c$ are the sides opposite $A$, $B$ and $C$ respectively."}1060{"_id": "1199", "title": "Supremum of Subset", "text": "Let $\\left({U, \\preceq}\\right)$ be an ordered set. Let $S \\subseteq U$. Let $T \\subseteq S$. Let $S$ admit a supremum (in $U$). If $T$ also admits a supremum (in $U$), then $\\sup \\left({T}\\right) \\preceq\\sup \\left({S}\\right)$."}1061{"_id": "1200", "title": "Area of Triangle", "text": "This page gathers a variety of formulas for the area of a triangle."}1062{"_id": "1201", "title": "Intersecting Chord Theorem", "text": "Let $AC$ and $BD$ both be chords of the same circle. Let $AC$ and $BD$ intersect at $E$. Then $AE \\cdot EC = DE \\cdot EB$. {{EuclidSaid}} :''If in a circle two straight lines cut one another, the rectangle contained by the segments of the one is equal to the rectangle contained by the segments of the other.'' {{EuclidPropRef|III|35}}"}1063{"_id": "1202", "title": "Supremum Plus Constant", "text": "Let $S$ be a subset of the set of real numbers $\\R$. Let $S$ be bounded above. Let $\\xi \\in \\R$. Then: :$\\displaystyle \\map {\\sup_{x \\mathop \\in S} } {x + \\xi} = \\xi + \\map {\\sup_{x \\mathop \\in S} } x$ where $\\sup$ denotes supremum."}1064{"_id": "1203", "title": "Suprema and Infima of Combined Bounded Functions", "text": "Let $f$ and $g$ be real functions. Let $c$ be a constant."}1065{"_id": "1204", "title": "Multiple of Infimum", "text": "Let $T \\subseteq \\R: T \\ne \\varnothing$ be a non-empty subset of the set of real numbers $\\R$. Let $T$ be bounded below. Let $z \\in \\R: z > 0$ be a (strictly) positive real number. Then: :$\\displaystyle \\map {\\inf_{x \\mathop \\in T} } {z x} = z \\, \\map {\\inf_{x \\mathop \\in T} } x$ where $\\inf$ denotes infimum."}1066{"_id": "1205", "title": "Law of Sines", "text": "For any triangle $\\triangle ABC$: :$\\dfrac a {\\sin A} = \\dfrac b {\\sin B} = \\dfrac c {\\sin C} = 2 R$ where: :$a$, $b$, and $c$ are the sides opposite $A$, $B$ and $C$ respectively :$R$ is the circumradius of $\\triangle ABC$."}1067{"_id": "1207", "title": "Negative of Infimum is Supremum of Negatives", "text": "Let $T$ be a non-empty subset of the real numbers $\\R$. Let $T$ be bounded below. Then: : $(1): \\quad \\set {x \\in \\R: -x \\in T}$ is bounded above : $(2): \\quad \\displaystyle -\\inf_{x \\mathop \\in T} x = \\map {\\sup_{x \\mathop \\in T} } {-x}$ where $\\sup$ and $\\inf$ denote the supremum and infimum respectively."}1068{"_id": "1208", "title": "Infimum Plus Constant", "text": "Let $T$ be a subset of the set of real numbers. Let $T$ be bounded below. Let $\\xi \\in \\R$. Then: :$\\displaystyle \\map {\\inf_{x \\mathop \\in T} } {x + \\xi} = \\xi + \\map {\\inf_{x \\mathop \\in T} } x$ where $\\inf$ denotes infimum."}1069{"_id": "1209", "title": "Infimum of Subset", "text": "Let $\\struct {U, \\preceq}$ be an ordered set. Let $S \\subseteq U$. Let $T \\subseteq S$. Let $\\struct {S, \\preceq}$ admit an infimum in $U$. If $T$ also admits an infimum in $U$, then $\\map \\inf S \\preceq \\map \\inf T$."}1070{"_id": "1210", "title": "Construction of Equilateral Triangle", "text": "On a given straight line segment, it is possible to construct an equilateral triangle. {{:Euclid:Proposition/I/1}}"}1071{"_id": "1211", "title": "Construction of Equal Straight Line", "text": "At a given point, it is possible to construct a straight line segment of length equal to that of any given straight line segment. The given point will be an endpoint of the constructed straight line segment. {{:Euclid:Proposition/I/2}}"}1072{"_id": "1212", "title": "Construction of Equal Straight Lines from Unequal", "text": "Given two unequal straight line segments, it is possible to cut off from the greater a straight line segment equal to the lesser. {{:Euclid:Proposition/I/3}}"}1073{"_id": "1213", "title": "Triangle Side-Angle-Side Equality", "text": "If $2$ triangles have: : $2$ sides equal to $2$ sides respectively : the angles contained by the equal straight lines equal they will also have: : their third sides equal : the remaining two angles equal to their respective remaining angles, namely, those which the equal sides subtend."}1074{"_id": "1214", "title": "Isosceles Triangle has Two Equal Angles", "text": "In isosceles triangles, the angles at the base are equal to each other. Also, if the equal straight lines are extended, the angles under the base will also be equal to each other. {{:Euclid:Proposition/I/5}}"}1075{"_id": "1215", "title": "Triangle with Two Equal Angles is Isosceles", "text": "If a triangle has two angles equal to each other, the sides which subtend the equal angles will also be equal to one another. Hence, by definition, such a triangle will be isosceles. {{:Euclid:Proposition/I/6}}"}1076{"_id": "1216", "title": "Two Lines Meet at Unique Point", "text": "Let two straight line segments be constructed on a straight line segment from its endpoints so that they meet at a point. Then there cannot be two other straight line segments equal to the former two respectively, constructed on the same straight line segment and on the same side of it, meeting at a different point. {{:Euclid:Proposition/I/7}}"}1077{"_id": "1217", "title": "Triangle Side-Side-Side Equality", "text": "Let two triangles have all $3$ sides equal. Then they also have all $3$ angles equal. Thus two triangles whose sides are all equal are themselves congruent."}1078{"_id": "1218", "title": "Bisection of Angle", "text": "It is possible to bisect any given rectilineal angle. {{:Euclid:Proposition/I/9}}"}1079{"_id": "1219", "title": "Bisection of Straight Line", "text": "It is possible to bisect a straight line segment. {{:Euclid:Proposition/I/10}}"}1080{"_id": "1225", "title": "External Angle of Triangle Greater than Internal Opposite", "text": "The external angle of a triangle is greater than either of the opposite internal angles. {{:Euclid:Proposition/I/16}}"}1081{"_id": "1229", "title": "Limit iff Limits from Left and Right", "text": "Let $f$ be a real function defined on an open interval $\\openint a b$ except possibly at a point $c \\in \\openint a b$. Then: :$\\map f x \\to l$ as $x \\to c$ {{iff}}: :$\\map f x \\to l$ as $x \\to c^-$ and :$\\map f x \\to l$ as $x \\to c^+$"}1082{"_id": "1230", "title": "Limit of Function by Convergent Sequences", "text": "Let $M_1 = \\left({A_1, d_1}\\right)$ and $M_2 = \\left({A_2, d_2}\\right)$ be metric spaces. Let $S \\subseteq A_1$ be an open set of $M_1$. Let $f$ be a mapping defined on $S$, except possibly at the point $c \\in S$. Then $\\displaystyle \\lim_{x \\mathop \\to c} f \\left({x}\\right) = l$ iff: : for each sequence $\\left \\langle {x_n} \\right \\rangle$ of points of $S$ such that $\\forall n \\in \\N_{>0}: x_n \\ne c$ and $\\displaystyle \\lim_{n \\to \\infty} x_n = c$ it is true that: : $\\displaystyle \\lim_{n \\to \\infty} f \\left({x_n}\\right) = l$"}1083{"_id": "1231", "title": "Combination Theorem for Limits of Functions", "text": "Let $X$ be one of the standard number fields $\\Q, \\R, \\C$. Let $f$ and $g$  be functions defined on an open subset $S \\subseteq X$, except possibly at the point $c \\in S$. Let $f$ and $g$ tend to the following limits: :$\\displaystyle \\lim_{x \\to c} \\ f \\left({x}\\right) = l$ :$\\displaystyle \\lim_{x \\to c} \\ g \\left({x}\\right) = m$ Let $\\lambda, \\mu \\in X$ be arbitrary numbers in $X$. Then the following results hold:"}1084{"_id": "1232", "title": "Real Polynomial Function is Continuous", "text": "A (real) polynomial function is continuous at every point. Thus a (real) polynomial function is continuous on every interval of $\\R$."}1085{"_id": "1233", "title": "Linear Function is Continuous", "text": "Let $\\alpha, \\beta \\in \\R$ be real numbers. Let $f : \\R \\to \\R$ be the real function with: :$\\map f x = \\alpha x + \\beta$ for all $x \\in \\R$. Then $f$ is continuous at every real number $c \\in \\R$."}1086{"_id": "1234", "title": "Real Rational Function is Continuous", "text": "A real rational function is continuous at every point at which it is defined. Thus a real rational function is continuous on every interval of $\\R$ not containing a root of the denominator of the function."}1087{"_id": "1235", "title": "Limit of Composite Function", "text": "Let $f$ and $g$ be real functions. Let: :$\\displaystyle \\lim_{y \\mathop \\to \\eta} \\map f y = l$ :$\\displaystyle \\lim_{x \\mathop \\to \\xi} \\map g x = \\eta$ Then, if either: :'''Hypothesis 1:''' $f$ is continuous at $\\eta$ (that is $l = \\map f \\eta$) or: :'''Hypothesis 2:''' for some open interval $I$ containing $\\xi$, it is true that $\\map g x \\ne \\eta$ for any $x \\in I$ except possibly $x = \\xi$ then: : $\\displaystyle \\lim_{x \\mathop \\to \\xi} \\map f {\\map g x} = l$"}1088{"_id": "1237", "title": "Limit of Image of Sequence", "text": "Let $M_1 = \\struct {A_1, d_1}$ and $M_2 = \\struct {A_2, d_2}$ be metric spaces. Let $f: A_1 \\to A_2$ be a mapping which is continuous at $a \\in A_1$. Let $\\sequence {x_n}$ be a sequence of points in $A_1$ such that: :$\\displaystyle \\lim_{n \\mathop \\to \\infty} x_n = a$ where $\\displaystyle \\lim_{n \\mathop \\to \\infty} x_n$ is the limit of $x_n$. Then: :$\\displaystyle \\lim_{n \\mathop \\to \\infty} \\map f {x_n} = \\map f a$ That is: :$\\displaystyle \\lim_{n \\mathop \\to \\infty} \\map f {x_n} = \\map f {\\lim_{n \\mathop \\to \\infty} x_n}$ That is, for a continuous mapping, the limit and function symbols commute."}1089{"_id": "1238", "title": "Image of Interval by Continuous Function is Interval", "text": "Let $I$ be a real interval. Let $f: I \\to \\R$ be a continuous real function. Then the image of $f$ is a real interval."}1090{"_id": "1239", "title": "Interval Divided into Subsets", "text": "Let $\\mathbb I$ be a real interval. Let $S$ and $T$ be non-empty subsets of $\\mathbb I$ such that $\\mathbb I \\subseteq S \\cup T$. Then one of $S$ or $T$ contains an element at zero distance from the other."}1091{"_id": "1240", "title": "Distance from Subset of Real Numbers", "text": "Let $S$ be a subset of the set of real numbers $\\R$. Let $x \\in \\R$ be a real number. Let $\\map d {x, S}$ be the distance between $x$ and $S$. Then:"}1092{"_id": "1241", "title": "Limit of Sequence to Zero Distance Point", "text": "Let $S$ be a non-empty subset of $\\R$. Let the distance $\\map d {\\xi, S} = 0$ for some $\\xi \\in \\R$. Then there exists a sequence $\\sequence {x_n}$ in $S$ such that $\\displaystyle \\lim_{n \\mathop \\to \\infty} x_n = \\xi$."}1093{"_id": "1242", "title": "Image of Closed Real Interval is Bounded", "text": "Let $f$ be a real function which is continuous on the closed interval $\\closedint a b$. Then $f$ is bounded on $\\closedint a b$."}1094{"_id": "1243", "title": "Convergent Subsequence in Closed Interval", "text": "Let $\\closedint a b$ be a closed real interval. Then every sequence of points of $\\closedint a b$ contains a subsequence which converges to a point in $\\closedint a b$."}1095{"_id": "1244", "title": "Max and Min of Function on Closed Real Interval", "text": "Let $f$ be a real function which is continuous on the closed interval $\\closedint a b$. Then $f$ reaches a maximum and a minimum on $\\closedint a b$."}1096{"_id": "1245", "title": "Continuous Image of Closed Interval is Closed Interval", "text": "Let $f$ be a real function which is continuous on the closed interval $\\left[{a \\,.\\,.\\, b}\\right]$. Then the image of $\\left[{a \\,.\\,.\\, b}\\right]$ under $f$ is also a closed interval."}1097{"_id": "1247", "title": "Retraction Theorem", "text": "Let $M$ be a compact manifold with boundary $\\partial M$. Then there is no smooth mapping $f: M \\to \\partial M$ such that $\\partial f: \\partial M \\to \\partial M$ is the identity."}1098{"_id": "1248", "title": "Classification of Compact One-Manifolds", "text": "Every compact one-dimensional manifold is diffeomorphic to either a circle or a closed interval."}1099{"_id": "1249", "title": "Differentiable Function is Continuous", "text": "Let $f$ be a real function defined on an interval $I$. Let $x_0 \\in I$ such that $f$ is differentiable at $x_0$. Then $f$ is continuous at $x_0$."}1100{"_id": "1250", "title": "Quotient Rule for Derivatives", "text": "Let $\\map j x, \\map k x$ be real functions defined on the open interval $I$. Let $\\xi \\in I$ be a point in $I$ at which both $j$ and $k$ are differentiable. Define the real function $f$ on $I$ by: :$\\displaystyle \\map f x = \\begin{cases} \\dfrac {\\map j x} {\\map k x} & : \\map k x \\ne 0 \\\\ 0 & : \\text{otherwise} \\end{cases}$ Then, if $\\map k \\xi \\ne 0$, $f$ is differentiable at $\\xi$, and furthermore: :$\\map {f'} \\xi = \\dfrac {\\map {j'} \\xi \\map k \\xi - \\map j \\xi \\map {k'} \\xi} {\\paren {\\map k \\xi}^2}$ It follows from the definition of derivative that if $j$ and $k$ are both differentiable on the interval $I$, then: :$\\displaystyle \\forall x \\in I: \\map k x \\ne 0 \\implies \\map {f'} x = \\frac {\\map {j'} x \\map k x - \\map j x \\map {k'} x} {\\paren {\\map k x}^2}$"}1101{"_id": "1252", "title": "Preimage Theorem", "text": "Let $y$ be a regular value of a smooth submersion $f:X \\to Y$. Then the preimage $f^{-1}(y)$ is a smooth submanifold of $X$, with $\\dim f^{-1}(y) = \\dim X - \\dim Y$."}1102{"_id": "1255", "title": "Backwards Induction", "text": "Let $P$ be a propositional function on the natural numbers $\\N$. Suppose that: :$(1): \\quad \\forall n \\in \\N: \\map P {2^n}$ holds. :$(2): \\quad \\map P n \\implies \\map P {n - 1}$. Then $\\map P n$ holds for all $\\forall n \\in \\N$. The proof technique based on this result is called '''backwards induction'''."}1103{"_id": "1256", "title": "Inequalities Concerning Roots", "text": "Let $\\closedint X Y$ be a closed real interval such that $0 < X \\le Y$. Let $x, y \\in \\closedint X Y$. Then: :$\\forall n \\in \\N_{> 0}: X Y^{1/n} \\size {x - y} \\le n X Y \\size {x^{1/n} - y^{1/n} } \\le Y X^{1/n} \\size {x - y}$"}1104{"_id": "1257", "title": "Continuity of Root Function", "text": "Let $n \\in \\N_{>0}$ be a non-zero natural number. Let $f: \\hointr 0 \\infty \\to \\R$ be the real function defined by $\\map f x = x^{1/n}$. Then $f$ is continuous at each $\\xi > 0$ and continuous on the right at $\\xi = 0$."}1105{"_id": "1259", "title": "P-adic Norm is Norm", "text": "The $p$-adic norm forms a norm on the rational numbers $\\Q$."}1106{"_id": "1260", "title": "Fundamental Theorem of Algebra", "text": "Every non-constant polynomial with coefficients in $\\C$ has a root in $\\C$."}1107{"_id": "1261", "title": "Extendability Theorem for Intersection Numbers", "text": "Let $X = \\partial W$ be a smooth manifold which is the boundary of a smooth compact manifold $W$. Let $Y$ be a smooth manifold, $Z$ be a closed smooth submanifold of $Y$, and $f: X \\to Y$ a smooth map. Let there exist a smooth map $g: W \\to Y$ such that $g \\restriction_X = f$. Then: :$I \\left({f, Z}\\right) = 0$ where $I \\left({f, Z}\\right)$ is the intersection number. {{explain|what $I \\left({f, Z}\\right)$ is the intersection number of: presumably the words will go something like \"... the intersection number of $f$ with respect to $Z$\", or something.}}"}1108{"_id": "1264", "title": "Homotopy Group is Group", "text": "The set of all homotopy classes of continuous mappings: :$c: \\closedint 0 1^n \\to X$ satisfying: :$\\map c {\\partial \\closedint 0 1^n} = x_0$ in a space $X$ at a base point $x_0$, under the operation of concatenation on class members, forms a group. {{finish|This operation needs to be shown well-defined; probably a nasty proof}} This group is called the $n$th homotopy group."}1109{"_id": "1265", "title": "Derivative at Maximum or Minimum", "text": "Let $f$ be a real function which is differentiable on the open interval $\\openint a b$. Let $f$ have a local minimum or local maximum at $\\xi \\in \\openint a b$. Then: :$\\map {f'} \\xi = 0$"}1110{"_id": "1266", "title": "Behaviour of Function Near Limit", "text": "Let $f$ be a real function. Let $f \\left({x}\\right) \\to l$ as $x \\to \\xi$. Then: * If $l > 0$, then $\\exists h > 0: \\forall x: \\xi - h < x < \\xi + h, x \\ne \\xi: f \\left({x}\\right) > 0$ * If $l < 0$, then $\\exists h > 0: \\forall x: \\xi - h < x < \\xi + h, x \\ne \\xi: f \\left({x}\\right) < 0$"}1111{"_id": "1267", "title": "Fundamental Group is Independent of Base Point for Path-Connected Space", "text": "Let $X$ be a path-connected space. For $x \\in X$ let $\\pi_1(X,x)$ denote the fundamental group. For $x, y \\in X$, there is an isomorphism: :$\\phi: \\pi_1 \\left({X, x}\\right) \\to \\pi_1 \\left({X, y}\\right)$"}1112{"_id": "1268", "title": "List of Fundamental Groups for 2-Manifolds", "text": "For the following two-manifolds, the fundamental group for any point in $X$, written $\\pi_1 \\left({X}\\right)$ is isomorphic to the listed group: :$\\pi_1 \\left({\\Bbb S^1 \\times \\left[{0\\,.\\,.\\,1}\\right]}\\right) = \\Z$ :$\\pi_1 \\left({\\Bbb S^2}\\right) = \\left\\{{e}\\right\\}$, the trivial group. :$\\pi_1 \\left({\\Bbb T^2}\\right) = \\pi_1 \\left({\\Bbb S^1 \\times \\Bbb S^1}\\right) = \\Z \\times \\Z$ :$\\pi_1 \\left({\\Bbb {RP}^2}\\right) = \\Z_2$"}1113{"_id": "1269", "title": "Rolle's Theorem", "text": "Let $f$ be a real function which is: :continuous on the closed interval $\\closedint a b$ and: :differentiable on the open interval $\\openint a b$. Let $\\map f a = \\map f b$. Then: :$\\exists \\xi \\in \\openint a b: \\map {f'} \\xi = 0$"}1114{"_id": "1270", "title": "Mean Value Theorem", "text": "Let $f$ be a real function which is continuous on the closed interval $\\closedint a b$ and differentiable on the open interval $\\openint a b$. Then: :$\\exists \\xi \\in \\openint a b: \\map {f'} \\xi = \\dfrac {\\map f b - \\map f a} {b - a}$"}1115{"_id": "1271", "title": "Zero Derivative implies Constant Function", "text": "Let $f$ be a real function which is continuous on the closed interval $\\left[{a \\,.\\,.\\, b}\\right]$ and differentiable on the open interval $\\left({a \\,.\\,.\\, b}\\right)$. Suppose that: :$\\forall x \\in \\left({a \\,.\\,.\\, b}\\right): f' \\left({x}\\right) = 0$ Then $f$ is constant on $\\left[{a \\,.\\,.\\, b}\\right]$."}1116{"_id": "1272", "title": "Ostrowski's Theorem", "text": "Every non-trivial norm on the rational numbers $\\Q$ is equivalent to either: :the $p$-adic norm $\\norm {\\, \\cdot \\,}_p$ for some prime $p$ or: :the absolute value, $\\size {\\, \\cdot \\,}$."}1117{"_id": "1273", "title": "Cauchy Mean Value Theorem", "text": "Let $f$ and $g$ be a real functions which are continuous on the closed interval $\\left[{a \\,.\\,.\\, b}\\right]$ and differentiable on the open interval $\\left({a \\,.\\,.\\, b}\\right)$. Suppose that: :$\\forall x \\in \\left({a \\,.\\,.\\, b}\\right): g' \\left({x}\\right) \\ne 0$ Then: :$\\exists \\xi \\in \\left({a \\,.\\,.\\, b}\\right): \\dfrac {f' \\left({\\xi}\\right)} {g' \\left({\\xi}\\right)} = \\dfrac {f \\left({b}\\right) - f \\left({a}\\right)} {g \\left({b}\\right) - g \\left({a}\\right)}$"}1118{"_id": "1274", "title": "Homology Group is Group", "text": "The $p^{th}$ singular homology group of a space $X$ is a group."}1119{"_id": "1275", "title": "Limit of Absolute Value", "text": "Let $x, \\xi \\in \\R$ be real numbers. Then: :$\\left\\vert{x - \\xi}\\right\\vert \\to 0$ as $x \\to \\xi$ where $\\left\\vert{x - \\xi}\\right\\vert$ denotes the Absolute Value."}1120{"_id": "1276", "title": "Limit of Function in Interval", "text": "Let $f$ be a real function which is defined on the open interval $\\openint a b$. Let $\\xi \\in \\openint a b$ Suppose that, $\\forall x \\in \\openint a b$, either: :$\\xi \\le \\map f x \\le x$ or: :$x \\le \\map f x \\le \\xi$ Then $\\map f x \\to \\xi$ as $x \\to \\xi$."}1121{"_id": "1278", "title": "Derivative of Monotone Function", "text": "Let $f$ be a real function which is continuous on the closed interval $\\closedint a b$ and differentiable on the open interval $\\openint a b$."}1122{"_id": "1279", "title": "Strictly Monotone Real Function is Bijective", "text": "Let $f$ be a real function which is defined on $I \\subseteq \\R$. Let $f$ be strictly monotone on $I$. Let the image of $f$ be $J$.  Then $f: I \\to J$ is a bijection."}1123{"_id": "1280", "title": "Inverse of Strictly Monotone Function", "text": "Let $f$ be a real function which is defined on $I \\subseteq \\R$. Let $f$ be strictly monotone on $I$. Let the image of $f$ be $J$.  Then $f$ always has an inverse function $f^{-1}$ and: : if $f$ is strictly increasing then so is $f^{-1}$ : if $f$ is strictly decreasing then so is $f^{-1}$."}1124{"_id": "1281", "title": "Rokhlin's Theorem (Intersection Forms)", "text": "Let $M$ be a smooth 4-manifold. Then: :$\\map {\\omega_2} {\\map T M} = 0 \\implies \\operatorname {sign} Q_M = 0 \\pmod {16}$ where: :$Q_M$ is the intersection form :$\\map T M$ is the tangent bundle :$\\omega_2$ is the second Stiefel-Whitney class."}1125{"_id": "1283", "title": "Convex Real Function is Continuous", "text": "Let $f$ be a real function which is convex on the open interval $\\left({a \\,.\\,.\\, b}\\right)$. Then $f$ is continuous on $\\left({a \\,.\\,.\\, b}\\right)$."}1126{"_id": "1284", "title": "Convex Real Function is Left-Hand and Right-Hand Differentiable", "text": "Let $f$ be a real function which is either convex on the open interval $\\left({a \\,.\\,.\\, b}\\right)$. Then the left-hand derivative $f'_- \\left({x}\\right)$ and right-hand derivative $f'_+ \\left({x}\\right)$ both exist for all $x \\in \\left({a \\,.\\,.\\, b}\\right)$."}1127{"_id": "1285", "title": "Real Function is Convex iff Derivative is Increasing", "text": "Let $f$ be a real function which is differentiable on the open interval $\\openint a b$. Then: :$f$ is convex on $\\openint a b$ {{iff}}: :its derivative $f'$ is increasing on $\\openint a b$. Thus the intuitive result that a convex function \"gets steeper\"."}1128{"_id": "1286", "title": "Inverse of Strictly Increasing Convex Real Function is Concave", "text": "Let $f$ be a real function which is convex on the open interval $I$. Let $J = f \\left[{I}\\right]$. If $f$ be strictly increasing on $I$, then $f^{-1}$ is concave on $J$."}1129{"_id": "1287", "title": "Upper and Lower Bounds of Integral", "text": "Let $f$ be a real function which is continuous on the closed interval $\\closedint a b$. Let $\\displaystyle \\int_a^b \\map f x \\rd x$ be the definite integral of $\\map f x$ over $\\closedint a b$. Then: :$\\displaystyle m \\paren {b - a} \\le \\int_a^b \\map f x \\rd x \\le M \\paren {b - a}$ where: :$M$ is the maximum of $f$ :$m$ is the minimum of $f$ on $\\closedint a b$."}1130{"_id": "1288", "title": "Wilson's Theorem", "text": "A (strictly) positive integer $p$ is a prime {{iff}}: :$\\paren {p - 1}! \\equiv -1 \\pmod p$"}1131{"_id": "1289", "title": "Integral of Constant", "text": "Let $c$ be a constant."}1132{"_id": "1290", "title": "Sum of Integrals on Adjacent Intervals for Continuous Functions", "text": "Let $f$ be a real function which is continuous on any closed interval $I$. Let $a, b, c \\in I$. Then: :$\\ds \\int_a^c \\map f t \\rd t + \\int_c^b \\map f t \\rd t = \\int_a^b \\map f t \\rd t$"}1133{"_id": "1291", "title": "Primitives which Differ by Constant", "text": "Let $F$ be a primitive for a real function $f$ on the closed interval $\\closedint a b$. Let $G$ be a real function defined on $\\closedint a b$. Then $G$ is a primitive for $f$ on $\\closedint a b$ {{iff}}: :$\\exists c \\in \\R: \\forall x \\in \\closedint a b: \\map G x = \\map F x + c$ That is, {{iff}} $F$ and $G$ differ by a constant on the whole interval."}1134{"_id": "1293", "title": "Definite Integral of Function plus Constant", "text": "Let $f$ be a real function which is continuous on the closed interval $\\closedint a b$. Let $c$ be a constant. Then: :$\\ds \\int_a^b \\paren {\\map f t + c} \\rd t = \\int_a^b \\map f t \\rd t + c \\paren {b - a}$"}1135{"_id": "1294", "title": "Continuous Real Function is Darboux Integrable", "text": "Let $f$ be a real function which is continuous on the closed interval $\\closedint a b$. Then $f$ is Darboux integrable on $\\closedint a b$."}1136{"_id": "1295", "title": "Definite Integral on Zero Interval", "text": "Let $f$ be a real function which is defined on the closed interval $\\Bbb I := \\closedint a b$, where $a < b$. Then: :$\\displaystyle \\forall c \\in \\Bbb I: \\int_c^c \\map f t \\rd t = 0$"}1137{"_id": "1296", "title": "Linear Combination of Derivatives", "text": "Let $f \\left({x}\\right), g \\left({x}\\right)$ be real functions defined on the open interval $I$. Let $\\xi \\in I$ be a point in $I$ at which both $f$ and $g$ are differentiable. Then: :$D \\left({\\lambda f + \\mu g}\\right) = \\lambda D f + \\mu D g$ at the point $\\xi$. It follows from the definition of derivative that if $f$ and $g$ are both differentiable on the interval $I$, then: :$\\forall x \\in I: D \\left({\\lambda f \\left({x}\\right) + \\mu g \\left({x}\\right)}\\right) = \\lambda D f \\left({x}\\right) + \\mu D g \\left({x}\\right)$"}1138{"_id": "1297", "title": "Linear Combination of Integrals", "text": "Let $f$ and $g$ be real functions which are integrable on the closed interval $\\closedint a b$. Let $\\lambda$ and $\\mu$ be real numbers. Then the following results hold:"}1139{"_id": "1298", "title": "Integration by Parts", "text": "Let $f$ and $g$ be real functions which are continuous on the closed interval $\\closedint a b$. Let $f$ and $g$ have primitives $F$ and $G$ respectively on $\\closedint a b$. Then:"}1140{"_id": "1299", "title": "Integration by Substitution", "text": "Let $\\phi$ be a real function which has a derivative on the closed interval $\\closedint a b$. Let $I$ be an open interval which contains the image of $\\closedint a b$ under $\\phi$. Let $f$ be a real function which is continuous on $I$."}1141{"_id": "1300", "title": "Relative Sizes of Definite Integrals", "text": "Let $f$ and $g$ be real functions which are continuous on the closed interval $\\closedint a b$, where $a < b$. If: :$\\forall t \\in \\closedint a b: \\map f t \\le \\map g t$ then: :$\\displaystyle \\int_a^b \\map f t \\rd t \\le \\int_a^b \\map g t \\rd t$ Similarly, if: :$\\forall t \\in \\closedint a b: \\map f t < \\map g t$ then: :$\\displaystyle \\int_a^b \\map f t \\rd t < \\int_a^b \\map g t \\rd t$"}1142{"_id": "1301", "title": "Absolute Value of Definite Integral", "text": "Let $f$ be a real function which is continuous on the closed interval $\\closedint a b$. Then: :$\\displaystyle \\size {\\int_a^b \\map f t \\rd t} \\le \\int_a^b \\size {\\map f t} \\rd t$"}1143{"_id": "1302", "title": "Integral Test", "text": "Let $f$ be a real function which is continuous, positive and decreasing on the interval $\\hointr 1 {+\\infty}$. Let the sequence $\\sequence {\\Delta_n}$ be defined as: :$\\displaystyle \\Delta_n = \\sum_{k \\mathop = 1}^n \\map f k - \\int_1^n \\map f x \\rd x$ Then $\\sequence {\\Delta_n} $ is decreasing and bounded below by zero. Hence it converges."}1144{"_id": "1304", "title": "Properties of Natural Logarithm", "text": "Let $x \\in \\R$ be a real number such that $x > 0$. Let $\\ln x$ be the natural logarithm of $x$. Then:"}1145{"_id": "1306", "title": "Harmonic Series is Divergent", "text": "The harmonic series: :$\\displaystyle \\sum_{n \\mathop = 1}^\\infty \\frac 1 n$  diverges."}1146{"_id": "1307", "title": "Sum of Reciprocals of Primes is Divergent", "text": "Let $n \\in \\N: n \\ge 1$. There exists a (strictly) positive real number $C \\in \\R_{>0}$ such that: :$(1): \\quad \\displaystyle \\sum_{\\substack {p \\mathop \\in \\Bbb P \\\\ p \\mathop \\le n} } \\frac 1 p > \\map \\ln {\\ln n} - C$ where $\\Bbb P$ is the set of all prime numbers. :$(2): \\quad \\displaystyle \\lim_{n \\mathop \\to \\infty} \\paren {\\map \\ln {\\ln n} - C} = +\\infty$"}1147{"_id": "1308", "title": "Measure of Interval is Length", "text": "Let $I$ be a real interval whose endpoints are $a$ and $b$. Then $I$ is Lebesgue measurable, and the value of the measure is the length of the interval $b - a$."}1148{"_id": "1309", "title": "Measurable Sets form Algebra of Sets", "text": "Let $\\mu^*$ be an outer measure on a set $X$. Then the set of $\\mu^*$-measurable sets is an algebra of sets."}1149{"_id": "1310", "title": "Classification of Compact Three-Manifolds Supporting Zero-Curvature Geometry", "text": "Every closed, orientable, path connected $3$-dimensional Riemannian manifold which supports a geometry of zero curvature is homeomorphic to one of the following: * Torus $\\mathbb T^3$ * Half-Twist Cube * Quarter-Twist Cube * Hantschze-Wendt Manifold * $\\frac 1 6$-Twist Hexagonal Prism * $\\frac 1 3$-Twist Hexagonal Prism The $3$-torus is described on the torus page. The other manifolds can be described using quotient spaces on familiar prisms, with the equivalence relations described below. <gallery perRow=\"5\"> File:Halftwistcube.JPG|The Half-Twist Cube File:Hantschzewendt.JPG|The Hantschze-Wendt Manifold File:Quartertwistcube.JPG|The Quarter-Twist Cube File:Sixthtwisthexagon.JPG|The $\\frac 1 6$-Twist Hexagonal Prism File:Thirdtwisthexagon.JPG|The $\\frac 1 3$-Twist Hexagonal Prism </gallery>"}1150{"_id": "1311", "title": "Sum of Logarithms", "text": "{{:Sum of Logarithms/General Logarithm}}"}1151{"_id": "1312", "title": "Heine-Borel Theorem/Real Line", "text": "Let $\\R$ be the real number line considered as a Euclidean space. Let $C \\subseteq \\R$. Then $C$ is closed and bounded in $\\R$ {{iff}} $C$ is compact."}1152{"_id": "1314", "title": "Measurable Image", "text": "Let $\\mathfrak M$ be the set of measurable sets of $\\R$. For any extended real-valued function $f: \\R \\to \\R \\cup \\left\\{{-\\infty \\,.\\,.\\, +\\infty}\\right\\}$ whose domain is measurable, the following statements are equivalent: : $(1): \\quad \\forall \\alpha \\in \\R: \\left\\{{x: f \\left({x}\\right) > \\alpha}\\right\\} \\in \\mathfrak M$ : $(2): \\quad \\forall \\alpha \\in \\R: \\left\\{{x: f \\left({x}\\right) \\ge \\alpha}\\right\\} \\in \\mathfrak M$ : $(3): \\quad \\forall \\alpha \\in \\R: \\left\\{{x: f \\left({x}\\right) < \\alpha}\\right\\} \\in \\mathfrak M$ : $(4): \\quad \\forall \\alpha \\in \\R: \\left\\{{x: f \\left({x}\\right) \\le \\alpha}\\right\\} \\in \\mathfrak M$ These statements imply: : $(5): \\quad \\forall \\alpha \\in \\R \\cup \\left\\{{-\\infty \\,.\\,.\\, +\\infty}\\right\\}: \\left\\{{x:f \\left({x}\\right) = \\alpha}\\right\\} \\in \\mathfrak M$ {{refactor}} {{proofread}}"}1153{"_id": "1315", "title": "Lebesgue Integral is Extension of Darboux Integral", "text": "Let $f: \\closedint a b \\to \\R$ be a Darboux integrable function. Then it is also Lebesgue integrable, and furthermore: :$\\displaystyle R \\int_a^b \\map f x \\rd x = \\int_{\\closedint a b} f \\rd \\lambda$ where $\\displaystyle R \\int_a^b$ is the Darboux integral and $\\displaystyle \\int_{\\closedint a b}$ is the Lebesgue integral."}1154{"_id": "1316", "title": "Properties of Exponential Function", "text": "Let $x \\in \\R$ be a real number. Let $\\exp x$ be the exponential of $x$. Then:"}1155{"_id": "1317", "title": "Euclidean Metric on Real Vector Space is Metric", "text": "The Euclidean metric on a real vector space $\\R^n$ is a metric."}1156{"_id": "1318", "title": "Metric Induces Topology", "text": "Let $M = \\struct {A, d}$ be a metric space. Then the topology $\\tau$ induced by the metric $d$ is a topology on $M$."}1157{"_id": "1319", "title": "Properties of Algebras of Sets", "text": "Let $X$ be a set. Let $\\mathfrak A$ be an algebra of sets on $X$. Then the following hold: :$(1): \\quad$ The intersection of two sets in $\\mathfrak A$ is in $\\mathfrak A$. :$(2): \\quad$ The difference of two sets in $\\mathfrak A$ is in $\\mathfrak A$. :$(3): \\quad$ $X \\in \\mathfrak A$. :$(4): \\quad$ The empty set $\\O$ is in $\\mathfrak A$."}1158{"_id": "1320", "title": "Countable Sets Have Measure Zero", "text": "Let $S$ be a countable set. {{explain|Is it assumed that $S \\subseteq \\R$?}} Then the measure of $S$ is $\\map m S = 0$."}1159{"_id": "1321", "title": "Mean Value of Convex Real Function", "text": "Let $f$ be a real function which is continuous on the closed interval $\\closedint a b$ and differentiable on the open interval $\\openint a b$. Let $f$ be convex on $\\openint a b$. Then: :$\\forall \\xi \\in \\openint a b: \\map f x - \\map f \\xi \\ge \\map {f'} \\xi \\paren {x - \\xi}$"}1160{"_id": "1322", "title": "Upper Bound of Natural Logarithm", "text": "Let $\\ln y$ be the natural logarithm of $y$ where $y \\in \\R_{>0}$. Then: :$\\ln y \\le y - 1$"}1161{"_id": "1323", "title": "Exponential of Sum/Real Numbers", "text": "Let $x, y \\in \\R$ be real numbers. Let $\\exp x$ be the exponential of $x$. Then: :$\\map \\exp {x + y} = \\paren {\\exp x} \\paren {\\exp y}$"}1162{"_id": "1324", "title": "Exponential of Product", "text": "Let $x, y \\in \\R$ be real numbers. Let $\\exp x$ be the exponential of $x$. Then: :$\\map \\exp {x y} = \\paren {\\exp y}^x$"}1163{"_id": "1325", "title": "Exponent Combination Laws", "text": "Let $a, b \\in \\R_{>0}$ be strictly positive real numbers. Let $x, y \\in \\R$ be real numbers. Let $a^x$ be defined as $a$ to the power of $x$. Then:"}1164{"_id": "1326", "title": "Derivative of Exponential at Zero", "text": "Let $\\exp x$ be the exponential of $x$ for real $x$. Then: : $\\displaystyle \\lim_{x \\mathop \\to 0} \\frac {\\exp x - 1} x = 1$"}1165{"_id": "1327", "title": "Classification of Groups of Order up to 15", "text": "Up to isomorphism, every group of order $\\order G \\le 15$ is one of the below: {| class=\"sortable wikitable\" |- bgcolor=\"#ececec\" ! Order !! Abelian !! Non-Abelian |- |1 || $\\Z_1$ ||  |- |2 || $\\Z_2$ || |- |3 || $\\Z_3$ || |- |4 || $\\Z_4, \\Z_2 \\oplus \\Z_2$ ||  |- |5 || $\\Z_5$ || |- |6 || $\\Z_6$ || $D_3 = S_3$ |-  |7 || $\\Z_7$ || |- |8 || $\\Z_8, \\Z_4 \\oplus \\Z_2, \\Z_2 \\oplus \\Z_2 \\oplus \\Z_2$ || $D_4, \\Dic 2$ |- |9 || $\\Z_9, \\Z_3 \\oplus \\Z_3$ ||  |- |10 || $\\Z_{10}$ || $D_5$ |- |11 || $\\Z_{11}$ || |- |12 || $\\Z_{12}, \\Z_6 \\oplus \\Z_2$ || $D_6, A_4, \\Dic 3$ |- |13 || $\\Z_{13}$ || |- |14 || $\\Z_{14}$ || $D_7$ |- |15 || $\\Z_{15}$ ||  |} where: : $D_n$ is the dihedral group of order $2 n$ : $S_n$ is the $n$th symmetric group : $A_n$ is the alternating group on $n$ points : $\\Dic 2$ is the dicyclic group of order $4 n$."}1166{"_id": "1329", "title": "Cyclic Groups of Order p q", "text": "Let $p, q$ be primes such that $p < q$ and $p$ does not divide $q - 1$. Let $G$ be a group of order $p q$. Then $G$ is cyclic."}1167{"_id": "1330", "title": "Derivative of Logarithm at One", "text": "Let $\\ln x$ be the natural logarithm of $x$ for real $x$ where $x > 0$. Then: :$\\displaystyle \\lim_{x \\mathop \\to 0} \\frac {\\map \\ln {1 + x} } x = 1$"}1168{"_id": "1331", "title": "Dicyclic Group is Group", "text": "The dicyclic group $Q_n$ is a non-abelian group on two generators."}1169{"_id": "1332", "title": "Existence of Interval of Convergence of Power Series", "text": "Let $\\xi \\in \\R$ be a real number. Let $\\displaystyle \\map S x = \\sum_{n \\mathop = 0}^\\infty a_n \\paren {x - \\xi}^n$ be a power series about $\\xi$. Then the interval of convergence of $\\map S x$ is a real interval whose midpoint is $\\xi$."}1170{"_id": "1334", "title": "Power Series is Differentiable on Interval of Convergence", "text": "Let $\\xi \\in \\R$ be a real number. Let $\\displaystyle \\map f x = \\sum_{n \\mathop = 0}^\\infty a_n \\paren {x - \\xi}^n$ be a power series about $\\xi$. Let $\\map f x$ have an interval of convergence $I$. Then $\\map f x$ is continuous on $I$, and differentiable on $I$ except possibly at its endpoints. Also: :$\\displaystyle \\map {D_x} {\\map f x} = \\sum_{n \\mathop = 1}^\\infty n a_n \\paren {x - \\xi}^{n - 1}$"}1171{"_id": "1336", "title": "Power Series Expansion for Exponential Function", "text": "Let $\\exp x$ be the exponential function. Then: {{begin-eqn}} {{eqn | ll= \\forall x \\in \\R:       | l = \\exp x       | r = \\sum_{n \\mathop = 0}^\\infty \\frac {x^n} {n!}       | c =  }} {{eqn | r = 1 + x + \\frac {x^2} {2!} + \\frac {x^3} {3!} + \\cdots       | c =  }} {{end-eqn}}"}1172{"_id": "1337", "title": "Properties of Real Sine Function", "text": "Let $x \\in \\R$ be a real number. Let $\\sin x$ be the sine of $x$. Then:"}1173{"_id": "1338", "title": "Properties of Real Cosine Function", "text": "Let $x \\in \\R$ be a real number. Let $\\cos x$ be the cosine of $x$. Then:"}1174{"_id": "1340", "title": "Derivative of Cosine Function", "text": ":$\\map {\\dfrac \\d {\\d x} } {\\cos x} = -\\sin x$"}1175{"_id": "1341", "title": "Limit of Sine of X over X", "text": ":$\\displaystyle \\lim_{x \\mathop \\to 0} \\frac {\\sin x} x = 1$"}1176{"_id": "1342", "title": "Sum of Squares of Sine and Cosine", "text": ":$\\cos^2 x + \\sin^2 x = 1$"}1177{"_id": "1343", "title": "Cosine of Sum", "text": ":$\\map \\cos {a + b} = \\cos a \\cos b - \\sin a \\sin b$"}1178{"_id": "1344", "title": "Equivalence of Definitions of Sine and Cosine", "text": "The definitions for sine and cosine are equivalent. That is: : $\\displaystyle \\sin x = \\sum_{n \\mathop = 0}^\\infty \\paren {-1}^n \\frac {x^{2 n + 1} } {\\paren {2 n + 1}!} \\iff \\sin x = \\frac {\\text{Opposite}} {\\text{Hypotenuse}}$ : $\\displaystyle \\cos x = \\sum_{n \\mathop = 0}^\\infty \\paren {-1}^n \\frac {x^{2 n} } {\\paren {2 n}!} \\iff \\cos x = \\frac {\\text{Adjacent}} {\\text{Hypotenuse}}$"}1179{"_id": "1345", "title": "Real Cosine Function is Bounded", "text": ": $\\size {\\cos x} \\le 1$"}1180{"_id": "1346", "title": "Boundedness of Sine X over X", "text": "Let $x \\in \\R$. Then: :$\\size {\\dfrac {\\sin x} x} \\le 1$"}1181{"_id": "1347", "title": "Sine and Cosine are Periodic on Reals", "text": "The sine and cosine functions are periodic on the set of real numbers $\\R$: :$(1): \\quad \\map \\cos {x + 2 \\pi} = \\cos x$ :$(2): \\quad \\map \\sin {x + 2 \\pi} = \\sin x$ :800px"}1182{"_id": "1348", "title": "Differentiable Bounded Convex Real Function is Constant", "text": "Let $f$ be a real function which is: :$(1): \\quad$ Differentiable on $\\R$ :$(2): \\quad$ Bounded on $\\R$ :$(3): \\quad$ Convex on $\\R$. Then $f$ is constant."}1183{"_id": "1349", "title": "Banach-Tarski Paradox", "text": "The unit ball $\\mathbb D^3 \\subset \\R^3$ is equidecomposable to the union of two unit balls."}1184{"_id": "1350", "title": "Equidecomposability is Equivalence Relation", "text": "The property of being equidecomposable is an equivalence relation on the power set $\\mathcal P \\left({\\R^n}\\right)$."}1185{"_id": "1351", "title": "Equidecomposability Unaffected by Union", "text": "{{explain|I guess the sets should be disjoint, in the sense that $S_i \\cap S_j$ is empty for $i \\neq j$. Similar for $T$.}} Let $\\left\\{{S_1, \\ldots, S_m}\\right\\}, \\left\\{{T_1, \\ldots, T_m }\\right\\}$ be sets of sets in $\\R^n$ such that: : for each $k \\in \\left\\{{1, \\dots, m}\\right\\}, S_k$ and $T_k$ are equidecomposable.  Then the set $\\displaystyle S = \\bigcup_{i \\mathop = 1}^m S_i$ is equidecomposable with $\\displaystyle T = \\bigcup_{i \\mathop = 1}^m T_i$."}1186{"_id": "1352", "title": "Subsets of Equidecomposable Subsets are Equidecomposable", "text": "Let $A, B \\subseteq \\R^n$ be equidecomposable. Let $S \\subseteq A$. Then there exists $T \\subseteq B$ such that $S$ and $T$ are equidecomposable."}1187{"_id": "1353", "title": "Shape of Cosine Function", "text": "The cosine function is: :$(1): \\quad$ strictly decreasing on the interval $\\closedint 0 \\pi$ :$(2): \\quad$ strictly increasing on the interval $\\closedint \\pi {2 \\pi}$ :$(3): \\quad$ concave on the interval $\\closedint {-\\dfrac \\pi 2} {\\dfrac \\pi 2}$ :$(4): \\quad$ convex on the interval $\\closedint {\\dfrac \\pi 2} {\\dfrac {3 \\pi} 2}$"}1188{"_id": "1354", "title": "Tangent Function is Periodic on Reals", "text": "The tangent function is periodic on the set of real numbers $\\R$ with period $\\pi$. This can be written: :$\\tan x = \\tan \\left({x \\bmod \\pi}\\right)$ where $x \\bmod \\pi$ denotes the modulo operation."}1189{"_id": "1355", "title": "Derivative of Tangent Function", "text": ":$\\map {\\dfrac \\d {\\d x} } {\\tan x} = \\sec^2 x = \\dfrac 1 {\\cos^2 x}$ when $\\cos x \\ne 0$."}1190{"_id": "1356", "title": "Stirling's Formula", "text": "The factorial function can be approximated by the formula: :$n! \\sim \\sqrt {2 \\pi n} \\paren {\\dfrac n e}^n$ where $\\sim$ denotes asymptotically equal."}1191{"_id": "1357", "title": "Derivative of Cotangent Function", "text": ":$\\map {\\dfrac \\d {\\d x} } {\\cot x} = -\\csc^2 x = \\dfrac {-1} {\\sin^2 x}$ where $\\sin x \\ne 0$."}1192{"_id": "1358", "title": "Shape of Sine Function", "text": "The sine function is: :$(1): \\quad$ strictly increasing on the interval $\\closedint {-\\dfrac \\pi 2} {\\dfrac \\pi 2}$ :$(2): \\quad$ strictly decreasing on the interval $\\closedint {\\dfrac \\pi 2} {\\dfrac {3 \\pi} 2}$ :$(3): \\quad$ concave on the interval $\\closedint 0 \\pi$ :$(4): \\quad$ convex on the interval $\\closedint \\pi {2 \\pi}$"}1193{"_id": "1359", "title": "Derivative of Secant Function", "text": ":$\\map {\\dfrac \\d {\\d x} } {\\sec x} = \\sec x \\tan x$ where $\\cos x \\ne 0$."}1194{"_id": "1360", "title": "Derivative of Cosecant Function", "text": ":$\\map {\\dfrac \\d {\\d x} } {\\csc x} = -\\csc x \\cot x$ where $\\sin x \\ne 0$."}1195{"_id": "1361", "title": "Shape of Tangent Function", "text": "The nature of the tangent function on the set of real numbers $\\R$ is as follows: : $\\tan x$ is continuous and strictly increasing on the interval $\\left({-\\dfrac \\pi 2 \\,.\\,.\\, \\dfrac \\pi 2}\\right)$ : $\\tan x \\to + \\infty$ as $x \\to \\dfrac \\pi 2 ^-$ : $\\tan x \\to - \\infty$ as $x \\to -\\dfrac \\pi 2 ^+$ : $\\tan x$ is not defined on $\\forall n \\in \\Z: x = \\left({n + \\dfrac 1 2}\\right) \\pi$, at which points it is discontinuous : $\\forall n \\in \\Z: \\tan \\left({n \\pi}\\right) = 0$."}1196{"_id": "1362", "title": "Derivative of Arcsine Function", "text": ":$\\dfrac {\\map \\d {\\arcsin x} } {\\d x} = \\dfrac 1 {\\sqrt {1 - x^2} }$"}1197{"_id": "1363", "title": "Derivative of Arccosine Function", "text": ":$\\map {D_x} {\\arccos x} = \\dfrac {-1} {\\sqrt {1 - x^2}}$"}1198{"_id": "1364", "title": "Sum of Arcsine and Arccosine", "text": "Let $x \\in \\R$ be a real number such that $-1 \\le x \\le 1$. Then: : $\\arcsin x + \\arccos x = \\dfrac \\pi 2$ where $\\arcsin$ and $\\arccos$ denote arcsine and arccosine respectively."}1199{"_id": "1365", "title": "Derivative of Arctangent Function", "text": ":$\\dfrac {\\map \\d {\\arctan x} } {\\d x} = \\dfrac 1 {1 + x^2}$"}1200{"_id": "1367", "title": "Complex Numbers form Field", "text": "Consider the algebraic structure $\\struct {\\C, +, \\times}$, where: :$\\C$ is the set of all complex numbers :$+$ is the operation of complex addition :$\\times$ is the operation of complex multiplication Then $\\struct {\\C, +, \\times}$ forms a field."}

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