Characterisation of Cauchy Sequence in Non-Archimedean Norm

Let $\struct {R, \norm {\,\cdot\,} }$ be a normed division ring with non-Archimedean norm $\norm {\,\cdot\,}$. Let $\sequence {x_n}$ be a sequence in $R$. Then: :$\sequence {x_n}$ is a Cauchy sequence {{iff}} $\displaystyle \lim_{n \mathop \to \infty} \norm {x_{n + 1} - x_n} = 0$.


Characterisation of Cauchy Sequence in Non-Archimedean Norm/Necessary Condition

Let $\struct {R, \norm {\,\cdot\,} }$ be a normed division ring with non-Archimedean norm $\norm {\,\cdot\,}$. Let $\sequence {x_n}$ be a Cauchy sequence in $R$. Then: :$\lim_{n \mathop \to \infty} \norm {x_{n + 1} - x_n} = 0$


Characterisation of Cauchy Sequence in Non-Archimedean Norm/Sufficient Condition

Let $\struct {R, \norm {\,\cdot\,} }$ be a normed division ring with non-Archimedean norm $\norm {\,\cdot\,}$. Let $\sequence {x_n}$ be a sequence in $R$. Let $\displaystyle \lim_{n \mathop \to \infty} \norm {x_{n + 1} - x_n} = 0$. Then: :$\sequence {x_n}$ is a Cauchy sequence.


Power Series of Sine of Odd Theta

Let $r \in \R$ such that $\size r < 1$. Let $\theta \in \R$ such that $\theta \ne m \pi$ for any $m \in \Z$. Then: {{begin-eqn}} {{eqn | l = \sum_{k \mathop \ge 0} \map \sin {2 k + 1} \theta r^k       | r = \sin \theta + r \sin 3 \theta + r^2 \sin 5 \theta + \cdots       | c =  }} {{eqn | r = \dfrac {\paren {1 + r} \sin \theta} {1 - 2 r \cos 2 \theta + r^2}       | c =  }} {{end-eqn}}


Modulus of Limit/Normed Division Ring

Let $\struct {R, \norm { \, \cdot \, } }$ be a normed division ring. Let $\sequence {x_n}$ be a convergent sequence in $R$ to the limit $l$. That is, let $\displaystyle \lim_{n \mathop \to \infty} x_n = l$. Then :$\displaystyle \lim_{n \mathop \to \infty} \norm {x_n} = \norm l$


Three Points in Ultrametric Space have Two Equal Distances

Let $\struct {X, d}$ be an ultrametric space. Let $x, y, z \in X$ with $\map d {x, z} \ne \map d {y, z}$. Then: :$\map d {x, y} = \max \set {\map d {x, z}, \map d {y, z} }$


Three Points in Ultrametric Space have Two Equal Distances/Corollary

Let $\struct {X, d}$ be an ultrametric space. Let $x, y, z \in X$. Then: :at least two of the distances $\map d {x, y}$, $\map d {x, z}$ and $\map d {y, z}$ are equal.


Non-Archimedean Norm iff Non-Archimedean Metric/Necessary Condition

Let $\struct {R, \norm {\,\cdot\,}}$ be a non-Archimedean normed division ring. Let $d$ be the metric induced by $\norm {\,\cdot\,}$. Then $d$ is a non-Archimedean metric.


Non-Archimedean Norm iff Non-Archimedean Metric/Sufficient Condition

Let $\struct {R, \norm {\,\cdot\,} }$ be a normed division ring with zero $0$. Let $d$ be the metric induced by $\norm {\, \cdot \,}$. Let $d$ be non-Archimedean. Then: :$\norm {\, \cdot \,}$ is a non-Archimedean norm.


Non-Archimedean Norm iff Non-Archimedean Metric

Let $\struct {R, \norm {\, \cdot \,} }$ be a normed division ring with zero $0$. Let $d$ be the metric induced by $\norm {\,\cdot\,}$. Then: :$\norm {\, \cdot \,}$ is a non-Archimedean norm {{iff}} $d$ is a non-Archimedean metric.


Quotients of 3 Unequal Numbers are Unequal

Let $x, y, z \in \R_{\ne 0}$ be non-zero real numbers which are not all equal. Then $\dfrac x y, \dfrac y z, \dfrac z x$ are also not all equal.


Union of Power Sets not always Equal to Powerset of Union

The union of the power sets of two sets $S$ and $T$ is not necessarily equal to the power set of their union.


Limit of Intersection of Closed Intervals from Zero to Positive Integer Reciprocal

For all (strictly) positive integers $n \in \Z_{>0}$, let $A_n$ be the closed real interval: :$A_n = \closedint 0 {\dfrac 1 n}$ Let $A \subseteq \R$ be the subset of the real numbers defined as: :$A = \displaystyle \lim_{n \mathop \to \infty} \bigcap A_n$ Then: :$A = \set 0$


Distributive Laws/Examples/A cap B cap (C cup D) subset of (A cap D) cup (B cap C)

Let: :$P = A \cap B \cap \paren {C \cup D}$ :$Q = \paren {A \cap D} \cup \paren {B \cap C}$ Then: :$P \subseteq Q$


Total Number of Set Partitions

Let $S$ be a finite set of cardinality $n$. Then the number of different partitions of $S$ is $B_n$, where $B_n$ is the $n$th Bell number.


Combination Theorem for Sequences/Normed Division Ring/Inverse Rule

Suppose $l \ne 0$. Then: :$\exists k \in \N : \forall n \in \N: x_{k + n} \ne 0$ and the subsequence $\sequence { x_{k+n}^{-1} }$ is well-defined and convergent with: :$\displaystyle \lim_{n \mathop \to \infty} {x_{k + n} }^{-1} = l^{-1}$.


Bell Number as Summation over Lower Index of Stirling Numbers of the Second Kind

Let $B_n$ be the Bell number for $n \in \Z_{\ge 0}$. Then: :$B_n = \displaystyle \sum_{k \mathop = 0}^n {n \brace k}$ where $\displaystyle {n \brace k}$ denotes a Stirling number of the second kind.


Quotient Group of Solvable Group is Solvable

Let $G$ be a solvable group. Let $N$ be a normal subgroup of $G$. Then, $G / N$, the quotient group of $G$ by $N$ is solvable.


Equivalence of Well-Ordering Principle and Induction/Proof/PFI implies PCI

The Principle of Finite Induction implies the Principle of Complete Finite Induction. That is: :Principle of Finite Induction: Given a subset $S \subseteq \N$ of the natural numbers which has these properties: ::$0 \in S$ ::$n \in S \implies n + 1 \in S$ :then $S = \N$. implies: :Principle of Complete Finite Induction: Given a subset $S \subseteq \N$ of the natural numbers which has these properties: ::$0 \in S$ ::$\set {0, 1, \ldots, n} \subseteq S \implies n + 1 \in S$ :then $S = \N$.


Equivalence of Well-Ordering Principle and Induction/Proof/PCI implies WOP

The Principle of Complete Induction implies the Well-Ordering Principle. That is: :Principle of Complete Induction: Given a subset $S \subseteq \N$ of the natural numbers which has these properties: ::$0 \in S$ ::$\set {0, 1, \ldots, n} \subseteq S \implies n + 1 \in S$ :then $S = \N$. implies: :Well-Ordering Principle: Every nonempty subset of $\N$ has a minimal element.


Equivalence of Well-Ordering Principle and Induction/Proof/WOP implies PFI

The Well-Ordering Principle implies the Principle of Finite Induction. That is: :Well-Ordering Principle: Every non-empty subset of $\N$ has a minimal element implies: :Principle of Finite Induction: Given a subset $S \subseteq \N$ of the natural numbers which has these properties: ::$0 \in S$ ::$n \in S \implies n + 1 \in S$ :then $S = \N$.


Floor Function/Examples/Floor of 5 over 2

:$\floor {\dfrac 5 2} = 2$


Floor Function/Examples/Floor of Minus 5 over 2

:$\floor {-\dfrac 5 2} = -3$


Floor Function/Examples/Floor of 14

:$\floor {14} = 14$


Floor Function/Examples/Floor of Root 10

:$\floor {\sqrt {10} } = 3$


Combination Theorem for Cauchy Sequences/Inverse Rule

Suppose $\sequence {x_n}$ does not converge to $0$. Then: :$\exists K \in \N: \forall n > K : x_n \ne 0$ and the sequence: :$\sequence {\paren {x_{K + n} }^{-1} }_{n \mathop \in \N}$ is well-defined and a Cauchy sequence.


Subsequence of Cauchy Sequence in Normed Division Ring is Cauchy Sequence

Let $\struct {R, \norm {\, \cdot \,} }$ be a normed division ring with zero: $0$. Let $\sequence {x_n}$ be a Cauchy sequence in $R$. Let $\sequence {x_{n_r} }$ be a subsequence of $\sequence {x_n}$. Then: :$\sequence {x_{n_r} }$ is a Cauchy sequence in $R$.


Coprime Divisors of Square Number are Square

Let $r$ be a square number. Let $r = s t$ where $s$ and $t$ are coprime. Then both $s$ and $t$ are square.


Combination Theorem for Cauchy Sequences/Constant Rule

:the constant sequence $\tuple {a, a, a, \dots}$ is a Cauchy sequence.


Combination Theorem for Cauchy Sequences/Multiple Rule

:$\sequence {a x_n}$ is a Cauchy sequence.


Combination Theorem for Cauchy Sequences/Difference Rule

:$\sequence {x_n - y_n}$ is a Cauchy sequence.


Congruent Integers in Same Residue Class

Let $m \in \Z_{>0}$ be a (strictly) positive integer. Let $\Z_m$ be the set of residue classes modulo $m$: :$Z_m = \set {\eqclass 0 m, \eqclass 1 m, \dotsc, \eqclass {m - 1} m}$ Let $a, b \in \set {0, 1, \ldots, m -1 }$. Then: :$\eqclass a m = \eqclass b m \iff a \equiv b \pmod m$


Residue Classes form Partition of Integers

Let $m \in \Z_{>0}$ be a (strictly) positive integer. Let $\Z_m$ be the set of residue classes modulo $m$: :$\Z_m = \set {\eqclass 0 m, \eqclass 1 m, \dotsc, \eqclass {m - 1} m}$ Then $\Z_m$ forms a partition of $\Z$.


Cardinality of Set of Residue Classes

Let $m \in \Z_{>0}$ be a (strictly) positive integer. Let $\Z_m$ be the set of residue classes modulo $m$. Then: :$\card {Z_m} = m$ where $\card { \, \cdot \,}$ denotes cardinality.


Structure Induced by Ring with Unity Operations is Ring with Unity

Let $\struct {R, +, \circ}$ be a ring with unity whose unity is $1_R$. Let $S$ be a set. Let $\struct {R^S, +', \circ'}$ be the structure on $R^S$ induced by $+'$ and $\circ'$. Then $\struct {R^S, +', \circ'}$ is a ring with unity whose unity is $f_{1_R}: S \to R$, defined by: :$\forall s \in S: \map {f_{1_R} } s = 1_R$


Cauchy Sequences form Ring with Unity

Let $\struct {R, +, \circ, \norm {\, \cdot \,} }$ be a normed division ring. Let $\struct {R^\N, +, \circ}$ be the ring of sequences over $R$ with unity $\tuple {1, 1, 1, \dotsc}$. Let $\CC \subset R^\N$ be the set of Cauchy sequences on $R$. Then: :$\struct {\CC, +, \circ}$ is a subring of $R^\N$ with unity $\tuple {1, 1, 1, \dotsc}$.


Equivalence Relation on Power Set induced by Intersection with Subset

Let $A, T$ be sets such that $A \subseteq T$. Let $S = \powerset T$ denote the power set of $T$. Let $\alpha$ denote the relation defined on $S$ by: :$\forall X, Y \in S: X \mathrel \alpha Y \iff X \cap A = Y \cap A$ Then $\alpha$ is an equivalence relation.


Equivalence Relation on Power Set induced by Intersection with Subset/Equivalence Class of Empty Set

The equivalence class of $\O$ in $S$ with respect to $\alpha$ is given by: :$\eqclass \O \alpha = \powerset {T \setminus A}$


Reflexive and Symmetric Relation is not necessarily Transitive

Let $S$ be a set. Let $\alpha \subseteq S \times S$ be a relation on $S$. Let $\alpha$ be both reflexive and symmetric. Then it is not necessarily the case that $\alpha$ is also transitive.


Symmetric and Transitive Relation is not necessarily Reflexive

Let $S$ be a set. Let $\alpha \subseteq S \times S$ be a relation on $S$. Let $\alpha$ be both symmetric and transitive. Then it is not necessarily the case that $\alpha$ is also reflexive.


Hilbert Proof System Instance 2 Independence Results

Let $\mathscr H_2$ be Instance 2 of the Hilbert proof systems. Then the following independence results hold:


Equivalence Relation on Natural Numbers such that Quotient is Power of Two

Let $\alpha$ denote the relation defined on the natural numbers $\N$ by: :$\forall x, y \in \N: x \mathrel \alpha y \iff \exists n \in \Z: x = 2^n y$ Then $\alpha$ is an equivalence relation.


Equivalence Relation on Natural Numbers such that Quotient is Power of Two/Equivalence Class of Prime

Let $\eqclass p \alpha$ be the $\alpha$-equivalence class of a prime number $p$. Then $\eqclass p \alpha$ contains no other prime number other than $p$.


Equivalence Relation on Natural Numbers such that Quotient is Power of Two/Smallest Equivalence Class with no Prime

Let $\eqclass x \alpha$ denote the $\alpha$-equivalence class of a natural number $x$. Let $r$ be the smallest natural number such that $\eqclass r \alpha$ contains no prime number. Then $r = 9$.


Equivalence Relation on Square Matrices induced by Positive Integer Powers

Let $n \in \Z_{>0}$ be a (strictly) positive integer. Let $S$ be the set of all square matrices of order $n$. Let $\alpha$ denote the relation defined on $S$ by: :$\forall \mathbf A, \mathbf B \in S: \mathbf A \mathrel \alpha \mathbf B \iff \exists r, s \in \N: \mathbf A^r = \mathbf B^s$ Then $\alpha$ is an equivalence relation.


Equivalence Relation on Integers Modulo 5 induced by Squaring

Let $\beta$ denote the relation defined on the integers $\Z$ by: :$\forall x, y \in \Z: x \mathrel \beta y \iff x^2 \equiv y^2 \pmod 5$ Then $\beta$ is an equivalence relation.


Equivalence Relation on Integers Modulo 5 induced by Squaring/Number of Equivalence Classes

The number of distinct $\beta$-equivalence classes is $3$: {{begin-eqn}} {{eqn | l = \eqclass 0 \beta       | o =  }} {{eqn | l = \eqclass 1 \beta       | r = \eqclass 4 \beta       | c =  }} {{eqn | l = \eqclass 2 \beta       | r = \eqclass 3 \beta       | c =  }} {{end-eqn}}


Equivalence Relation on Integers Modulo 5 induced by Squaring/Addition Modulo Beta is not Well-Defined

Let the $+_\beta$ operator ("addition") on the $\beta$-equivalence classes be defined as: :$\eqclass a \beta +_\beta \eqclass b \beta := \eqclass {a + b} \beta$ Then such an operation is not well-defined.


Constant Sequence Converges to Constant in Normed Division Ring

:the constant sequence $\tuple {\lambda, \lambda, \lambda, \dots}$ is convergent and $\displaystyle \lim_{n \mathop \to \infty} \lambda = \lambda$


Product of Sequence Converges to Zero with Cauchy Sequence Converges to Zero

Let $\struct {R, \norm {\,\cdot\,} }$ be a normed division ring with zero $0$. Let $\sequence {x_n}$, $\sequence {y_n} $ be sequences in $R$. Let $\sequence {x_n}$ converge to $0$. Let $\sequence {y_n}$ be a Cauchy sequence. Then: :$\sequence {x_n y_n}$ and $\sequence {y_n x_n}$ converge to $0$.


Cauchy Sequence is Bounded/Normed Division Ring

Let $\struct {R, \norm {\,\cdot\,} }$ be a normed division ring. Every Cauchy sequence in $R$ is bounded.


Cauchy Sequence with Finite Elements Prepended is Cauchy Sequence

Let $\struct {R, \norm {\, \cdot \,} }$ be a normed division ring. Let $\sequence {x_n}$ be a sequence in $R$. Let $N \in \N$ Let $\sequence {y_n}$ be the sequence defined by: :$\forall n, y_n = x_{N + n}$ Let $\sequence {y_n}$ be a Cauchy sequence in $R$. Then: :$\sequence {x_n}$ is a Cauchy sequence in $R$.


Convergent Sequence with Finite Elements Prepended is Convergent Sequence

Let $\struct {R, \norm { \, \cdot \, } }$ be a normed division ring. Let $\sequence {x_n}$ be a sequence in $R$. Let $N \in \N$ Let $\sequence {y_n}$ be the sequence defined by: :$\forall n, y_n = x_{N+n}$ Let $\sequence {y_n}$ be a convergent sequence in $R$ with limit $l$. Then: :$\sequence {x_n}$ is a convergent sequence in $R$ with limit $l$.


Maximal Left and Right Ideal iff Quotient Ring is Division Ring

Let $R$ be a ring with unity. Let $J$ be an ideal of $R$. Then the following are equivalent: :$(1): \quad J$ is a maximal left ideal :$(2): \quad J$ is a maximal right ideal :$(3): \quad$ the quotient ring $R / J$ is a division ring.


Definition:Constructed Semantics/Instance 2/Rule of Addition

The Rule of Addition: :$q \implies (q \lor p)$ is a tautology in Instance 2 of constructed semantics.


Definition:Constructed Semantics/Instance 2/Rule of Commutation

The Rule of Commutation: :$\left({p \lor q}\right) \implies \left({q \lor p}\right)$ is a tautology in Instance 2 of constructed semantics.


Definition:Constructed Semantics/Instance 2/Factor Principle

The Factor Principle: :$\left({p \implies q}\right) \implies \left({\left({r \lor p}\right) \implies \left ({r \lor q}\right)}\right)$ is a tautology in Instance 2 of constructed semantics.


Test for Left Ideal

Let $J$ be a subset of a ring $\struct {R, +, \circ}$. Then $J$ is an left ideal of $\struct{R, +, \circ}$ {{iff}} these all hold: :$(1): \quad J \ne \O$ :$(2): \quad \forall x, y \in J: x + \paren {-y} \in J$ :$(3): \quad \forall j \in J, r \in R: r \circ j \in J$


Test for Right Ideal

Let $J$ be a subset of a ring $\struct {R, +, \circ}$. Then $J$ is a right ideal of $\struct {R, +, \circ}$ {{iff}} these all hold: :$(1): \quad J \ne \O$ :$(2): \quad \forall x, y \in J: x + \paren {-y} \in J$ :$(3): \quad \forall j \in J, r \in R: j \circ r \in J$


Maximal Left and Right Ideal iff Quotient Ring is Division Ring/Maximal Left Ideal implies Quotient Ring is Division Ring

Let $R$ be a ring with unity. Let $J$ be an ideal of $R$. If $J$ is a maximal left ideal then the quotient ring $R / J$ is a division ring.


Maximal Left and Right Ideal iff Quotient Ring is Division Ring/Maximal Right Ideal implies Quotient Ring is Division Ring

Let $R$ be a ring with unity. Let $J$ be an ideal of $R$. If $J$ is a maximal right ideal then the quotient ring $R / J$ is a division ring.


Inverse of Injective and Surjective Mapping is Mapping

Let $S$ and $T$ be sets. Let $f: S \to T$ be a mapping such that: :$(1): \quad f$ is an injection :$(2): \quad f$ is a surjection. Then the inverse $f^{-1}$ of $f$ is itself a mapping.


Inverse is Mapping implies Mapping is Injection and Surjection

Let $S$ and $T$ be sets. Let $f: S \to T$ be a mapping. Let the inverse $f^{-1} \subseteq T \times S$ itself be a mapping. Then: :$(1): \quad f$ is an injection :$(2): \quad f$ is a surjection.


Maximal Left and Right Ideal iff Quotient Ring is Division Ring/Quotient Ring is Division Ring implies Maximal Right Ideal

Let $R$ be a ring with unity. Let $J$ be an ideal of $R$. If the quotient ring $R / J$ is a division ring then $J$ is a maximal right ideal.


Maximal Left and Right Ideal iff Quotient Ring is Division Ring/Quotient Ring is Division Ring implies Maximal Left Ideal

Let $R$ be a ring with unity. Let $J$ be an ideal of $R$. If the quotient ring $R / J$ is a division ring then $J$ is a maximal left ideal.


Quotient Ring of Cauchy Sequences is Division Ring

Let $\struct {R, \norm {\, \cdot \,} }$ be a normed division ring. Let $\CC$ be the ring of Cauchy sequences over $R$. Let $\NN$ be the set of null sequences. Then the quotient ring $\CC / \NN$ is a division ring.


Hilbert Proof System Instance 2 Independence Results/Independence of A1

Axiom $(A1)$ is independent from $(A2)$, $(A3)$, $(A4)$.


Embedding Ring into Ring Structure Induced by Ring Operations

Let $\struct {R, +, \circ}$ be a ring. Let $S$ be a non-empty set. Let $\struct {R^S, +', \circ'}$ be the ring of mappings, where $+'$ and $\circ'$ are the pointwise operations induced on $R^S$ by $+$ and $\circ$. For each $r \in R$, let $f_r: S \to R$ be the mapping defined by: :$\forall s \in S, \map {f_r} s = r$ That is, $f_r$ is the constant mapping from $S$ to $r$. Let $\phi: R \to R^S$ be the mapping from the ring $R$ to the ring $R^S$ defined by: :$\forall r \in R: \map \phi r = f_r$ Then: :$\phi$ is a ring monomorphism.


Embedding Normed Division Ring into Ring of Cauchy Sequences

Let $\struct {R, \norm {\, \cdot \,} }$ be a normed division ring. Let $\CC$ be the ring of Cauchy sequences over $R$ Let $\phi: R \to \CC$ be the mapping from $R$ to $\CC$ defined as: :$\forall a \in R: \map \phi a = \tuple {a, a, a, \dots}$ where $\tuple {a, a, a, \dots}$ is the constant sequence. Then $\phi$ is a ring monomorphism.


Hilbert Proof System Instance 2 Independence Results/Independence of A2

Axiom $(A2)$ is independent from $(A1)$, $(A3)$, $(A4)$.


Definition:Constructed Semantics/Instance 3/Rule of Idempotence

The Rule of Idempotence: :$(p \lor p) \implies p$ is a tautology in Instance 3 of constructed semantics.


Definition:Constructed Semantics/Instance 3/Rule of Commutation

The Rule of Commutation: :$\left({p \lor q}\right) \implies \left({q \lor p}\right)$ is a tautology in Instance 3 of constructed semantics.


Definition:Constructed Semantics/Instance 3/Factor Principle

The Factor Principle: :$\left({p \implies q}\right) \implies \left({\left({r \lor p}\right) \implies \left ({r \lor q}\right)}\right)$ is a tautology in Instance 3 of constructed semantics.


Hilbert Proof System Instance 2 Independence Results/Independence of A3

Axiom $(A3)$ is independent from $(A1)$, $(A2)$, $(A4)$.


Definition:Constructed Semantics/Instance 4/Rule of Idempotence

The Rule of Idempotence: :$(p \lor p) \implies p$ is a tautology in Instance 4 of constructed semantics.


Definition:Constructed Semantics/Instance 4/Rule of Addition

The Rule of Addition: :$q \implies (q \lor p)$ is a tautology in Instance 4 of constructed semantics.


Definition:Constructed Semantics/Instance 4/Factor Principle

The Factor Principle: :$\left({p \implies q}\right) \implies \left({\left({r \lor p}\right) \implies \left ({r \lor q}\right)}\right)$ is a tautology in Instance 4 of constructed semantics.


Cardinality of Set of Surjections

Let $S$ and $T$ be finite sets. Let $\card S = m, \card T = n$. Let $C$ be the number of surjections from $S$ to $T$. Then: :$C = n! \displaystyle {m \brace n}$ where $\displaystyle {m \brace n}$ denotes a Stirling number of the second kind.


Hilbert Proof System Instance 2 Independence Results/Independence of A4

Axiom $(\text A 4)$ is independent from $(\text A 1)$, $(\text A 2)$, $(\text A 3)$.


Image of Set Difference under Mapping/Corollary 3

Let $f: S \to T$ be a surjection. Let $A \subseteq S$ be a subset of $S$. Then: :$T \setminus f \sqbrk A \subseteq f \sqbrk {S \setminus A}$ where $\setminus$ denotes set difference.


Definition:Constructed Semantics/Instance 5/Rule of Idempotence

The Rule of Idempotence: :$(p \lor p) \implies p$ is a tautology in Instance 5 of constructed semantics.


Additive Function is Linear for Rational Factors

Let $f: \R \to \R$ be an additive function. Then: :$\forall r \in \Q, x \in \R: \map f {x r} = r \map f x$


Definition:Constructed Semantics/Instance 5/Rule of Addition

The Rule of Addition: :$q \implies (q \lor p)$ is a tautology in Instance 5 of constructed semantics.


Definition:Constructed Semantics/Instance 5/Rule of Commutation

The Rule of Commutation: :$\left({p \lor q}\right) \implies \left({q \lor p}\right)$ is a tautology in Instance 5 of constructed semantics.


Additive Function of Zero is Zero

Let $f: \R \to \R$ be an additive function. Then: :$f \paren 0 = 0$


Additive Function is Odd Function

Let $f: \R \to \R$ be an additive function. Then $f$ is an odd function.


Cross-Relation on Real Numbers is Equivalence Relation

Let $\R^2$ denote the cartesian plane. Let $\alpha$ denote the relation defined on $\R^2$ by: :$\tuple {x_1, y_1} \mathrel \alpha \tuple {x_2, y_2} \iff x_1 + y_2 = x_2 + y_1$ Then $\alpha$ is an equivalence relation on $\R^2$.


Cross-Relation on Real Numbers is Equivalence Relation/Geometrical Interpretation

The equivalence classes of $\alpha$, when interpreted as points in the plane, are the straight lines of slope $1$.


Subtraction on Numbers is Anticommutative/Natural Numbers

The operation of subtraction on the natural numbers $\N$ is anticommutative, and defined only when $a = b$: That is: :$a - b = b - a \iff a = b$


Subtraction on Numbers is Anticommutative/Integral Domains

The operation of subtraction on the numbers is anticommutative. That is: :$a - b = b - a \iff a = b$


Natural Number Subtraction is not Closed

The operation of subtraction on the natural numbers is not closed.


Integers under Addition form Semigroup

The set of integers under addition $\struct {\Z, +}$ forms a semigroup.


Natural Numbers under Multiplication form Subsemigroup of Integers

Let $\struct {\N, \times}$ denote the set of natural numbers under multiplication. Let $\struct {\Z, \times}$ denote the set of integers under multiplication. Then $\struct {\N, \times}$ is a subsemigroup of $\struct {\Z, \times}$.


Cauchy Sequence Is Eventually Bounded Away From Non-Limit

Let $\struct {R, \norm {\, \cdot \,} }$ be a normed division ring. Let $\sequence {x_n}$ be a Cauchy sequence in $R$. Suppose $\sequence {x_n}$ does not converge to $l \in R$, then: :$\exists K \in \N$ and $C \in \R_{>0}: \forall n > K: C < \norm {x_n - l}$


Embedding Division Ring into Quotient Ring of Cauchy Sequences

Let $\struct {R, \norm {\, \cdot \,} }$ be a normed division ring. Let $\CC$ be the ring of Cauchy sequences over $R$ Let $\NN = \set {\sequence {x_n}: \displaystyle \lim_{n \mathop \to \infty} x_n = 0}$ Let $\norm {\, \cdot \,}: \CC \, \big / \NN \to \R_{\ge 0}$ be the norm on the quotient ring $\CC \, \big / \NN$ defined by: :$\displaystyle \forall \sequence {x_n} + \NN: \norm {\sequence {x_n} + \NN} = \lim_{n \mathop \to \infty} \norm {x_n}$ Let $\phi: R \to \CC \, \big / \NN$ be the mapping from $R$ to the quotient ring $\CC \, \big / \NN$ defined by: :$\forall a \in R: \map \phi a = \sequence {a, a, a, \dotsc} + \NN$ where $\sequence {a, a, a, \dotsc} + \NN$ is the left coset in $\CC \, \big / \NN$ that contains the constant sequence $\sequence {a, a, a, \dotsc}$. Then: :$\phi$ is a distance-preserving ring monomorphism.


Non-Zero Real Numbers under Multiplication form Group

Let $\R_{\ne 0}$ be the set of real numbers without zero: :$\R_{\ne 0} = \R \setminus \set 0$ The structure $\struct {\R_{\ne 0}, \times}$ forms a group.


Symmetric Group on n Letters is Isomorphic to Symmetric Group

The symmetric group on $n$ letters $\struct {S_n, \circ}$ is isomorphic to the symmetric group on the $n$ elements of any set $T$ whose cardinality is $n$. That is: :$\forall T \subseteq \mathbb U, \card T = n: \struct {S_n, \circ} \cong \struct {\Gamma \paren T, \circ}$


Symmetric Groups of Same Order are Isomorphic

Let $n \in \Z_{>0}$ be a (strictly) positive integer. Let $T_1$ and $T_2$ be sets whose cardinality $\card {T_1}$ and $\card {T_2}$ are both $n$. Let $\struct {\map \Gamma {T_1}, \circ}$ and $\struct {\map \Gamma {T_2}, \circ}$ be the symmetric group on $S$ and $T$ respectively. Then $\struct {\map \Gamma {T_1}, \circ}$ and $\struct {\map \Gamma {T_2}, \circ}$ are isomorphic.


Quotient Ring of Cauchy Sequences is Normed Division Ring

Let $\struct {R, \norm {\, \cdot \,} }$ be a normed division ring. Let $\CC$ be the ring of Cauchy sequences over $R$ Let $\NN$ be the set of null sequences. For all $\sequence {x_n} \in \CC$, let $\eqclass {x_n} {}$ denote the left coset $\sequence {x_n} + \NN$ Let $\norm {\, \cdot \,}_1: \CC \,\big / \NN \to \R_{\ge 0}$ be defined by: :$\displaystyle \forall \eqclass {x_n} {} \in \CC \,\big / \NN: \norm {\eqclass {x_n} {} }_1 = \lim_{n \mathop \to \infty} \norm {x_n}$ Then: :$\struct {\CC \,\big / \NN, \norm {\, \cdot \,}_1 }$ is a normed division ring.


Inequality Rule for Real Sequences

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 there exist $N \in \N$ such that: :$\forall n \ge N: x_n \le y_n$ Then: :$l \le m$


Rule of Association/Disjunction/Formulation 2/Reverse Implication

:$\vdash \left({p \lor \left({q \lor r}\right)}\right) \impliedby \left({\left({p \lor q}\right) \lor r}\right)$


Left Regular Representation of Subset Product

Let $\struct {S, \circ}$ be a magma. Let $T \subseteq S$ be a subset of $S$. Let $\lambda_a: S \to S$ be the left regular representation of $S$ with respect to $a$. Then: :$\lambda_a \sqbrk T = \set a \circ T = a \circ T$ where $a \circ T$ denotes subset product with a singleton.


Right Regular Representation of Subset Product

Let $\struct {S, \circ}$ be a magma. Let $T \subseteq S$ be a subset of $S$. Let $\rho_a: S \to S$ be the right regular representation of $S$ with respect to $a$. Then: :$\rho_a \sqbrk T = T \circ \set a = T \circ a$ where $T \circ a$ denotes subset product with a singleton.


Order of Cycle is Length of Cycle

Let $S_n$ denote the symmetric group on $n$ letters. Let $\pi \in S_n$ be a cyclic permutation of length $k$. Then: :$\order \pi = k$ where: :$\order \pi$ denotes the order of $\pi$ in $S_n$.


Norm Sequence of Cauchy Sequence has Limit

Let $\struct {R, \norm {\, \cdot \,} }$ be a normed division ring. Let $\sequence {x_n}$ be a Cauchy sequence in $R$. Then $\sequence {\norm {x_n} }$ has a limit in $\R$. That is, :$\exists l \in \R: \displaystyle \lim_{n \mathop \to \infty} \norm {x_n} = l$


Equivalent Cauchy Sequences have Equal Limits of Norm Sequences

Let $\struct {R, \norm {\, \cdot \,} }$ be a normed division ring. Let $\sequence {x_n}$ and $\sequence {y_n}$ be Cauchy sequences in $R$. Let $\displaystyle \lim_{n \mathop \to \infty} {x_n - y_n} = 0$. Then: :$\displaystyle \lim_{n \mathop \to \infty} \norm {x_n} = \lim_{n \mathop \to \infty} \norm {y_n}$


Rule of Conjunction/Sequent Form/Formulation 2

:$\vdash p \implies \paren{ q \implies \paren{ p \land q } }$


Subgroup Generated by One Element is Cyclic

Let $G$ be a group. Let $a \in G$. Then $\gen a$, the subgroup generated by $a$, is cyclic:


Order of Elements in Quaternion Group

Let $Q = \Dic 2$ be the quaternion group, whose group presentation is given by: :$\Dic 2 = \gen {a, b: a^4 = e, b^2 = a^2, a b a = b}$ Then $\Dic 2$ has: : $1$ element of order $2$ and: : $6$ elements of order $4$.


Reduced Residues Modulo 5 under Multiplication form Cyclic Group

Let $\struct {\Z'_5, \times_5}$ denote the multiplicative group of reduced residues modulo $5$. Then $\struct {\Z'_5, \times_5}$ is cyclic.


Number of Generators of Cyclic Group whose Order is Power of 2

Let $G$ be a finite cyclic group. Let the order of $G$ be $2^k$ for some $k \in \Z_{>0}$. Then $G$ has $2^{n - 1}$ distinct generators.


Element of Group is in its own Coset/Left

Let: : $x H$ be the left coset of $x$ modulo $H$. Then: : $x \in x H$


Element of Group is in its own Coset/Right

Let: : $H x$ be the right coset of $x$ modulo $H$. Then: : $x \in H x$


Element of Group is in Unique Coset of Subgroup/Right

There exists a exactly one right coset of $H$ containing $x$, that is: $H x$


Equivalence of Definitions of Infinite Order Element

{{TFAE|def = Infinite Order Element}} Let $G$ be a group whose identity is $e_G$. Let $x \in G$ be an element of $G$.


Equivalence of Definitions of Finite Order Element

{{TFAE|def = Finite Order Element}} Let $G$ be a group whose identity is $e_G$. Let $x \in G$ be an element of $G$.


Right Cosets are Equal iff Left Cosets by Inverse are Equal

Let $G$ be a group whose identity is $e$. Let $H$ be a subgroup of $G$. Let $g_1, g_2 \in G$. Then: :$H g_1 = H g_2 \iff {g_1}^{-1} H = {g_2}^{-1} H$ where: :${g_1}^{-1}$ and ${g_2}^{-1}$ denote the inverses of $g_1$ and $g_2$ in $G$ :$H g_1$ and $H g_2$ denote the right cosets of $H$ by $g_1$ and $g_2$ respectively :${g_1}^{-1} H$ and ${g_2}^{-1} H$ denote the left cosets of $H$ by ${g_1}^{-1}$ and ${g_2}^{-1}$ respectively.


Subgroup of Subgroup with Prime Index

Let $\struct {G, \circ}$ be a group. Let $H$ be a subgroup of $G$. Let $K$ be a subgroup of $H$. Let: :$\index G K = p$ where: :$p$ denotes a prime number :$\index G K$ denotes the index of $K$ in $G$. Then either: :$H = K$ or: :$H = G$


Completeness Theorem for Hilbert Proof System Instance 2 and Boolean Interpretations

Instance 2 of the Hilbert proof systems is a complete proof system for boolean interpretations. That is, for every WFF $\mathbf A$: :$\models_{\mathrm{BI}} \mathbf A$ implies $\vdash_{\mathscr H_2} \mathbf A$


Klein Four-Group as Subgroup of S4

Let $G$ be the following subset of the symmetric group on $4$ letters $S_4$, expressed in two-row notation: {{begin-eqn}} {{eqn | l = e       | r = \begin{bmatrix} 1 & 2 & 3 & 4 \\ 1 & 2 & 3 & 4 \end{bmatrix}       | c =  }} {{eqn | l = a       | r = \begin{bmatrix} 1 & 2 & 3 & 4 \\ 2 & 1 & 4 & 3 \end{bmatrix}       | c =  }} {{eqn | l = b       | r = \begin{bmatrix} 1 & 2 & 3 & 4 \\ 3 & 4 & 1 & 2 \end{bmatrix}       | c =  }} {{eqn | l = c       | r = \begin{bmatrix} 1 & 2 & 3 & 4 \\ 4 & 3 & 2 & 1 \end{bmatrix}       | c =  }} {{end-eqn}} Then $G$ is an example of the Klein $4$-group.


Non-Abelian Order 10 Group has Order 5 Element

Let $G$ be a non-abelian group of order $10$. Then $G$ has at least one element of order $5$.


Quotient of Cauchy Sequences is Metric Completion

Let $\struct {R, \norm {\, \cdot \,} }$ be a normed division ring. Let $d$ be the metric induced by $\struct {R, \norm {\, \cdot \,} }$. Let $\mathcal C$ be the ring of Cauchy sequences over $R$. Let $\mathcal N$ be the set of null sequences in $R$. Let $\mathcal C \,\big / \mathcal N$ be the quotient ring of Cauchy sequences of $\mathcal C$ by the maximal ideal $\mathcal N$. Let $\norm {\, \cdot \,}: \mathcal C \,\big / \mathcal N \to \R_{\ge 0}$ be the norm on the quotient ring $\mathcal C \,\big / \mathcal N$ defined by: :$\displaystyle \forall \sequence {x_n} + \mathcal N: \norm {\sequence {x_n} + \mathcal N} = \lim_{n \mathop \to \infty} \norm{x_n}$ Let $d'$ be the metric induced by $\struct {\mathcal C \,\big / \mathcal N, \norm {\, \cdot \,} }$ Let $\phi: R \to \mathcal C \,\big / \mathcal N$ be the mapping from $R$ to the quotient ring $\mathcal C \,\big / \mathcal N$ defined by: :$\forall a \in R: \map \phi a = \sequence {a, a, a, \ldots} + \mathcal N$ where $\sequence {a, a, a, \ldots} + \mathcal N$ is the left coset in $\mathcal C \, \big / \mathcal N$ that contains the constant sequence $\sequence {a, a, a, \ldots}$. Then: :$\struct {\mathcal C \,\big / \mathcal N, d'}$ is the metric completion of $\struct {R,d}$ and: :$\map \phi R$ is a dense subset of $\mathcal C \,\big / \mathcal N$


Completion of Normed Division Ring

Let $\struct {R, \norm {\, \cdot \,} }$ be a normed division ring. Then: :$\struct {R, \norm {\, \cdot \,} }$ has a normed division ring completion $\struct {R', \norm {\, \cdot \,}' }$


Group of Order 27 has Subgroup of Order 3

Let $G$ be a group whose identity element is $e$. Let $G$ be of order $27$. Then $G$ has at least one subgroup of order $3$.


Group does not Necessarily have Subgroup of Order of Divisor of its Order

Let $G$ be a finite group whose order is $n$. Let $d$ be a divisor of $n$. Then it is not necessarily the case that $G$ has a subgroup of order $d$.


Characteristic Function of Normal Distribution

The characteristic function of the normal distribution with mean $\mu$ and variance $\sigma^2$ is :$\map \phi t = e^{i t \mu - \frac 1 2 t^2 \sigma^2}$


Product of Subset with Intersection/Equality does not Hold

While it is the case that: :$X \circ \paren {Y \cap Z} \subseteq \paren {X \circ Y} \cap \paren {X \circ Z}$ it is not necessarily the case that: :$X \circ \paren {Y \cap Z} = \paren {X \circ Y} \cap \paren {X \circ Z}$


Left Cosets are Equal iff Element in Other Left Coset

Let $x H$ denote the left coset of $H$ by $x$. Then: :$x H = y H \iff x \in y H$


Right Cosets are Equal iff Element in Other Right Coset

Let $H x$ denote the right coset of $H$ by $x$. Then: :$H x = H y \iff x \in H y$


Order of Element divides Order of Centralizer

Let $G$ be a finite group. Let $x \in G$ be an element of $G$. Let $\map {C_G} x$ denote the centralizer of $x$. Then: :$\order x \divides \order {\map {C_G} x}$ where: :$\order x$ denotes the order of $x$ in $G$ :$\divides$ denotes divisibility :$\order {\map {C_G} x}$ denotes the order of $\map {C_G} x$.


Left Coset of Stabilizer in Group of Transformations

Let $S$ be a non-empty set. Let $G$ be a group of permutations of $S$. Let $t \in G$. Let $G_t$ be the set defined as: :$G_t = \set {g \in G: \map g t = t}$ Then each left coset of $G_t$ in $G$ consists of the elements of $G$ that map $t$ to some element of $S$. {{explain|The source work does not discuss group actions, but still defines $G_t$ as the stabilizer of $t$ in $G$. This needs to be reviewed and put into the language of group actions as a result related to transformation group action -- but this area of group theory is not as well covered in {{ProofWiki}} as it ought to be. I need to dig out my college notes on group actions, which were more comprehensive and understandable than any of the other works I have on my shelf, which will also need to be exploited properly.<br>Hence the second part of this question in Whitelaw is not covered yet.}}


Normed Division Ring is Field iff Completion is Field

Let $\struct {R, \norm {\, \cdot \,} }$ be a normed division ring. Let $\struct {R', \norm {\, \cdot \,}' }$ be a normed division ring completion of $\struct {R, \norm {\, \cdot \,} }$ Then: :$R$ is a field {{iff}} $R'$ is a field.


Non-Abelian Order 8 Group has Order 4 Element

Let $G$ be a non-abelian group of order $8$. Then $G$ has at least one element of order $4$.


Group of Prime Order p has p-1 Elements of Order p

Let $p$ be a prime number. Let $G$ be a group with identity $e$ whose order is $p$. Then $G$ has $p - 1$ elements of order $p$.


Number of Order p Elements in Group with m Order p Subgroups

Let $G$ be a group whose identity is $e$. Let $G$ have $m$ subgroups of order $p$. The total number of elements of $G$ of order $p$ is $m \paren {p - 1}$.


Order of Finite Abelian Group with p+ Order p Elements is Divisible by p^2

Let $p$ be a prime number. Let $G$ be a finite abelian group whose identity is $e$. Let $G$ have at least $p$ elements of order $p$. Then: : $p^2 \divides \order G$ where: :$\divides$ denotes divisibility :$\order G$ denotes the order of $G$.


Abelian Group of Semiprime Order is Cyclic

Let $p$ and $q$ be distinct prime numbers. Let $G$ be an abelian group such that: :$\order G = p q$ where $\order G$ denotes the order of $G$. Then $G$ is cyclic.


General Linear Group to Determinant is Homomorphism/Corollary

The kernel of the $\det$ mapping is the special linear group $\SL {n, \R}$.


Quotient Group of General Linear Group by Special Linear Group

Let $\GL {n, \R}$ denote the general linear group of degree $n$ over $\R$. Let $\SL {n, \R}$ denote the special linear group of degree $n$ over $\R$. Then the quotient group $\GL {n, \R} / \SL {n, \R}$ is the multiplicative group of real numbers $\struct {\R_{\ne 0}, \times}$.


Normed Division Ring Completions are Isometric and Isomorphic/Lemma 1

Let $\psi' = \phi_2 \circ \phi_1^{-1}:\phi_1 \paren R \to \phi_2 \paren R$ be the composition of $\phi_1^{-1}$ with $\phi_2$.  Then $\psi': \struct {\map {\phi_1} R, \norm {\, \cdot \,}_1 } \to \struct {\map {\phi_2} R, \norm {\, \cdot \,}_2 }$ is an isometric ring isomorphism.


Normed Division Ring Completions are Isometric and Isomorphic/Lemma 2

Let $\psi: S_1 \to S_2$ be defined by: :$\forall x \in S_1: \map \psi x = \displaystyle \lim_{n \mathop \to \infty} \map {\psi'} {x_n}$ where $x = \displaystyle \lim_{n \mathop \to \infty} x_n$ for some sequence $\sequence {x_n} \subseteq R_1$ Then $\psi$ is a well-defined mapping.


Normed Division Ring Completions are Isometric and Isomorphic/Lemma 3

:$\psi$ is a surjective mapping.


Normed Division Ring Completions are Isometric and Isomorphic/Lemma 4

:$\psi$ is an isometry.


Normed Division Ring Completions are Isometric and Isomorphic/Lemma 5

:$\psi$ is a ring isomorphism.


Normed Division Ring Completions are Isometric and Isomorphic

Let $\struct {R, \norm {\, \cdot \,} }$ be a normed division ring. Let $\struct {S_1, \norm {\, \cdot \,}_1 }$ and $\struct {S_2, \norm {\, \cdot \,}_2 }$ be normed division ring completions of $\struct {R, \norm {\, \cdot \,} }$ Then there exists an isometric isomorphism $\psi: \struct {S_1, \norm {\, \cdot \,}_1 } \to \struct {S_2, \norm {\, \cdot \,}_2 }$


Distance-Preserving Image Isometric to Domain for Metric Spaces

Let $M_1 = \struct {A_1, d_1}$ and $M_2 = \struct {A_2, d_2}$ be metric spaces. Let $\phi: M_1 \to M_2$ be a distance-preserving mapping. Then: :$\phi: M_1 \to \Img \phi$ is an isometry.


Distance-Preserving Mapping is Injection of Metric Spaces

Let $M_1 = \struct {A_1, d_1}$ and $M_2 = \struct {A_2, d_2}$ be metric spaces. Let $\phi: M_1 \to M_2$ be a distance-preserving mapping. Then $\phi$ is an injection.


Odd Power Function is Surjective

Let $n \in \Z_{\ge 0}$ be an odd positive integer. Let $f_n: \R \to \R$ be the real function defined as: :$\map {f_n} x = x^n$ Then $f_n$ is a surjection.


Residue at Multiple Pole

Let $f: \C \to \C$ be a function meromorphic on some region, $D$, containing $a$.  Let $f$ have a single pole in $D$, of order $N$, at $a$. Then the residue of $f$ at $a$ is given by: :$\displaystyle \Res f a = \frac 1 {\paren {N - 1}!} \lim_{z \mathop \to a} \frac { \d^{N - 1} } { \d z^{N - 1} } \paren {\paren {z - a}^N \map f z}$


Bijection from Cartesian Product of Initial Segments to Initial Segment

Let $\N_k$ be used to denote the set of the first $k$ non-zero natural numbers: :$\N_k := \set {1, 2, \ldots, k}$ Then a bijection can be established between $\N_k \times \N_l$ and $\N_{k l}$, where $\N_k \times \N_l$ denotes the Cartesian product of $\N_k$ and $\N_l$.


Bijection between S x T and T x S

Let $S$ and $T$ be sets. Let $S \times T$ be the Cartesian product of $S$ and $T$. Then there exists a bijection from $S \times T$ to $T \times S$.


Trivial Group is Group

The trivial group is a group.


Order of Dihedral Group

The dihedral group $D_n$ is of order $2 n$.


Matrix Entrywise Addition is Commutative

Let $\map \MM {m, n}$ be a $m \times n$ matrix space over one of the standard number systems. For $\mathbf A, \mathbf B \in \map \MM {m, n}$, let $\mathbf A + \mathbf B$ be defined as the matrix entrywise sum of $\mathbf A$ and $\mathbf B$. The operation $+$ is commutative on $\map \MM {m, n}$. That is: :$\mathbf A + \mathbf B = \mathbf B + \mathbf A$ for all $\mathbf A$ and $\mathbf B$ in $\map \MM {m, n}$.


Special Linear Group is not Abelian

Let $K$ be a field whose zero is $0_K$ and unity is $1_K$. Let $\SL {n, K}$ be the special linear group of order $n$ over $K$. Then $\SL {n, K}$ is not an abelian group.


Summation Formula (Complex Analysis)/Lemma

Let $C_N$ be the square with vertices $\left({N + \frac 1 2}\right) \left({\pm 1 \pm i}\right)$ for $N \in \N$.  Then there exists a constant $A$ independent of $N$ such that:  :$\displaystyle \left\vert{\cot \left({\pi z}\right)}\right\vert < A$  for all $z$ on $C_N$.


Product of Generating Elements of Dihedral Group

Let $D_n$ be the dihedral group of order $2 n$. Let $D_n$ be defined by its group presentation: :$D_n = \gen {\alpha, \beta: \alpha^n = \beta^2 = e, \beta \alpha \beta = \alpha^{−1} }$ Then for all $k \in \Z_{\ge 0}$: :$\beta \alpha^k = \alpha^{n - k} \beta$


Center of Dihedral Group

Let $n \in \N$ be a natural number such that $n \ge 3$. Let $D_n$ be the dihedral group of order $2 n$, given by: :$D_n = \gen {\alpha, \beta: \alpha^n = \beta^2 = e, \beta \alpha \beta = \alpha^{−1} }$ Let $\map Z {D_n}$ denote the center of $D_n$. Then: :$\map Z {D_n} = \begin{cases} e & : n \text { odd} \\ \set {e, \alpha^{n / 2} } & : n \text { even} \end{cases}$


Intersection of Additive Groups of Integer Multiples

Let $m, n \in \Z_{> 0}$ be (strictly) positive integers. Let $\struct {m \Z, +}$ and $\struct {n \Z, +}$ be the corresponding additive groups of integer multiples. Then: :$\struct {m \Z, +} \cap \struct {n \Z, +} = \struct {\lcm \set {m, n} \Z, +}$


Subgroup of Additive Group of Integers Generated by Two Integers

Let $m, n \in \Z_{> 0}$ be (strictly) positive integers. Let $\struct {\Z, +}$ denote the additive group of integers. Let $\gen {m, n}$ be the subgroup of $\struct {\Z, +}$ generated by $m$ and $n$. Then: :$\gen {m, n} = \struct {\gcd \set {m, n} \Z, +}$ That is, the additive groups of integer multiples of $\gcd \set {m, n}$, where $\gcd \set {m, n}$ is the greatest common divisor of $m$ and $n$.


Subgroups of Cartesian Product of Additive Group of Integers

Let $\struct {\Z, +}$ denote the additive group of integers. Let $m, n \in \Z_{> 0}$ be (strictly) positive integers. Let $\struct {\Z \times \Z, +}$ denote the Cartesian product of $\struct {\Z, +}$ with itself. The subgroups of $\struct {\Z \times \Z, +}$ are not all of the form: :$\struct {m \Z, +} \times \struct {n \Z, +}$ where $\struct {m \Z, +}$ denotes the additive group of integer multiples of $m$.


Equivalence Relation on Symmetric Group by Image of n is Congruence Modulo Subgroup

Let $S_n$ denote the symmetric group on $n$ letters $\set {1, \dots, n}$. Let $\sim$ be the relation on $S_n$ defined as: :$\forall \pi, \tau \in S_n: \pi \sim \tau \iff \map \pi n = \map \tau n$ Then $\sim$ is an equivalence relation which is congruence modulo a subgroup. {{explain|Work needed to be done to explain exactly what is happening here.}}


Order of Multiplicative Group of Reduced Residues

Let $\struct {\Z'_m, \times_m}$ denote the multiplicative group of reduced residues modulo $m$. The order of $\struct {\Z'_m, \times_m}$ is $\map \phi m$, where $\phi$ denotes the Euler $\phi$ function.


Circle Group is Uncountably Infinite

The circle group $\struct {K, \times}$ is an uncountably infinite group.


Subgroups of Symmetry Group of Regular Hexagon

Let $\mathcal H = ABCDEF$ be a regular hexagon. Let $D_6$ denote the symmetry group of $\mathcal H$. :520px Let $e$ denote the identity mapping Let $\alpha$ denote rotation of $\mathcal H$ anticlockwise through $\dfrac \pi 3$ radians ($60 \degrees$) Let $\beta$ denote reflection of $\mathcal H$ in the $AD$ axis. The subsets of $D_6$ which form its subgroups are as follows: ;Order $1$: :$\set e$ ;Order $2$: :$\set {e, \alpha^3}$ :$\set {e, \beta}$ :$\set {e, \alpha \beta}$ :$\set {e, \alpha^2 \beta}$ :$\set {e, \alpha^3 \beta}$ :$\set {e, \alpha^4 \beta}$ :$\set {e, \alpha^5 \beta}$ ;Order $3$: :$\set {e, \alpha^2, \alpha^4}$ ;Order $4$: :$\set {e, \alpha^3, \beta, \alpha^3 \beta}$ :$\set {e, \alpha^3, \alpha \beta, \alpha^4 \beta}$ :$\set {e, \alpha^3, \alpha^2 \beta, \alpha^5 \beta}$ ;Order $6$: :$\set {e, \alpha, \alpha^2, \alpha^3, \alpha^4, \alpha^5}$ :$\set {e, \alpha^2, \alpha^4, \beta, \alpha^2 \beta, \alpha^4 \beta}$ :$\set {e, \alpha^2, \alpha^4, \alpha \beta, \alpha^3 \beta, \alpha^5 \beta}$ ;Order $12$: :$D_6$ itself.


Subgroup of Symmetric Group that Fixes n

Let $S_n$ denote the symmetric group on $n$ letters. Let $H$ denote the subgroup of $S_n$ which consists of all $\pi \in S_n$ such that: :$\map \pi n = n$ Then: :$H = S_{n - 1}$ and the index of $H$ in $S_n$ is given by: :$\index {S_n} H = n$


Product of Orders of Abelian Group Elements Divides LCM of Order of Product

Let $G$ be an abelian group. Let $a, b \in G$. Then: :$\order {a b} \divides \lcm \set {\order a, \order b}$ where: :$\order a$ denotes the order of $a$ :$\divides$ denotes divisibility :$\lcm$ denotes the lowest common multiple.


Groups of Order 4

There exist exactly $2$ groups of order $4$, up to isomorphism: :$C_4$, the cyclic group of order $4$ :$K_4$, the Klein $4$-group.


Subgroup of Circle Group Generated by Distinct Roots of Unity

Let $K$ be the circle group. Let $m, n \in \Z_{>0}$ be (strictly) positive integers. Let $d = \lcm \set {m, n}$ be the least common multiple of $m$ and $n$. Let $\alpha$ be a primitive $n$th root of unity. Let $\beta$ be a primitive $m$th root of unity. Let $\gamma$ be a primitive $d$th root of unity. Let $H = \gen {\alpha, \beta}$ be the subgroup of $K$ generated by $\alpha, \beta$. Then $H = \gen \gamma$.


Multiplicative Group of Complex Roots of Unity is Subgroup of Circle Group

Let $n \in \Z$ be an integer such that $n > 0$. Let $\struct {U_n, \times}$ denote the multiplicative group of complex $n$th roots of unity. Let $\struct {K, \times}$ denote the circle group. Then $\struct {U_n, \times}$ is a subgroup of $\struct {K, \times}$.


Intersection of Multiplicative Groups of Complex Roots of Unity

Let $\struct {K, \times}$ denote the circle group. Let $m, n \in \Z_{>0}$ be (strictly) positive integers. Let $c = \lcm \set {m, n}$ be the lowest common multiple of $m$ and $n$. Let $\struct {U_n, \times}$ denote the multiplicative group of complex $n$th roots of unity. Let $\struct {U_m, \times}$ denote the multiplicative group of complex $m$th roots of unity. Let $H = U_m \cap U_n$. Then $H = U_c$.


Direct Product of Normal Subgroups is Normal

Let $G$ and $G'$ be groups. Let: :$H \lhd G$ :$H' \lhd G'$ where $\lhd$ denotes the relation of being a normal subgroup. Then: :$\paren {H \times H'} \lhd \paren {G \times G'}$ where $H \times H'$ denotes the group direct product of $H$ and $H'$


Normalizer of Reflection in Dihedral Group

Let $n \in \N$ be a natural number such that $n \ge 3$. Let $D_n$ be the dihedral group of order $2 n$, given by: :$D_n = \gen {\alpha, \beta: \alpha^n = \beta^2 = e, \beta \alpha \beta = \alpha^{−1} }$ Let $\map {N_{D_n} } {\set \beta}$ denote the normalizer of the singleton containing the reflection element $\beta$. Then: :$\map {N_{D_n} } {\set \beta} = \begin{cases} \set {e, \beta} & : n \text { odd} \\ \set {e, \beta, \alpha^{n / 2}, \alpha^{n / 2} \beta} & : n \text { even} \end{cases}$


Center of Quaternion Group

Let $Q = \Dic 2 = \gen {a, b: a^4 = e, b^2 = a^2, a b a = b}$ be the quaternion group. Let $\map Z {\Dic 2}$ denote the center of $\Dic 2$. Then: :$\map Z {\Dic 2} = \set {e, a^2}$


Product of Generating Elements of Quaternion Group

Let $Q = \Dic 2$ be the quaternion group: :$\Dic 2 = \gen {a, b: a^4 = e, b^2 = a^2, a b a = b}$ Then for all $k \in \Z_{\ge 0}$: :$b a^k = a^{-k} b$


Conjugacy Classes of Quaternion Group

Let $Q = \Dic 2 = \gen {a, b: a^4 = e, b^2 = a^2, a b a = b}$ be the quaternion group. The conjugacy classes of $\Dic 2$ are: :$\set e, \set {a^2}, \set {a, a^3}, \set {b, a^2 b}, \set {a b, a^3 b}$


Conjugacy Action on Subsets is Group Action

Let $\powerset G$ be the set of all subgroups of $G$. For any $S \in \powerset G$ and for any $g \in G$, the conjugacy action: :$g * S := g \circ S \circ g^{-1}$ is a group action.


Normed Vector Space Requires Multiplicative Norm on Division Ring

Let $R$ be a normed division ring with a submultiplicative norm $\norm {\, \cdot \,}_R$. Let $V$ be a vector space that is not a trivial vector space. Let $\norm {\, \cdot \,}: V \to \R_{\ge 0}$ be a mapping from $V$ to the positive real numbers satisfying the vector space norm axioms. Then $\norm {\, \cdot \,}_R$ is a multiplicative norm. That is: :$\forall r, s \in R: \norm {r s}_R = \norm r_R \norm s_R$


Ring with Multiplicative Norm has No Proper Zero Divisors

Let $\struct {R, +, \circ}$ be a ring. Let its zero be denoted by $0_R$. Let $\norm {\,\cdot\,}$ be a multiplicative norm on $R$. Then $R$ has no proper zero divisors.  That is: :$\forall x, y \in R^*: x \circ y \ne 0_R$ where $R^*$ is defined as $R \setminus \set {0_R}$.


Finite Ring with Multiplicative Norm is Field

Let $R$ be a finite ring with a multiplicative norm. Then $R$ is a field.


Composition of Isometries is Isometry

Let: :$\struct {X_1, d_1}$ :$\struct {X_2, d_2}$ :$\struct {X_3, d_3}$ be metric spaces. Let: :$\phi: \struct {X_1, d_1} \to \struct {X_2, d_2}$ :$\psi: \struct {X_2, d_2} \to \struct {X_3, d_3}$ be isometries. Then the composite of $\phi$ and $\psi$ is also an isometry.


Subgroup Action is Group Action

Let $\struct {G, \circ}$ be a group. Let $\struct {H, \circ}$ be a subgroup of $G$. Let $*: H \times G \to G$ be the subgroup action defined for all $h \in H, g \in G$ as: :$\forall h \in H, g \in G: h * g := h \circ g$ Then $*$ is a group action.


Orbit of Subgroup Action is Coset

Let $\struct {G, \circ}$ be a group. Let $\struct {H, \circ}$ be a subgroup of $G$. Let $*: H \times G \to G$ be the subgroup action defined for all $h \in H, g \in G$ as: :$\forall h \in H, g \in G: h * g := h \circ g$ The orbit of $x \in G$ is the right coset by $x$ of $H$: :$\Orb x = H x$


Removable Singularity at Infinity implies Constant Function

Let $f : \C \to \C$ be an entire function. Let $f$ have an removable singularity at $\infty$.  Then $f$ is constant.


Stabilizers of Elements in Same Orbit are Conjugate Subgroups

Let $G$ be a group acting on a set $X$. Let: :$y, z \in \Orb x$ where $\Orb x$ denotes the orbit of some $x \in X$. Then their stabilizers $\Stab y$ and $\Stab z$ are conjugate subgroups.


Stabilizer of Subgroup Action on Left Coset Space

Let $G$ be a group. Let $H$ and $K$ be subgroups of $G$. Let $K$ act on the left coset space $G / H^l$ by: :$\forall \tuple {k, g H} \in K \times G / H^l: k * g H := \paren {k g} H$ The stabilizer of $g H$ is $K \cap H^g$, where $H^g$ denotes the $G$-conjugate of $H$ by $g$.


Cardinality of Set Difference with Subset

Let $S$ and $T$ be sets such that $T$ is finite. Let $T \subseteq S$. Then: :$\card {S \setminus T} = \card S - \card T$ where $\card S$ denotes the cardinality of $S$.


Riemann Zeta Function at Non-Positive Integers

Let $n \ge 0$ be a integer.  Then:  :$\map \zeta {-n} = \paren {-1}^n \dfrac {B_{n + 1} } {n + 1}$  where: :$B_n$ is the $n$th Bernoulli number :$\zeta$ is the Riemann Zeta function


Remainder on Division is Least Positive Residue

Let $a, b \in \Z$ be integers such that $a \ge 0$ and $b \ne 0$. Let $r$ be the remainder resulting from the operation of integer division of $a$ by $b$: $a = q b + r, 0 \le r < \size b$ Then $r$ is equal to the least positive residue of $a \pmod b$.


Floor Function/Examples/Floor of 3

:$\floor 3 = 3$


Exponential on Real Numbers is Injection

Let $\exp: \R \to \R$ be the exponential function: :$\map \exp x = e^x$ Then $\exp$ is an injection.


Surjective Restriction of Real Exponential Function

Let $\exp: \R \to \R$ be the exponential function: :$\map \exp x = e^x$ Then the restriction of the codomain of $\exp$ to the strictly positive real numbers: :$\exp: \R \to \R_{>0}$ is a surjective restriction. Hence: :$\exp: \R \to \R_{>0}$ is a bijection.


Maximum Rule for Real Sequences

:$\displaystyle \lim_{n \mathop \to \infty} \max \set {x_n, y_n} = \max \set {l, m}$


Division Subring of Normed Division Ring

Let $\struct {R, \norm {\, \cdot \,} }$ be a normed division ring. Let $S$ be a division subring of $R$. Then: :$\struct {S, \norm {\, \cdot \,}_S}$ is a normed division subring of $\struct {R, \norm {\, \cdot \,} }$ where $\norm {\, \cdot \,}_S$ is the norm $\norm{\,\cdot\,}$ restricted to $S$.


Normed Division Ring is Dense Subring of Completion

Let $\struct {R, \norm {\, \cdot \,} }$ be a normed division ring. Let $\struct {R', \norm {\, \cdot \,}' }$ be a normed division ring completion of $\struct {R, \norm {\, \cdot \,} }$ Then: :$\struct {R, \norm {\, \cdot \,} }$ is isometrically isomorphic to a dense normed division subring of $\struct {R', \norm {\, \cdot \,}' }$.


Inverse of Isometric Isomorphism

Let $\struct {R, \norm {\,\cdot\,}_R}$ and $\struct {S, \norm {\,\cdot\,}_S}$ be normed division rings. Let $\phi:R \to S$ be a mapping. Then $\phi:R \to S$ is an isometric isomorphism {{iff}} $\phi^{-1}: S \to R$ is also an isometric isomorphism.


Isometric Isomorphism is Norm-Preserving

Let $\struct {R, \norm {\,\cdot\,}_R}$ and $\struct {S, \norm {\,\cdot\,}_S}$ be normed division rings. Let $\phi: R \to S$ be a ring isomorphism. Then $\phi: R \to S$ is an isometric isomorphism {{iff}} $\phi$ satisfies: :$\forall x \in R: \norm {\map \phi x}_S = \norm x_R $


Taylor Series of Holomorphic Function

:$\displaystyle \map f z = \sum_{n \mathop = 0}^\infty \frac {\map {f^n} a} {n!} \paren {z - a}^n$


Equivalence of Definitions of Square Function

Let $\F$ denote one of the standard classes of numbers: $\N$, $\Z$, $\Q$, $\R$, $\C$. {{TFAE|def = Square Function}}


Subring of Non-Archimedean Division Ring

Let $\struct {R, \norm {\, \cdot \,} }$ be a normed division ring with non-archimedean norm $\norm {\, \cdot \,}$. Let $\struct {S, \norm {\, \cdot \,}_S }$ be a normed division subring of $R$. Then: :$\norm {\, \cdot \,}_S$ is a non-archimedean norm.


Isometrically Isomorphic Non-Archimedean Division Rings

Let $\struct {R, \norm {\,\cdot\,}_R}$ and $\struct {S, \norm {\,\cdot\,}_S}$ be normed division rings. Let $\phi:R \to S$ be an isometric isomorphism. Then: :$\norm {\,\cdot\,}_R$ is a non-archimedean norm {{iff}} $\norm {\,\cdot\,}_S$ is a non-archimedean norm.


Non-Archimedean Division Ring Iff Non-Archimedean Completion

Let $\struct {R, \norm {\, \cdot \,} }$ be a normed division ring. Let $\struct {R', \norm {\, \cdot \,}' }$ be a normed division ring completion of $\struct {R, \norm {\, \cdot \,} }$ Then: :$\norm {\, \cdot \,}$ is non-archimedean {{iff}} $\norm {\, \cdot \,}'$ is non-archimedean.


Domain of Integer Square Function

The domain of the integer square function is the entire set of integers $\Z$.


Image of Integer Square Function

The image of the integer square function is the set of square numbers.


Restriction of Real Square Mapping to Positive Reals is Bijection

Let $f: \R \to \R$ be the real square function: :$\forall x \in \R: \map f x = x^2$ Let $g: \R_{\ge 0} \to R_{\ge 0} := f {\restriction_{\R_{\ge 0} \times R_{\ge 0} } }$ be the restriction of $f$ to the positive real numbers $\R_{\ge 0}$. Then $g$ is a bijective restriction of $f$.


Inverse of Real Square Function on Positive Reals

Let $f: \R_{\ge 0} \to R_{\ge 0}$ be the restriction of the real square function to the positive real numbers $\R_{\ge 0}$. The inverse of $f$ is $f^{-1}: \R_{\ge 0} \times R_{\ge 0}$ defined as: :$\forall x \in \R_{\ge 0}: \map {f^{-1} } x = \sqrt x$ where $\sqrt x$ is the positive square root of $x$.


Real Square Function is not Bijective

Let $f: \R \to \R$ be the real square function: :$\forall x \in \R: \map f x = x^2$ Then $f$ is not a bijection.


Inverse of Linear Function on Real Numbers

Let $a, b \in \R$ be real numbers such that $a \ne 0$. Let $f: \R \to \R$ be the real function defined as: :$\forall x \in \R: \map f x = a x + b$ Then the inverse of $f$ is given by: :$\forall y \in \R: \map {f^{-1} } y = \dfrac {y - b} a$


Linear Function on Real Numbers is Bijection

Let $a, b \in \R$ be real numbers. Let $f: \R \to \R$ be the real function defined as: :$\forall x \in \R: \map f x = a x + b$ Then $f$ is a bijection {{iff}} $a \ne 0$.


Composition of Linear Real Functions

Let $a, b, c, d \in \R$ be real numbers. Let $\theta_{a, b}: \R \to \R$  be the real function defined as: :$\forall x \in \R: \map {\theta_{a, b} } x = a x + b$ Let $\theta_{c, d} \circ \theta_{a, b}$ denote the composition of $\theta_{c, d}$ with $\theta_{a, b}$. Then: :$\theta_{c, d} \circ \theta_{a, b} = \theta_{a c, b c + d}$


Condition for Composition of Linear Real Functions to be Commutative

Let $a, b, c, d \in \R$ be real numbers. Let $\theta_{a, b}: \R \to \R$  be the real function defined as: :$\forall x \in \R: \map {\theta_{a, b} } x = a x + b$ Let $\theta_{c, d} \circ \theta_{a, b}$ denote the composition of $\theta_{c, d}$ with $\theta_{a, b}$. Then: :$\theta_{c, d} \circ \theta_{a, b} = \theta_{a, b} \circ \theta_{c, d}$ {{iff}}: :$b c + d = a d + b$


Composition of Right Inverse with Mapping is Idempotent

Let $f: S \to T$ be a mapping. Let $g: T \to S$ be a right inverse mapping of $f$. Then: :$\paren {g \circ f} \circ \paren {g \circ f} = g \circ f$


Sets of Permutations of Equivalent Sets are Equivalent

Let $A$ and $B$ be sets such that: :$A \sim B$ where $\sim$ denotes set equivalence. Let $\map \Gamma A$ denote the set of permutations on $A$. Then: :$\map \Gamma A \sim \map \Gamma B$


Three Points in Ultrametric Space have Two Equal Distances/Corollary 2

Let $\struct {R, \norm {\,\cdot\,} }$ be a normed division ring with non-Archimedean norm $\norm{\,\cdot\,}$, Let $x, y \in R$ and $\norm x \ne \norm y$. Then: :$\norm {x + y} = \norm {x - y} = \norm {y - x} = \max \set {\norm x, \norm y}$


Topological Properties of Non-Archimedean Division Rings/Centers of Open Balls

:$y \in \map {B_r} x \implies \map {B_r} y = \map {B_r} x$


Topological Properties of Non-Archimedean Division Rings/Centers of Closed Balls

:$y \in \map { {B_r}^-} x \implies \map { {B_r}^-} y = \map { {B_r}^-} x$


Topological Properties of Non-Archimedean Division Rings/Open Balls are Clopen

:The open $r$-ball of $x$, $\map {B_r} x$, is both open and closed in the metric induced by $\norm {\,\cdot\,}$.


Topological Properties of Non-Archimedean Division Rings/Closed Balls are Clopen

:The closed $r$-ball of $x$, $\map { {B_r}^-} x$, is both open and closed in the metric induced by $\norm {\,\cdot\,}$.


Mittag-Leffler Expansion for Hyperbolic Cotangent Function

:$\displaystyle \pi \, \map \coth {\pi z} = \frac 1 z + 2 \sum_{n \mathop = 1}^\infty \frac z {z^2 + n^2}$ where:  :$z \in \C$ is not an integer multiple of $i$ :$\coth$ is the hyperbolic cotangent function.


Mittag-Leffler Expansion for Secant Function

:$\displaystyle \pi \, \map \sec {\pi z} = 4 \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {2 n + 1} {\paren {2 n + 1}^2 - 4 z^2}$ where:  :$z \in \C$ is not a half-integer :$\sec$ is the secant function.


Pringsheim's Theorem

Let $f$ be a holomorphic function defined on a unit disc centered at the origin of the complex plane and is denoted by its Taylor series: :$\map f z = \displaystyle \sum_{n \mathop = 0}^{\infty} c_n z^n$ Let: :$(1): \quad \forall n \ge 0: c_n \ge 0$ :$(2): \quad$ the radius of convergence of the Taylor series of function $f$ is $1$. Then $z = 1$ is an isolated singularity of $f$.


Mittag-Leffler Expansion for Tangent Function

:$\displaystyle \pi \map \tan {\pi z} = 8 \sum_{n \mathop = 0}^\infty \frac z {\paren {2 n + 1}^2 - 4 z^2}$ where:  :$z \in \C$ is not a half-integer :$\tan$ is the tangent function.


Mittag-Leffler Expansion for Hyperbolic Tangent Function

:$\displaystyle \pi \map \tanh {\pi z} = 8 \sum_{n \mathop = 0}^\infty \frac z {4 z^2 + \paren {2 n + 1}^2}$ where: :$z \in \C$ is not a half-integer multiple of $i$ :$\tanh$ is the hyperbolic tangent function.


Mittag-Leffler Expansion for Hyperbolic Secant Function

:$\displaystyle \pi \, \map \sech {\pi z} = 4 \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {2 n + 1} {\paren {2 n + 1}^2 + 4 z^2}$ where:  :$z \in \C$ is not a half-integer multiple of $i$  :$\sech$ is the hyperbolic secant function.


Mittag-Leffler Expansion for Hyperbolic Cosecant Function

:$\ds \pi \map \csch {\pi z} = \frac 1 z + 2 \sum_{n \mathop = 1}^\infty \paren {-1}^n \frac z {z^2 + n^2}$ where: :$z \in \C$ is not an integer multiple of $i$ :$\csch$ is the hyperbolic cosecant function.


Real Numbers under Subtraction do not form Semigroup

The set of real numbers under subtraction $\struct {\R, -}$ does not form a semigroup.


Count of Binary Operations on Set/Sequence

The sequence of $N$ for each $n$ begins: $\begin{array} {c|rr} n & n^2 & N = n^{\paren {n^2} } \\ \hline 1 & 1 & 1 \\ 2 & 4 & 16 \\ 3 & 9 & 19 \ 683 \\ 4 & 16 & 4 \ 294 \ 967 \ 296 \\ \end{array}$ There are still only $4$ elements in a set, and already there are over $4$ thousand million different possible algebraic structures.


Count of Commutative Binary Operations on Set/Sequence

The sequence of $N$ for each $n$ begins: $\begin{array} {c|cr} n & \dfrac {n \paren {n + 1} }2 & n^{\frac {n \paren {n + 1} } 2} \\ \hline 1 & 1 & 1 \\ 2 & 3 & 8 \\ 3 & 6 & 729 \\ 4 & 10 & 1 \ 048 \ 576 \\ \end{array}$ and so on.


Mittag-Leffler’s Expansion Theorem

Let $f$ be a meromorphic function with only simple poles continuous, or with a removable singularity, at $0$.  Let $X$ be the set of poles of $f$.  For $N \in \N$, let $C_N$ be a circle, centred at the origin, of radius $R_N$, where $R_N \to \infty$ as $N \to \infty$, such that $\partial C_N$ contains no poles of $f$ for any $N$.  Let $M > 0$ be a real number independent of $N$ such that for all $z \in \partial C_N$, $\cmod {\map f z} < M$ , for all $N \in \N$. Then: :$\displaystyle \map f z = \map f 0 + \sum_{n \mathop \in X} \Res f n \paren {\frac 1 {z - n} + \frac 1 n}$  where: :$\Res f n$ is the residue of $f$ at $n$ :$z$ is not a pole of $f$ :$\displaystyle \map f 0 = \lim_{z \mathop \to 0} \map f z$ if $f$ has a removable singularity at $0$.


Normed Division Ring Operations are Continuous

Let $\struct {R, +, *, \norm {\,\cdot\,} }$ be a normed division ring. Let $d$ be the metric induced by the norm $\norm {\,\cdot\,}$. Let $p \in \R_{\ge 1} \cup \set \infty$. Let $d_p$ be the $p$-product metric on $R \times R$. Let $R^* = R \setminus \set 0$ Let $d^*$ be the restriction of $d$ to $R^*$. Then the following results hold:


Normed Division Ring Operations are Continuous/Addition

:$+ : \struct {R \times R, d_p} \to \struct{R,d}$  is continuous.


Normed Division Ring Operations are Continuous/Negation

:$\eta: \struct {R, d} \to \struct {R, d}: \map \eta x = -x$  is continuous.


Normed Division Ring Operations are Continuous/Multiplication

:$* : \struct {R \times R, d_p} \to \struct {R, d}$  is continuous.


Normed Division Ring Operations are Continuous/Corollary

Let $\tau$ be the topology induced by the metric $d$. Then: :$\struct {R, \tau}$ is a topological division ring.


Count of Binary Operations with Identity/Sequence

The sequence of $N$ for each $n$ begins: $\begin{array} {c|cr} n & \paren {n - 1}^2 + 1 & n^{\paren {n - 1}^2 + 1}\\ \hline 1 & 1 & 1 \\ 2 & 2 & 4 \\ 3 & 5 & 243 \\ 4 & 10 & 1 \ 048 \ 576 \\ \end{array}$


Structure with Element both Identity and Zero has One Element

Let $\struct {S, \circ}$ be an algebraic structure. Let $z \in S$ such that $z$ is both an identity element and a zero element. Then: :$S = \set z$


Group/Examples/Linear Functions

Let $G$ be the set of all real functions $\theta_{a, b}: \R \to \R$ defined as: :$\forall x \in \R: \map {\theta_{a, b} } x = a x + b$ where $a, b \in \R$ such that $a \ne 0$. The algebraic structure $\struct {G, \circ}$, where $\circ$ denotes composition of mappings, is a group. $\struct {G, \circ}$ is specifically non-abelian.


Summation Formula for Alternating Series

Let $C_N$ be the square with vertices $\paren {N + \dfrac 1 2} \paren {\pm 1 \pm i}$ for some real $N \in \N$.  Let $f$ be a function meromorphic on $C_N$. Let $\cmod {\map f z} < \dfrac M {\cmod z^k}$, for constants $k > 1$ and $M$ independent of $N$, for all $z \in \partial C_N$.  Let $X$ be the set of poles of $f$. Then:  :$\displaystyle \sum_{n \mathop \in \Z \mathop \setminus X} \paren {-1}^n \map f n = -\sum_{z_0 \mathop \in X} \Res {\pi \map \csc {\pi z} \map f z} {z_0}$ If $X \cap \Z = \O$, this becomes:  :$\displaystyle \sum_{n \mathop = -\infty}^\infty \paren {-1}^n \map f n = -\sum_{z_0 \mathop \in X} \Res {\pi \map \csc {\pi z} \map f z} {z_0}$


Integer Multiples under Addition form Subgroup of Integers

Let $\struct {\Z, +}$ denote the additive group of integers. Let $n \Z$ be the set of integer multiples of $n$. Then $\struct {n \Z, +}$ is a subgroup of $\struct {\Z, +}$. Hence $\struct {n \Z, +}$ can be justifiably referred to as the additive group of integer multiples.


Group is Generated by Itself

Let $G$ be a group. Then: :$G = \gen G$ where $\gen G$ denotes the group generated by $G$.


Summation Formula for Alternating Series over Half-Integers

Let $C_N$ be the square with vertices $N \paren {\pm 1 \pm i}$ for real $N \in \N$.  Let $f$ be a function meromorphic on $C_N$. Let $\cmod {\map f z} < \dfrac M {\cmod z^k}$, for constants $k > 1$ and $M$ independent of $N$, for all $z \in \partial C_N$.  Let $X$ be the set of poles of $f$. Let $Y$ be the set of poles of $\map f {\dfrac {2 z + 1} 2}$.  Then:  :$\displaystyle \sum_{n \in \Z \setminus Y} \paren {-1}^n \map f {\frac {2 n + 1} 2} = \sum_{z_0 \in X} \Res {\pi \sec \paren {\pi z} \map f z} {z_0}$ If $Y \cap \Z = \O$, this becomes:  :$\displaystyle \sum_{n \mathop = -\infty}^\infty \paren {-1}^n \map f {\frac {2 n + 1} 2} = \sum_{z_0 \in X} \Res {\pi \sec \paren {\pi z} \map f z} {z_0}$


Summation Formula over Half-Integers

Let $C_N$ be the square with vertices $N \paren {\pm 1 \pm i}$ for real $N \in \N$.  Let $f$ be a function meromorphic on $C_N$. Let $\cmod {\map f z} < \dfrac M {\cmod z^k}$, for constants $k > 1$ and $M$ independent of $N$, for all $z \in \partial C_N$.  Let $X$ be the set of poles of $f$. Let $Y$ be the set of poles of $\map f {\dfrac {2 z + 1} 2}$. Then:  :$\displaystyle \sum_{n \in \Z \setminus Y} \map f {\frac {2 n + 1} 2} = \sum_{z_0 \in X} \Res {\pi \tan \paren {\pi z} \map f z} {z_0}$ If $Y \cap \Z = \O$, this becomes:  :$\displaystyle \sum_{n \mathop = -\infty}^\infty \map f {\frac {2 n + 1} 2} = \sum_{z_0 \in X} \Res {\pi \tan \paren {\pi z} \map f z} {z_0}$


Heaviside Expansion Formula

Let $P, Q$ be polynomials with coefficients in $\C$.  Let $\deg Q \ge \deg P + 1$. Let $\map Q z$ have a simple zero for $z \in X$.  Let $\map {\laptrans f} z = \dfrac {\map P z} {\map Q z}$.  Then: :$\displaystyle \map f t = \sum_{z \mathop \in X} e^{z t} \frac {\map P z} {\map {Q'} z}$


Action of Inverse of Group Element

Let $\struct {G, \circ}$ be a group. Let $S$ be a sets. Let $*: G \times S \to S$ be a group action. Then: :$g * a = b \iff g^{-1} * b = a$


Union Operation on Supersets of Subset is Closed

Let $S$ be a set. Let $T \subseteq S$ be a given subset of $S$. Let $\powerset S$ denote the power set of $S$ Let $\mathscr S$ be the subset of $\powerset S$ defined as: :$\mathscr S = \set {Y \in \powerset S: T \subseteq Y}$ Then the algebraic structure $\struct {\mathscr S, \cup}$ is closed.


Existence of Magma with no Proper Submagma

Let $n \in \Z_{>0}$ be a strictly positive integer. Let $S$ be a set of cardinality $n$: :$\card S = n$ Then there exists an operation $\circ$ on $S$ such that: :$\struct {S, \circ}$ is a magma :$\struct {S, \circ}$ has no submagma $\struct {T, \circ}$ such that $T$ is a non-empty proper subset of $S$.


Subset of Abelian Group Generated by Product of Element with Inverse Element is Subgroup

Let $\struct {G, \circ}$ be an abelian group. Let $S \subset G$ be a non-empty subset of $G$ such that $\struct {S, \circ}$ is closed. Let $H$ be the set defined as: :$H := \set {x \circ y^{-1}: x, y \in S}$ Then $\struct {H, \circ}$ is a subgroup of $\struct {G, \circ}$.


Local Basis Test

Let $\struct {S, \tau}$ be a topological space. Let $x \in S$. Let $\BB$ be a local basis for $x$ in $\struct {S, \tau}$. Let $\CC$ be a set of open neighborhoods of $x$. Then: :$\CC$ is a local basis {{iff}}: ::$\forall B \in \BB \implies \exists C \in \CC: C \subseteq B$


Union of Subgroups/Corollary 1

Let $H \cup K$ be a subgroup of $G$. Then either $H \subseteq K$ or $K \subseteq H$.


Stabilizer of Element after Group Action

Let $\struct {G, \circ}$ be a group. Let $S$ be a set. Let $*_S: G \times S \to S$ be a group actions. Let $x \in S, a \in G$. Then: :$\Stab {a * x} = a^{-1} \circ \Stab x \circ a$


Group Action of Symmetric Group/Subset

Let $r \in \N: 0 < r \le n$. Let $B_r$ denote the set of all subsets of $\N_n$ of cardinality $r$: :$B_r  := \set {S \subseteq \N_n: \card S = r}$ Let $*$ be the mapping $*: S_n \times B_r \to B_r$ defined as: :$\forall \pi \in S_n, \forall S \in B_r: \pi * B_r = \pi \sqbrk S$ where $\pi \sqbrk S$ denotes the image of $S$ under $\pi$. Then $*$ is a group action.


Group Action of Symmetric Group on Subset is Transitive

Let $r \in \N: 0 < r \le n$. Let $B_r$ denote the set of all subsets of $\N_n$ of cardinality $r$: :$B_r  := \set {S \subseteq \N_n: \card S = r}$ Let $*$ be the mapping $*: S_n \times B_r \to B_r$ defined as: :$\forall \pi \in S_n, \forall S \in B_r: \pi * B_r = \pi \sqbrk S$ where $\pi \sqbrk S$ denotes the image of $S$ under $\pi$. Then $*$ is a transitive group action.


Coset Product on Non-Normal Subgroup is not Well-Defined

Let $\struct {G, \circ}$ be a group. Let $H$ be a subgroup of $G$ which is not normal. Let $a, b \in G$. Then it is not necessarily the case that the coset product: :$\paren {a \circ H} \circ \paren {b \circ H} = \paren {a \circ b} \circ H$ is well-defined.


Klein Four-Group is Normal in A4

Let $A_4$ denote the alternating group on $4$ letters, whose Cayley table is given as: {{:Alternating Group on 4 Letters/Cayley Table}} <onlyinclude> The subsets of $A_4$ which form subgroups of $A_4$ are as follows: Consider the order $4$ subgroup $V$ of $A_4$, presented by Cayley table: :$\begin{array}{c|cccc} \circ & e & t & u & v \\ \hline e & e & t & u & v \\ t & t & e & v & u \\ u & u & v & e & t \\ v & v & u & t & e \\ \end{array}$ Then $V$ is normal in $A_4$. Its index is: :$\index {A_4} V = \dfrac {\order {A_4} } {\order V} = \dfrac {12} 4 = 3$ The (left) cosets of $V$ are: :$V$ :$A := a V$ :$P := p V$ and the Cayley table of the quotient group $A_4 / V$ is given by: :$\begin{array}{c|ccc} \circ & V & A & P \\ \hline V & V & A & P \\ A & A & P & V \\ P & P & V & A \\ \end{array}$ Note that while $A_4 / V$ is Abelian, $A_4$ is not.


Coset of Trivial Subgroup is Singleton

Let $\struct {G, \circ}$ be a group whose identity is $e$. Let $E := \struct {\set e, \circ}$ denote the trivial subgroup of $\struct {G, \circ}$. Let $g \in G$. Then the left coset and right coset of $E$ by $g$ is $\set g$.


Inverse Elements of Right Transversal is Left Transversal

Let $G$ be a group. Let $H$ be a subgroup of $G$. Let $S \subseteq G$ be a right transversal for $H$ in $G$. Let $T$ be the set defined as: :$T := \set {x^{-1}: x \in S}$ where $x^{-1}$ is the inverse of $x$ in $G$. Then $T$ is a left transversal for $H$ in $G$.


Condition for Subset of Group to be Right Transversal

Let $G$ be a group. Let $H$ be a subgroup of $G$ whose index in $G$ is $n$: :$\index G H = n$ Let $S \subseteq G$ be a subset of $G$ of cardinality $n$. Then $S$ is a right transversal for $H$ in $G$ {{iff}}: :$\forall x, y \in S: x \ne y \implies x y^{-1} \notin H$


Group Action on Coset Space is Transitive

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 transitive group action.


Stabilizer of Coset under Group Action on Coset Space

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 the stabilizer of $a H$ under $*$ is given by: :$\Stab {a H} = a H a^{-1}$


Index of Subgroup equals Index of Conjugate

Let $G$ be a group. Let $H$ be a subgroup of $G$. Then: :$\index G H = \index G {a H a^{-1} }$ where $\index G H$ denotes the index of $H$ in $G$.


Normality Relation is not Transitive

Let $G$ be a group. Let $N$ be a normal subgroup of $G$. Let $K$ be a normal subgroup of $N$. Then it is not necessarily the case that $K$ is a normal subgroup of $G$.


Power of Coset Product is Coset of Power

Let $\struct {G, \circ}$ be a group. Let $N$ be a normal subgroup of $G$. Let $a \in G$. Then: :$\forall n \in \Z: \paren {a \circ N} = \paren {a^n} \circ N$


Condition for Power of Element of Quotient Group to be Identity

Let $G$ be a group whose identity is $e$. Let $N$ be a normal subgroup of $G$. Let $a \in G$. Then: :$\paren {a N}^n$ is the identity of the quotient group $G / N$ {{iff}}: :$a^n \in N$


Additive Group of Integers is Normal Subgroup of Reals

Let $\struct {\Z, +}$ be the additive group of integers. Let $\struct {\R, +}$ be the additive group of real numbers. Then $\struct {\Z, +}$ is a normal subgroup of $\struct {\R, +}$.


Mapping from Additive Group of Integers to Powers of Group Element is Homomorphism

Let $\struct {G, \circ}$ be a group. Let $g \in G$. Let $\struct {\Z, +}$ denote the additive group of integers. Let $\phi_g: \struct {\Z, +} \to \struct {G, \circ}$ be the mapping defined as: :$\forall k \in \Z: \map {\phi_g} k = g^k$ Then $\phi_g$ is a (group) homomorphism.


Inner Automorphisms form Subgroup of Symmetric Group

Let $G$ be a group. Let $\struct {\map \Gamma G, \circ}$ be the symmetric group on $G$. Let $\Inn G$ denote the inner automorphism group of $G$. Then: :$\Inn G \le \struct {\map \Gamma G, \circ}$ where $\le$ denotes the relation of being a subgroup.


Order is Preserved by Group Isomorphism

Let $G$ and $H$ be groups. Let $\phi: G \to H$ be a (group) isomorphism. Then: :$\order G = \order H$ where $\order {\, \cdot \,}$ denotes the order of a group.


Additive Group of Reals is Subgroup of Complex

Let $\struct {\R, +}$ be the additive group of real numbers. Let $\struct {\C, +}$ be the additive group of complex numbers. Then $\struct {\R, +}$ is a subgroup of $\struct {\C, +}$.


Mapping to Power is Endomorphism iff Abelian

Let $\struct {G, \circ}$ be a group. Let $n \in \Z$ be an integer. Let $\phi: G \to G$ be defined as: :$\forall g \in G: \map \phi g = g^n$ Then $\struct {G, \circ}$ is abelian {{iff}} $\phi$ is a (group) endomorphism.


Additive Groups of Integers and Integer Multiples are Isomorphic

Let $n \in \Z_{> 0}$ be a strictly positive integer. Let $\struct {n \Z, +}$ denote the additive group of integer multiples. Let $\struct {\Z, +}$ denote the additive group of integers. Then $\struct {n \Z, +}$ is isomorphic to $\struct {\Z, +}$.


Additive Group of Real Numbers is Not Isomorphic to Multiplicative Group of Real Numbers

Let $\struct {\R, +}$ denote the additive group of real numbers. Let $\struct {\R_{\ne 0}, \times}$ denote the multiplicative group of real numbers. Then $\struct {\R, +}$ is not isomorphic to $\struct {\R_{\ne 0}, \times}$.


Normal Subgroup is Kernel of Group Homomorphism

Let $G$ be a group. Let $N$ be a normal subgroup of $G$. Then there exists a group homomorphism of which $N$ is the kernel.


Homomorphic Image of Cyclic Group is Cyclic Group

Let $G$ be a cyclic group with generator $g$. Let $H$ be a group. Let $\phi: G \to H$ be a (group) homomorphism. Let $\Img G$ denote the homomorphic image of $G$ under $\phi$. Then $\Img G$ is a cyclic group with generator $\map \phi g$. That is: :$\phi \sqbrk {\gen g} = \gen {\map \phi g}$


Kernel of Homomorphism on Cyclic Group

Let $G = \gen g$ be a cyclic group with generator $g$. Let $H$ be a group. Let $\phi: G \to H$ be a (group) homomorphism. Let $\map \ker \phi$ denote the kernel of $\phi$. Let $\Img G$ denote the homomorphic image of $G$ under $\phi$. Then: :$\map \ker \phi = \gen {g^m}$ where: :$m = 0$ if $\Img \phi$ is an infinite cyclic group :$m = \order {\Img \phi}$ if $\Img \phi$ is a finite cyclic group.


Mapping from Group Element to Inner Automorphism is Homomorphism

Let $G$ be a group. Let $\kappa: G \to \Aut G$ be the mapping from $G$ to the automorphism group of $G$ defined as: :$\forall x \in G: \map \kappa x := \kappa_x$ where $\kappa_x$ is the inner automorphism on $x$: :$\forall g \in G: \map {\kappa_x} g = x g x^{-1}$ Then $\kappa$ is a homomorphism.


Image of Mapping from Group Element to Inner Automorphism is Inner Automorphism Group

Let $G$ be a group. Let $\kappa: G \to \Aut G$ be the mapping from $G$ to the automorphism group of $G$ defined as: :$\forall x \in G: \map \kappa x := \kappa_x$ where $\kappa_x$ is the inner automorphism on $x$: :$\forall g \in G: \map {\kappa_x} g = x g x^{-1}$ Then $\Img \kappa$ is the inner automorphism group of $G$.


Order of Monomorphic Image of Group Element

Let $G$ and $H$be groups whose identities are $e_G$ and $e_H$ respectively. Let $\phi: G \to H$ be a monomorphism. Let $g \in G$ be of finite order. Then: :$\forall g \in G: \order {\map \phi g} = \order g$


Image under Epimorphism of Center is Subset of Center

Let $G$ and $H$ be groups. Let $\theta: G \to H$ be an epimorphism. Let $\map Z G$ denote the center of $G$. Then: :$\theta \sqbrk {\map Z G} \subseteq \map Z H$


Metric Subspace Induces Subspace Topology

Let $M = \struct {A,d}$ be a metric space. Let $H \subseteq A$. Let $\tau$ be the topology induced by the metric $d$. Let $\tau_H$ be the subspace topology induced by $\tau$ on $H$. Let $d_H$ be the subspace metric induced by $d$ on $H$. Let $\tau_{d_H}$ be the topology induced by the metric $d_H$. Then: :$\tau_{d_H} = \tau_H$


Quotient of Cauchy Sequences is Metric Completion/Lemma 1

:$\quad \mathcal C \,\big / \mathcal N = \tilde {\mathcal C}$


Quotient of Cauchy Sequences is Metric Completion/Lemma 2

:$\quad d' = \tilde d$


Generator for Quaternion Group

The Quaternion Group can be generated by the matrices: :$\mathbf a = \begin{bmatrix} 0 & 1 \\ -1 & 0 \end{bmatrix} \qquad \mathbf b = \begin{bmatrix} 0 & i \\ i & 0 \end{bmatrix}$ where $i$ is the imaginary unit: :$i^2 = -1$


Condition for Nu Function to be 1

Let: :$n = \displaystyle \prod_{i \mathop = 1}^s p_i^{m_i}$ where $p_1, p_2, \ldots, p_s$ are distinct primes. Then: :$(1): \quad m_1, m_2, \ldots, m_s = 1$, that is, $n$ is square-free :$(2): \quad \forall i, j \in \set {1, 2, \ldots, s}: p_i \not \equiv 1 \pmod {p_j}$ {{iff}}: :every group $G$ of order $n$ is cyclic and so $\map \nu n = 1$.


Group of Order 15 is Cyclic Group

Let $G$ be a group whose order is $15$. Then $G$ is cyclic.


Number of Abelian Groups

Let $n \in \Z_{\ge 1}$ be a (strictly) positive integer. Let: :$n = \displaystyle \prod_{i \mathop = 1}^s p_i^{m_i}$ where the $p_i$ are distinct primes. Let $\map {\nu_a} n$ denote the number of abelian groups of order $n$. Then: :$\map {\nu_a} n = \displaystyle \prod_{i \mathop = 1}^s \map {\nu_a} {p_i^{m_i} }$ where: :$\map {\nu_a} {p_i^{m_i} }$ is the number of integer partitions of $m_i$.


Order of Quotient Group

Let $G$ be a finite group. Let $N$ be a normal subgroup of $G$. Let $G / N$ be the quotient group of $G$ by $N$. Then: :$\dfrac {\order G} {\order N} = \order {G / N}$ where $\order G$ denotes the order of $G$.


Definition:Composition Series/Composition Length

Let $\HH$ be a composition series for $G$. The '''composition length''' of $G$ is the length of $\HH$.


Definition:Composition Series/Composition Factor

Let $\HH = \set e = G_0 \lhd G_1 \lhd \cdots \lhd G_{n - 1} \lhd G_n = G$ be a composition series for $G$. Each of the quotient groups: :$G_1 / G_0, G_2 / G_1, \ldots, G_n / G_{n - 1}$ are the '''composition factors''' of $G$.


Non-Abelian Simple Finite Groups are Infinitely Many

There exist infinitely many types of group which are non-abelian and finite.


Upper Bound of Order of Non-Abelian Finite Simple Group

Let $G$ be a non-abelian finite simple group. Let $t \in G$ be a self-inverse element of $G$. Let $\map {C_G} t$ denote the centralizer of $t$ in $G$. Let $m = \order {\map {C_G} t}$ be the order of $\map {C_G} t$. Then: :$\order G \le \paren {\dfrac {m \paren {m + 1} } 2}!$


Quotient Ring of Cauchy Sequences is Normed Division Ring/Lemma 1

:$\norm {\, \cdot \,}_1$ is well-defined. That is,  :$(1): \quad \forall \eqclass {x_n}{}: \lim_{n \mathop \to \infty} \norm{x_n}$ exists. :$(2): \quad \displaystyle \forall \eqclass {x_n}{}, \eqclass {y_n}{} \in \mathcal C \,\big / \mathcal N: \eqclass {x_n}{} = \eqclass {y_n}{} \implies \lim_{n \mathop \to \infty} \norm{x_n} = \lim_{n \mathop \to \infty} \norm{y_n}$


Quotient Ring of Cauchy Sequences is Normed Division Ring/Lemma 2

:$\norm {\, \cdot \,}_1$ satisfies {{NormAxiom|1}} That is: :$\forall \eqclass {x_n} {} \in \CC \,\big / \NN: \norm {\eqclass {x_n} {} }_1 = 0 \iff \eqclass {x_n} {} = \eqclass {0_R} {} $


Quotient Ring of Cauchy Sequences is Normed Division Ring/Lemma 3

:$\norm {\, \cdot \,}_1$ satisfies the {{NormAxiom|2}}. That is: :$\forall \eqclass {x_n} {}, \eqclass {y_n} {} \in \CC \,\big / \NN: \norm {\eqclass {x_n} {} \eqclass {y_n} {} }_1 = \norm {\eqclass {x_n} {} }_1 \times \norm {\eqclass {y_n} {} }_1$


Quotient Ring of Cauchy Sequences is Normed Division Ring/Lemma 4

:$\norm {\, \cdot \,}_1$ satisfies the {{NormAxiom|3}}. That is: :$\forall \eqclass {x_n} {}, \eqclass {y_n} {} \in \CC \,\big / \NN: \norm {\eqclass {x_n} {} +  \eqclass {y_n} {} }_1 \le \norm {\eqclass {x_n} {} }_1 + \norm {\eqclass {y_n} {} }_1$


Invertible Elements of Monoid form Subgroup

Let $\struct {S, \circ}$ be a monoid whose identity element is $e$. Let $U \subseteq S$ be the subset of $S$ consisting of the invertible elements of $S$. Then $\struct {U, \circ}$ forms a subgroup of $S$.


Combination Theorem for Sequences/Normed Division Ring/Inverse Rule/Lemma

:$\displaystyle \lim_{n \mathop \to \infty} y_n^{-1} = l^{-1}$


Division Ring Norm is Continuous on Induced Metric Space

Let $\struct {R, \norm {\,\cdot\,}}$ be a normed division ring. Let $d$ be the metric induced by the norm $\norm {\,\cdot\,}$. The mapping $\norm {\,\cdot\,} : \struct {R, d} \to \R$ is continuous.


Properties of Norm on Division Ring

Let $\struct {R, +, \circ}$ be a division ring with zero $0_R$ and unity $1_R$. Let $\norm {\,\cdot\,}$ be a norm on $R$. Let $x, y \in R$. Then the following hold:


Properties of Norm on Division Ring/Norm of Negative

:$\norm {-x} = \norm {x}$


Properties of Norm on Division Ring/Norm of Unity

:$\norm {1_R} = 1$.


Properties of Norm on Division Ring/Norm of Negative of Unity

:$\norm{-1_R} = 1$


Repunit is Zuckerman Number

Let $n$ be a repunit. Then $n$ is also a Zuckerman number.


Properties of Norm on Division Ring/Norm of Difference

:$\norm {x - y} \le \norm x + \norm y$


Properties of Norm on Division Ring/Norm of Inverse

:$x \ne 0_R \implies \norm {x^{-1} } = \dfrac 1 {\norm x}$


Properties of Norm on Division Ring/Norm of Quotient

:$y \ne 0_R \implies \norm{xy^{-1}} = \norm{y^{-1}x} = \dfrac {\norm{x}}{\norm{y}}$


Properties of Norm on Division Ring/Norm of Power Equals Unity

:$\forall n \in \N_{\gt 0}: \norm {x^n} = 1 \implies \norm x = 1$


Conversion from Hexadecimal to Binary

Let $n$ be a (positive) integer expressed in hexadecimal notation as: :$n = \sqbrk {a_r a_{r - 1} \dotso a_1 a_0}_H$ Then $n$ can be expressed in binary notation as: :$n = \sqbrk {b_{r 3} b_{r 2} b_{r 1} b_{r 0} b_{\paren {r - 1} 3} b_{\paren {r - 1} 2} b_{\paren {r - 1} 1} b_{\paren {r - 1} 0} \dotso b_{1 3} b_{1 2} b_{1 1} b_{1 0} b_{0 3} b_{0 2} b_{0 1} b_{0 0} }_2$ where $\sqbrk {b_{j 3} b_{j 2} b_{j 1} b_{j 0} }_2$ is the expression of the hexadecimal digit $a_j$ in binary notation. That is, you take the binary expression of each hexadecimal digit, padding them out with zeroes to make them $4$ bits long, and simply concatenate them.


Birthday Paradox/General/3

Let $n$ be a set of people. Let the probability that at least $3$ of them have the same birthday be greater than $50 \%$. Then $n \ge 88$.


Convergent Sequence is Cauchy Sequence/Normed Division Ring

Let $\struct {R, \norm {\,\cdot\,}} $ be a normed division ring. Every convergent sequence in $R$ is a Cauchy sequence.


Sum of 3 Squares in 2 Distinct Ways

$27$ is the smallest positive integer which can be expressed as the sum of $3$ square numbers in $2$ distinct ways: {{begin-eqn}} {{eqn | l = 27       | r = 3^2 + 3^2 + 3^2 }} {{eqn | r = 5^2 + 1^2 + 1^2 }} {{end-eqn}}


Triangular Numbers which are Sum of Two Cubes

The sequence of triangular numbers which are the sum of $2$ cubes begins: :$28, 91, 351, 2926, 8001, 46971, 58653, 93528, 97461, \dots$ {{OEIS|A113958}}


Product of Factors of Perfect Number

Let $P$ be the perfect number $2^{n - 1} \paren {2^n - 1}$. Then: :$\displaystyle \prod_{d \mathop \divides P} d = P^n$


Sequence is Bounded in Norm iff Bounded in Metric

Let $\struct {R, \norm {\,\cdot\,} } $ be a normed division ring. Let $d$ be the metric induced on $R$ be the norm $\norm {\,\cdot\,}$. Let $\sequence {x_n}$ be a sequence in $R$. Then: :$\sequence {x_n} $ is a bounded sequence in the normed division ring $\struct {R, \norm {\,\cdot\,} }$ {{iff}} $\sequence {x_n} $ is a bounded sequence in the metric space $\struct {R, d}$.


Sequence is Bounded in Norm iff Bounded in Metric/Necessary Condition

Let $\struct {R, \norm {\,\cdot\,}} $ be a normed division ring. Let $d$ be the metric induced on $R$ be the norm $\norm {\,\cdot\,}$. Let $\sequence {x_n}$ be a sequence in $R$. Let $\sequence {x_n} $ be a bounded sequence in the normed division ring $\struct {R, \norm {\,\cdot\,}}$  Then: :$\sequence {x_n} $ is a bounded sequence in the metric space $\struct {R, d}$


Sequence is Bounded in Norm iff Bounded in Metric/Sufficient Condition

Let $\struct {R, \norm {\,\cdot\,}} $ be a normed division ring. Let $d$ be the metric induced on $R$ be the norm $\norm {\,\cdot\,}$. Let $\sequence {x_n}$ be a sequence in $R$. Let $\sequence {x_n} $ be a bounded sequence in the metric space $\struct {R, d}$ Then: :$\sequence {x_n} $ is a bounded sequence in the normed division ring $\struct {R, \norm {\,\cdot\,} }$


Magic Constant of Magic Cube

The magic constant of a magic cube of order $n$ is given by: :$C_n = \dfrac {n \paren {n^3 + 1} } 2$


Sum of Terms of Magic Cube

The total of all the entries in a magic cube of order $n$ is given by: :$T_n = \dfrac {n^3 \paren {n^3 + 1} } 2$


Smallest Magic Cube is of Order 3

Apart from the trivial order $1$ magic cube: {{:Magic Cube/Examples/Order 1}} the smallest magic cube is the order $3$ magic cube: {{:Magic Cube/Examples/Order 3}}


Fourth Power as Summation of Groups of Consecutive Integers

Take the positive integers and group them in sets such that the $m$th set contains the next $m$ positive integers: :$\set 1, \set {2, 3}, \set {4, 5, 6}, \set {7, 8, 9, 10}, \set {11, 12, 13, 14, 15}, \ldots$ Remove all the sets with an even number of elements. Then the sum of all the integers in the first $n$ sets remaining equals $n^4$.


Smallest Number with 2^n Divisors

The smallest positive integer with $2^n$ divisors is found by multiplying together the first $n$ numbers in this sequence: :$2, 3, 4, 5, 7, 9, 11, 13, 16, 17, 19, \ldots$ which consists of all the primes and powers of primes.


Convergent Subsequence of Cauchy Sequence/Normed Division Ring

Let $\struct {R, \norm {\,\cdot\,} }$ be a normed division ring. Let $\sequence{x_n}_{n \mathop \in \N}$ be a Cauchy sequence in $\struct {R, \norm {\,\cdot\,} }$.  Let $x \in R$. Then $\sequence {x_n}$ converges to $x$ {{iff}} $\sequence {x_n}$ has a subsequence that converges to $x$.


Null Sequences form Maximal Left and Right Ideal

Let $\struct {R, \norm {\, \cdot \,} }$ be a normed division ring. Let $\CC$ be the ring of Cauchy sequences over $R$. Let $\NN$ be the set of null sequences. That is: :$\NN = \set {\sequence {x_n}: \displaystyle \lim_{n \mathop \to \infty} x_n = 0 }$ Then $\NN$ is a ring ideal of $\CC$ that is a maximal left ideal and a maximal right ideal.


Null Sequences form Maximal Left and Right Ideal/Lemma 1

:$\mathcal N$ is an ideal of $\mathcal C$.


Null Sequences form Maximal Left and Right Ideal/Lemma 2

:$\mathcal N$ is a maximal left ideal.


Null Sequences form Maximal Left and Right Ideal/Lemma 3

:$\NN$ is a maximal right ideal.


Prime Decomposition of 8th Fermat Number

The prime decomposition of the $8$th Fermat number is given by: {{begin-eqn}} {{eqn | l = 2^{\paren {2^8} } + 1       | r = 115 \, 792 \, 089 \, 237 \, 316 \, 195 \, 423 \, 570 \, 985 \, 008 \, 687 \, 907 \, 853 \, 269 \, 984 \, 665 \, 640 \, 564 \, 039 \, 457 \, 584 \, 007 \, 913 \, 129 \, 639 \, 937       | c =  }} {{eqn | r = 1 \, 238 \, 926 \, 361 \, 552 \, 897       | c =  }} {{eqn | o =       | ro= \times       | r = 93 \, 461 \, 639 \, 715 \, 357 \, 977 \, 769 \, 163 \, 558 \, 199 \, 606 \, 896 \, 584 \, 051 \, 237 \, 541 \, 638 \, 188 \, 580 \, 280 \, 321       | c =  }} {{eqn | r = \paren {2 \times 157 \times 3 \, 853 \, 149 \, 761 \times 2^{10} + 1}       | c =  }} {{eqn | o =       | ro= \times       | r = \paren {2 \times 3 \times 5 \times 7 \times 13 \times 31 \, 618 \, 624 \, 099 \, 079 \times 1 \, 057 \, 372 \, 046 \, 781 \, 162 \, 536 \, 274 \, 034 \, 354 \, 686 \, 893 \, 329 \, 625 \, 329 \times 2^{10} + 1}       | c =  }} {{end-eqn}}


Product of Sequence of Fermat Numbers plus 2

Let $F_n$ denote the $n$th Fermat number. Then: {{begin-eqn}} {{eqn | ll= \forall n \in \Z_{>0}:       | l = F_n       | r = \prod_{j \mathop = 0}^{n - 1} F_j + 2       | c =  }} {{eqn | r = F_0 F_1 \dotsm F_{n - 1} + 2       | c =  }} {{end-eqn}}


492 is Sum of 3 Cubes in 3 Ways

$492$ can be expressed as the sum of $3$ cubes, either positive or negative in $3$ known ways. {{begin-eqn}} {{eqn | l = 492       | r = 50^3 + \paren {-19}^3 + \paren {-49}^3 }} {{eqn | r = 123 \, 134^3 + 9179^3 + \paren {-123 \, 151}^3 }} {{eqn | r = 1 \, 793 \, 337 \, 644^3 + \paren {-81 \, 3701 \, 167}^3 + \paren {-1 \, 735 \, 662 \, 109}^3 }} {{end-eqn}}


Smallest n needing 6 Numbers less than n so that Product of Factorials is Square

Let $n \in \Z_{>0}$ be a positive integer. Then it is possible to choose at most $6$ positive integers less than $n$ such that the product of their factorials is square. The smallest $n$ that actually requires $6$ numbers to be chosen is $527$.


Null Sequences form Maximal Left and Right Ideal/Lemma 1/Lemma 1.1

:$\mathcal N \ne \O$


Null Sequences form Maximal Left and Right Ideal/Lemma 1/Lemma 1.2

:$\forall \sequence {x_n}, \sequence {y_n} \in \NN: \sequence {x_n} + \paren {-\sequence {y_n} } \in \NN$


Null Sequences form Maximal Left and Right Ideal/Lemma 1/Lemma 1.3

:$\quad \forall \sequence {x_n} \in \NN, \sequence {y_n} \in \CC: \sequence {x_n} \sequence {y_n} \in \NN, \sequence {y_n} \sequence {x_n} \in \NN$


Null Sequences form Maximal Left and Right Ideal/Lemma 2/Lemma 2.1

:$\NN \subsetneq \CC$.


Null Sequences form Maximal Left and Right Ideal/Lemma 2/Lemma 2.2

:There is no left ideal $\JJ$ of $\CC$ such that $\NN \subsetneq \JJ \subsetneq \CC$


Difference between Two Squares equal to Repdigit

{{begin-eqn}} {{eqn | l = 6^2 - 5^2       | r = 11       | c =  }} {{eqn | l = 56^2 - 45^2       | r = 1111       | c =  }} {{eqn | l = 556^2 - 445^2       | r = 111 \, 111       | c =  }} {{eqn | o = :       | c =  }} {{end-eqn}} {{begin-eqn}} {{eqn | l = 7^2 - 4^2       | r = 33       | c =  }} {{eqn | l = 67^2 - 34^2       | r = 3333       | c =  }} {{eqn | l = 667^2 - 334^2       | r = 333 \, 333       | c =  }} {{eqn | o = :       | c =  }} {{end-eqn}} {{begin-eqn}} {{eqn | l = 8^2 - 3^2       | r = 55       | c =  }} {{eqn | l = 78^2 - 23^2       | r = 5555       | c =  }} {{eqn | l = 778^2 - 223^2       | r = 555 \, 555       | c =  }} {{eqn | o = :       | c =  }} {{end-eqn}} {{begin-eqn}} {{eqn | l = 9^2 - 2^2       | r = 77       | c =  }} {{eqn | l = 89^2 - 12^2       | r = 7777       | c =  }} {{eqn | l = 889^2 - 112^2       | r = 777 \, 777       | c =  }} {{eqn | o = :       | c =  }} {{end-eqn}}


Largest Number Not Expressible as Sum of Fewer than 8 Cubes

$8042$ is (probably) the largest positive integer that cannot be expressed as the sum of fewer than $8$ cubes.


Sequence of 5 Consecutive Non-Primable Numbers by Changing 1 Digit

The following sequence of $5$ consecutive positive integers cannot be made into prime numbers by changing just one digit: :$872\,894, 872\,895, 872\,896, 872\,897, 872\,898$ {{OEIS|A192545}}


Smallest 17 Primes in Arithmetic Sequence

The smallest $17$ primes in arithmetic sequence are: :$3\,430\,751\,869 + 87\,297\,210 n$ for $n = 0, 1, \ldots, 16$.


Prime Gap of 654

There exists a prime gap of $654$ between $11\,000\,001\,446\,613\,353$ and $11\,000\,001\,446\,614\,007$.


Pair of Large Twin Primes

The integers defined as: :$1\,159\,142\,985 \times 2^{2304} \pm 1$ are a pair of twin primes each with $703$ digits.


Integers under Subtraction do not form Group

Let $\struct {\Z, -}$ denote the algebraic structure formed by the set of integers under the operation of subtraction. Then $\struct {\Z, -}$ is not a group.


Sequence of Powers of Number less than One/Necessary Condition

Let $x \in \R$ be such that $\size{x} < 1$. Let $\sequence {x_n}$ be the sequence in $\R$ defined as $x_n = x^n$. Then $\sequence {x_n}$ is a null sequence.


Sequence of Powers of Number less than One/Sufficient Condition

Let $x \in \R$. Let $\sequence {x_n}$ be the sequence in $\R$ defined as $x_n = x^n$. Let $\sequence {x_n}$ be a null sequence. Then $\size x < 1$.


Real Sine Function is neither Injective nor Surjective

The real sine function is neither an injection nor a surjection.


Sequence of Powers of Number less than One/Complex Numbers

Let $z \in \C$. Let $\sequence {z_n}$ be the sequence in $\C$ defined as $z_n = z^n$. Then: :$\size z < 1$ {{iff}} $\sequence {z_n}$ is a null sequence.


Sequence of Powers of Number less than One/Normed Division Ring

Let $\struct {R, \norm {\,\cdot\,}}$ be a normed division ring Let $x \in \R$. Let $\sequence {x_n}$ be the sequence in $\R$ defined as $x_n = x^n$. Then: :$\norm x < 1$ {{iff}} $\sequence {x_n}$ is a null sequence.


Sequence of Powers of Number less than One

Let $x \in \R$. Let $\sequence {x_n}$ be the sequence in $\R$ defined as $x_n = x^n$. Then: :$\size x < 1$ {{iff}} $\sequence {x_n}$ is a null sequence.


Set of Rotations is Subgroup of Symmetry Group

Let $G$ be a symmetry group. Let $H$ be the subset of $G$ consisting of the rotations in $G$ about a given axis. Then $H$ is a subgroup of $G$.


Equivalence of Definitions of Equivalent Division Ring Norms

Let $R$ be a division ring. Let $\norm {\, \cdot \,}_1: R \to \R_{\ge 0}$ and $\norm {\, \cdot \,}_2: R \to \R_{\ge 0}$ be norms on $R$. Let $d_1$ and $d_2$ be the metrics induced by the norms $\norm {\, \cdot \,}_1$ and $\norm {\, \cdot \,}_2$ respectively. {{TFAE|def = Equivalent Division Ring Norms}}


Equivalence of Definitions of Equivalent Division Ring Norms/Topologically Equivalent implies Convergently Equivalent

Let $d_1$ and $d_2$ be topologically equivalent metrics. Then: :$d_1$ and $d_2$ are convergently equivalent metrics.


Equivalence of Definitions of Equivalent Division Ring Norms/Convergently Equivalent implies Null Sequence Equivalent

Let $\norm {\, \cdot \,}_1$ and $\norm {\, \cdot \,}_2$ satisfy: :for all sequences $\sequence {x_n}$ in $R:\sequence {x_n}$ converges to $l$ in $\norm{\, \cdot \,}_1 \iff \sequence {x_n}$ is a converges to $l$ in $\norm {\, \cdot \,}_2$ Then for all sequences $\sequence {x_n}$ in $R$: :$\sequence {x_n}$ is a null sequence in $\norm {\, \cdot \,}_1 \iff \sequence {x_n}$ is a null sequence in $\norm {\, \cdot \,}_2$


Equivalence of Definitions of Equivalent Division Ring Norms/Null Sequence Equivalent implies Open Unit Ball Equivalent

Let $\norm {\, \cdot \,}_1$ and $\norm {\, \cdot \,}_2$ satisfy: :for all sequences $\sequence {x_n}$ in $R:\sequence {x_n}$ is a null sequence in $\norm {\, \cdot \,}_1  \iff \sequence {x_n}$ is a null sequence in $\norm {\, \cdot \,}_2$  Then $\forall x \in R$: :$\norm x_1 < 1 \iff \norm x_2 < 1$


Equivalence of Definitions of Equivalent Division Ring Norms/Open Unit Ball Equivalent implies Norm is Power of Other Norm

Let $\norm {\, \cdot \,}_1$ and $\norm {\, \cdot \,}_2$ satisfy: :$\forall x \in R: \norm x_1 < 1 \iff \norm x_2 < 1$  Then: :$\exists \alpha \in \R_{> 0}: \forall x \in R: \norm x_1 = \norm x_2^\alpha$


Product of Subgroups of Prime Power Order

Let $p$ be a prime number. Let $G$ be a group of order $p^a k$, where: :$a \in \Z_{>0}$ is a (strictly) positive integer :$p$ is not a divisor of $k$. Let $P \le G$ be a subgroup of $G$ of order $p^a$. Let $Q \le G$ be a subgroup of $G$ of order $p^b$, where $0 < b \le a$. Let it be the case that $Q$ is not a subgroup of $P$. Then $P Q$ is not a subgroup of $G$.


Linearly Independent Set is Contained in some Basis

There exists a basis $B$ for $E$ such that $H \subseteq B$.


Linearly Independent Set is Basis iff of Same Cardinality as Dimension

$H$ is a basis for $E$ {{iff}} it contains exactly $n$ elements.


ISBN-10 is Error-Correcting Code

ISBN-$10$ is an error-correcting code in the following sense:


ISBN-10 is Error-Correcting Code/Transposition Error

If any two of the first $9$ digits are transposed, the check digit will be wrong.


ISBN-10 is Error-Correcting Code/Transmission Error

If an error has been made in any one of the first $9$ digits, the check digit will be wrong.


Cardinality of Master Code

Let $\map V {n, p}$ be a master code of length $n$ modulo $p$. Then there are $p^n$ elements of $\map V {n, p}$.


Master Code forms Vector Space

Let $\map V {n, p}$ be a master code of length $n$ modulo $p$. Then $\map V {n, p}$ forms a vector space over $\Z_p$ of $n$ dimensions.


Minimum Distance of Linear Code is Smallest Weight of Non-Zero Codeword

Let $C$ be a linear $\tuple {n, k}$-code whose master code is $\map V {n, p}$. Let $\map d C$ denote the minimum distance of $C$. Then: :$\map d C = \displaystyle \min_{u \mathop \in C} \map w u$ where $\map w u$ denotes the weight of $u$.