hcju/proofwikips huggingface dataset:
Primitive of Reciprocal of x squared minus a squared/Inverse Hyperbolic Tangent Form/Proof
Primitive of Reciprocal of x squared minus a squared/Inverse Hyperbolic Tangent Form/Proof 0 91596 491322 2020-09-27T22:30:09Z Prime.mover 59 Created page with "== Theorem == {{:Primitive of Reciprocal of x squared minus a squared/Inverse Hyperbolic Tangent Form}}  == Proof ==  Let $\size x < a$.  Let:  {{begin-eqn}} {{eq..." wikitext text/x-wiki == Theorem == {{:Primitive of Reciprocal of x squared minus a squared/Inverse Hyperbolic Tangent Form}} == Proof ==  Let $\size x < a$. Let: {{begin-eqn}} {{eqn | l = u       | r = \tanh^{-1} {\frac x a}       | c = {{Defof|Real Inverse Hyperbolic Tangent}}, which is defined where $\size {\dfrac x a} < 1$ }} {{eqn | ll= \leadsto       | l = x       | r = a \tanh u       | c =  }} {{eqn | ll= \leadsto       | l = \frac {\d x} {\d u}       | r = a \sech^2 u       | c = Derivative of Hyperbolic Cotangent }} {{eqn | ll= \leadsto       | l = \int \frac 1 {x^2 - a^2} \rd x       | r = \int \frac {a \sech^2 u} {a^2 \tanh^2 u - a^2} \rd u       | c = Integration by Substitution }} {{eqn | r = \frac a {a^2} \int \frac {\sech^2 u} {-\paren {1 - \tanh^2 u} } \rd u       | c = Primitive of Constant Multiple of Function }} {{eqn | r = \frac 1 a \int \frac {\sech^2 u} {-\sech^2 u} \rd u       | c = Sum of Squares of Hyperbolic Secant and Tangent }} {{eqn | r = -\frac 1 a \int \rd u }} {{eqn | r = -\frac 1 a u + C       | c = Integral of Constant }} {{eqn | r = -\frac 1 a \tanh^{-1} {\frac x a} + C       | c = Definition of $u$ }} {{end-eqn}}
Leigh.Samphier/Sandbox/Matroid Satisfies Base Axiom/Sufficient Condition/Lemma/Lemma 1
 == Theoren == Let $U, V$ be finite sets. Let $\card V < \card U$. Then:  :$\card{V \setminus U} < \card{U \setminus V}$  == Proof == We have: {{begin-eqn}} {{eqn | l = \card{U \setminus V}       | r = \card U - \card{U \cap V}       | c = Cardinality of Set Difference }} {{eqn | o = >        | r = \card V - \card{U \cap V}       | c = As $\card V < \card U$ }} {{eqn | r = \card{V \setminus U}       | c = Cardinality of Set Difference }} {{end-eqn}}
Transitivity of Integrality/Lemma
Transitivity of Integrality/Lemma 0 92667 497829 2020-11-05T14:11:07Z Prime.mover 59 Created page with "== Lemma for Transitivity of Integrality ==  Let $A \subseteq B$ be a ring extension.  Let $x_1, \dotsc, x_n \in B$ be Definit..." wikitext text/x-wiki == Lemma for Transitivity of Integrality ==  Let $A \subseteq B$ be a ring extension. Let $x_1, \dotsc, x_n \in B$ be integral over $A$. Let $A \sqbrk {x_1, \dotsc, x_n}$ be the subring of $B$ generated by $A \cup \set {x_1, \dotsc, x_n}$ over $A$. Then $A \sqbrk {x_1, \dotsc, x_n}$ is integral over $A$.  == Proof == Let $C$ be the integral closure of $A$ in $B$. Since the $x_i$ are integral over $A$, they lie in $C$. So by Integral Closure is Subring, all sums of the form: :$\ds \sum_{\text {finite} } r x_1^{\alpha_1} \dotsm x_n^{\alpha_n}, \quad r \in A,\ \alpha_j \in \N \cup \set 0$ lie in $C$. That is, they are integral over $A$.  But the set of such sums is precisely $A \sqbrk {x_1, \dotsc, x_n}$.  == Linguistic Note == {{:Transitivity of Integrality/Linguistic Note}}
Schur-Zassenhaus Theorem
Let $G$ be a finite group and $N$ be a normal subgroup in $G$. Let $N$ be a Hall subgroup of $G$. Then there exists $H$, a complement of $N$, such that $G$ is the semidirect product of $N$ and $H$.
Non-Zero Integer has Unique Positive Integer Associate
Let $a \in \Z$ be an integer such that $a \ne 0$. Then $a$ has a unique associate $b \in \Z_{>0}$.
Element of Cyclic Group is not necessarily Generator
Let $\gen g = G$ be a cyclic group. Let $a \in G$ Then it is not necessarily the case that $a$ is also a generator of $G$.
Equivalent Statements for Congruence Modulo Subgroup/Left
Let $x \equiv^l y \pmod H$ denote that $x$ is left congruent modulo $H$ to $y$. Then the following statements are equivalent: {{begin-eqn}} {{eqn | n = 1       | l = x       | o = \equiv^l       | r = y \pmod H }} {{eqn | n = 2       | l = x^{-1} y       | o = \in       | r = H }} {{eqn | n = 3       | l = \exists h \in H: x^{-1} y       | r = h }} {{eqn | n = 4       | l = \exists h \in H: y       | r = x h }} {{end-eqn}}
Element of Group is in Unique Coset of Subgroup/Left
There exists a exactly one left coset of $H$ containing $x$, that is: $x H$
Order of Product of Commuting Group Elements of Coprime Order is Product of Orders
Let $G$ be a group. Let $g_1, g_2 \in G$ be commuting elements such that: {{begin-eqn}} {{eqn | l = \order {g_1}       | r = n_1 }} {{eqn | l = \order {g_1}       | r = n_2 }} {{end-eqn}} where $\order {g_1}$ denotes the order of $g_1$ in $G$. Let $n_1$ and $n_2$ be coprime. Then: :$\order {g_1 g_2} = n_1 n_2$
Trivial Norm on Division Ring is Norm
Let $\struct {R, +, \circ}$ be a division ring, and denote its ring zero by $0_R$. Then the trivial norm $\norm {\, \cdot \,}: R \to \R_{\ge 0}$, which is given by: :$\norm x = \begin{cases}   0 & \text { if } x = 0_R\\   1 & \text { otherwise} \end{cases}$ defines a norm on $R$.
Coset of Subgroup of Subgroup
Let $G$ be a group. Let $H, K \le G$ be subgroups of $G$. Let $K \subseteq H$. Let $x \in G$. Then either: :$x K \subseteq H$ or: :$x K \cap H = \O$ where $x K$ denotes the left coset of $K$ by $x$.
Equivalence of Definitions of Characteristic of Ring
Let $\struct {R, +, \circ}$ be a ring with unity whose zero is $0_R$ and whose unity is $1_R$. {{TFAE|def = Characteristic of Ring}}
Non-Cyclic Group of Order p^2 has p+3 Subgroups
Let $p$ be a prime number. Let $G$ be a non-cyclic group whose order is $p^2$. Then $G$ has exactly $p + 3$ subgroups.
General Morphism Property for Groups
Let $\struct {G, \circ}$ and $\struct {H, *}$ be groups. Let $\phi: G \to H$ be a homomorphism. Then: :$\forall g_k \in H: \map \phi {g_1 \circ g_2 \circ \cdots \circ g_n} = \map \phi {g_1} * \map \phi {g_2} * \cdots * \map \phi {g_n}$
Set Difference is Anticommutative
Set difference is an anticommutative operation: :$S = T \iff S \setminus T = T \setminus S = \varnothing$
Integers under Addition form Monoid
The set of integers under addition $\struct {\Z, +}$ forms a monoid.
Klein Four-Group is Group
The Klein $4$-group $K_4$ is a group.
Dihedral Group is Non-Abelian
Let $n \in \N$ be a natural number such that $n > 2$. Let $D_n$ denote the dihedral group of order $2 n$. Then $D_n$ is not abelian.
Summation Formula (Complex Analysis)
:$\displaystyle \sum_{n \in \Z \setminus X} \map f n = - \sum_{z_0 \mathop \in X} \Res {\pi \cot \paren {\pi z} \map f z} {z_0}$
Set with Complement forms Partition
Let $\varnothing \subset S \subset \mathbb U$. Then $S$ and its complement $\complement \left({S}\right)$ form a partition of the universal set $\mathbb U$.
Subgroups of Additive Group of Integers Modulo m
Let $n \in \Z_{> 0}$ be a (strictly) positive integer. Let $\struct {\Z_m, +_m}$ denote the additive group of integers modulo $m$. The subgroups of $\struct {\Z_m, +_m}$ are the additive groups of integers modulo $k$ where: :$k \divides m$
Finite Monoid with Left Cancellable Operation is Group
Let $\left({S, \circ}\right)$ be a finite monoid. Let $\circ$ be a left cancellable operation. Then $\left({S, \circ}\right)$ is a group.
Cartesian Product is Anticommutative
Let $S, T \ne \O$. Then: :$S \times T = T \times S \implies S = T$
Groups of Order 6
There exist exactly $2$ groups of order $6$, up to isomorphism: :$C_6$, the cyclic group of order $6$ :$S_3$, the symmetric group on $3$ letters.
Normalizer of Rotation 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 $\map {N_{D_n} } {\set \alpha}$ denote the normalizer of the singleton containing the rotation element $\alpha$. Then: :$\map {N_{D_n} } {\set \alpha} = \gen \alpha$ where $\gen \alpha$ is the subgroup generated by $\alpha$.
Möbius Strip has Euler Characteristic Zero
Let $M$ be a Möbius Strip. Then: :$\map \chi M = 0$ where $\map \chi M$ denotes the Euler characteristic of the graph $M$.
Group Action of Symmetric Group Acts Transitively
Let $S$ be a set. Let $\struct {\map \Gamma S, \circ}$ be the symmetric group on $S$. Let $*: \map \Gamma S \times S \to S$ be the group action defined as: :$\forall \pi \in \map \Gamma S, \forall s \in S: \pi * s = \map \pi s$ Then $*$ is a transitive group action. In other words, $\struct {\map \Gamma S, \circ}$ acts transitively on $S$ by $*$.
Orbit of Trivial Group Action is Singleton
Let $\left({G, \circ}\right)$ be a group whose identity is $e$. Let $S$ be a set. Let $*: G \times S \to S$ be the trivial group action: :$\forall \left({g, s}\right) \in G \times S: g * s = s$ Let $s \in S$. Then the orbit of $s$ under $*$ is $\left\{{s}\right\}$.
Length of Orbit 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 length of the orbit of $g H$ is $\index K {K \cap H^g}$.
Conjugacy Action on Abelian Group is Trivial
Let $\struct {G, \circ}$ be an abelian group whose identity is $e$. Let $*: G \times G \to G$ be the conjugacy group action: : $\forall g, h \in G: g * h = g \circ h \circ g^{-1}$ Then $*$ is a trivial group action.
Relation Symmetry
Every non-null relation has exactly one of these properties: it is either: :symmetric, :asymmetric or :non-symmetric.
Stabilizer of Cartesian Product of Group Actions
Let $\struct {G, \circ}$ be a group. Let $S$ and $T$ be sets. Let $*_S: G \times S \to S$ and $*_T: G \times T \to T$ be group actions. Let the group action $*: G \times \paren {S \times T} \to S \times T$ be defined as: :$\forall \tuple {g, \tuple {s, t} } \in G \times \paren {S \times T}: g * \tuple {s, t} = \tuple {g *_S s, g *_T t}$ Then the stabilizer of $\tuple {s, t} \in S \times T$ is given by: :$\Stab {s, t} = \Stab s \cap \Stab t$ where $\Stab s$ and $\Stab t$ are the stabilizers of $s$ and $t$ under $*_S$ and $*_T$ respectively.
Divisibility by 12
Let $N \in \N$ be expressed as: :$N = a_0 + a_1 10 + a_2 10^2 + \cdots + a_n 10^n$ Then $N$ is divisible by $12$  $a_0 - 2 a_1 + 4 \paren {\displaystyle \sum_{r \mathop = 2}^n a_r}$ is  divisible by $12$.
Composite Relation with Inverse is Symmetric
Let $\mathcal R \subseteq S \times T$ be a relation. Then the composition of $\mathcal R$ with its inverse $\mathcal R^{-1}$ is symmetric: * $\mathcal R^{-1} \circ \mathcal R$ is a symmetric relation on $S$ * $\mathcal R \circ \mathcal R^{-1}$ is a symmetric relation on $T$.
Minimum Rule for Real Sequences
:$\displaystyle \lim_{n \mathop \to \infty} \min \set {x_n, y_n} = \min \set {l, m}$
Union of Equivalences
The union of two equivalence relations is '''not''' necessarily an equivalence relation itself.
Topological Properties of Non-Archimedean Division Rings/Intersection of Open Balls
:$\map {B_r} x \cap \map {B_s} y \ne \O \iff \map {B_r} x \subseteq \map {B_s} y$ or $\map {B_s} y \subseteq \map {B_r} x$
Universal Negative implies Particular Negative iff First Predicate is not Vacuous
Consider the categorical statements: :$\map {\mathbf E} {S, P}: \quad$ The universal negative: $\forall x: \map S x \implies \neg \map P x$ :$\map {\mathbf O} {S, P}: \quad$ The particular negative: $\exists x: \map S x \land \neg \map P x$ Then: :$\map {\mathbf E} {S, P} \implies \map {\mathbf O} {S, P}$ : :$\exists x: \map S x$ Using the symbology of predicate logic: :$\exists x: \map S x \iff \paren {\paren {\forall x: \map S x \implies \neg \map P x} \implies \paren {\exists x: \map S x \land \neg \map P x} }$
Addition on Numbers has no Zero Element
On all the number systems: * natural numbers $\N$ * integers $\Z$ * rational numbers $\Q$ * real numbers $\R$ * complex numbers $\C$ there exists no zero element for addition.
Group has Latin Square Property/Additive Notation
Let $\struct {G, +}$ be a group. Then $G$ satisfies the Latin square property. That is, for all $a, b \in G$, there exists a unique $g \in G$ such that $a + g = b$. Similarly, there exists a unique $h \in G$ such that $h + a = b$.
Normed Division Ring Operations are Continuous/Inversion
:$\iota : \struct {R^* ,d^*} \to \struct {R, d} : \map \iota x = x^{-1}$  is continuous.
Intersection 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 $\struct {\mathscr S, \cap}$ is closed.
Composite of Quotient Mappings
Let $S$ be a set. Let $\mathcal R_1$ be an equivalence on $S$, and $\mathcal R_2$ be an equivalence on the quotient set $S / \mathcal R_1$. We can find an equivalence $\mathcal R_3$ on $S$ such that $\paren {S / \mathcal R_1} / \mathcal R_2$ is in one-to-one correspondence with $S / \mathcal R_3$ under the mapping: :$\phi: \paren {S / \mathcal R_1} / \mathcal R_2 \to S / \mathcal R_3: \eqclass {\eqclass x {\mathcal R_1} } {\mathcal R_2} \mapsto \eqclass x {\mathcal R_3}$.
Subgroup Generated by Commuting Elements is Abelian
Let $\struct {G, \circ}$ be a group. Let $S \subseteq G$ such that: :$\forall x, y \in S: x \circ y = y \circ x$ Then the subgroup generated by $S$ is abelian.
Stabilizer is Normal iff Stabilizer of Each Element of Orbit
Let $\struct {G, \circ}$ be a group. Let $S$ be a set. Let $*: G \times S \to S$ be a group action. Let $x \in S$. Let $\Stab x$ denote the stabilizer of $x$ under $*$. Let $\Orb x$ denote the orbit of $x$ under $*$. Then $\Stab x$ is normal in $G$  $\Stab x$ is also the stabilizer of every element in $\Orb x$.
Additive Group of Integers is Normal Subgroup of Rationals
Let $\struct {\Z, +}$ be the additive group of integers. Let $\struct {\Q, +}$ be the additive group of rational numbers. Then $\struct {\Z, +}$ is a normal subgroup of $\struct {\Q, +}$.
Empty Set and Set form Algebra of Sets
Let $S$ be any non-empty set. Then $\left\{{S, \varnothing}\right\}$ is (trivially) an algebra of sets, where $S$ is the unit.
Closure of Union and Complement imply Closure of Set Difference
Let $\RR$ be a system of sets on a universe $\mathbb U$ such that for all $A, B \in \RR$: :$(1): \quad A \cup B \in \RR$ :$(2): \quad \map \complement A \in \RR$ where $\cup$ denotes set union and $\complement$ denotes complement (relative to $\mathbb U$). Then: :$\forall A, B \in \RR: A \setminus B \in \RR$ where $\setminus$ denotes set difference.
Order of Alternating Group
Let $n \in \Z$ be an integer such that $n > 1$. Let $A_n$ be the alternating group on $n$ letters. Then: :$\order {A_n} = \dfrac {n!} 2$ where $\order {A_n}$ denotes the order of $A_n$.
Plane contains Infinite Number of Lines
A plane contains an infinite number of distinct lines.
Three Non-Collinear Planes have One Point in Common
Three planes which are not collinear have exactly one point in all three planes.
Union of Mappings which Agree is Mapping
Let $A, B, Y$ be sets. Let $f: A \to Y$ and $g: B \to Y$ be mappings. Let $X = A \cup B$. Let $f$ and $g$ agree on $A \cap B$. Then $f \cup g: X \to Y$ is a mapping.
Right Operation is Idempotent
The right operation is idempotent: :$\forall x: x \rightarrow x = x$
Index of Intersection of Subgroups/Corollary
Let $H$ be a subgroup of $G$. Let $K$ be a subgroup of finite index of $G$. Then: :$\index H {H \cap K} \le \index G K$
Centralizer of Self-Inverse Element of Non-Abelian Finite Simple Group is not That Group
Let $G$ be a non-abelian finite simple group. Let $t \in G$ be a self-inverse element of $G$. Then: :$\map {C_G} t \ne G$ where $\map {C_G} t$ denotes the centralizer of $t$ in $G$.
Sum of Sequence of Squares of Fibonacci Numbers
:$\forall n \ge 1: \displaystyle \sum_{j \mathop = 1}^n {F_j}^2 = F_n F_{n + 1}$ That is: :${F_1}^2 + {F_2}^2 + {F_3}^2 + \cdots + {F_n}^2 = F_n F_{n + 1}$
Properties of Norm on Division Ring/Norm of Integer
For all $n \in \N_{\gt 0}$, let $n \cdot 1_R$ denote the sum of $1_R$ with itself $n$-times. That is: :$n \cdot 1_R = \underbrace {1_R + 1_R + \dots + 1_R}_{n \, times}$ Then: :$\norm {n \cdot 1_R} \le n$.
Equivalence Relation is Congruence for Constant Operation
Every equivalence relation is a congruence relation for the constant operation.
Quotient Structure on Subset Product
Let $\left({S, \circ}\right)$ be an algebraic structure. Let $\mathcal R$ be a congruence for $\circ$ on $S$. Then: :$\forall X, Y \in S / \mathcal R: X \circ_\mathcal P Y \subseteq X \circ_\mathcal R Y$ where: : $S / \mathcal R$ is the quotient of $S$ by $\mathcal R$ : $\circ_\mathcal P$ is the operation induced on $\mathcal P \left({S}\right)$ by $\circ$ : $\circ_\mathcal R$ is the operation induced on $S / \mathcal R$ by $\circ$
Even Power of 3 as Sum of Consecutive Positive Integers
Take the positive integers and group them in sets such that the $n$th set contains the next $3^n$ positive integers: :$\set 1, \set {2, 3, 4}, \set {5, 6, \ldots, 13}, \set {14, 15, \cdots, 40}, \ldots$ Let the $n$th such set be denoted $S_{n - 1}$, that is, letting $S_0 := \set 1$ be considered as the zeroth. Then the sum of all the elements of $S_n$ is $3^{2 n}$.
Exterior of Union of Singleton Rationals is Empty
Let $B_\alpha$ be the singleton containing the rational number $\alpha$. Let $\struct {\R, \tau_d}$ be the real number line with the usual (Euclidean) topology $\tau_d$. Then the exterior in $\struct {\R, \tau_d}$ of the union of all $B_\alpha$ is the empty set: :$\displaystyle \paren {\bigcup_{\alpha \mathop \in \Q} B_\alpha}^e = \O$
Smallest 18 Primes in Arithmetic Sequence
The smallest $18$ primes in arithmetic sequence are: :$107\,928\,278\,317 + 9\,922\,782\,870 n$ for $n = 0, 1, \ldots, 16$.
Irrational Number Space is Non-Meager
Let $\struct {\R \setminus \Q, \tau_d}$ be the irrational number space under the Euclidean topology $\tau_d$. Then $\struct {\R \setminus \Q, \tau_d}$ is non-meager.
Quotient Structure of Inverse Completion
Let $\left({T, \circ'}\right)$ be an inverse completion of a commutative semigroup $\left({S, \circ}\right)$, where $C$ is the set of cancellable elements of $S$. Let $f: S \times C: T$ be the mapping defined as: :$\forall x \in S, y \in C: f \left({x, y}\right) = x \circ' y^{-1}$ Then the mapping $g: \left({S \times C}\right) / \mathcal R_f \to T$ defined by $g \left({\left[\!\left[{x, y}\right]\!\right]_{\mathcal R_f}}\right) = x \circ' y^{-1}$, where $\left({S \times C}\right) / \mathcal R_f$ is a quotient structure, is an isomorphism.
Square Matrices with +1 or -1 Determinant under Multiplication forms Group
Let $n \in \Z_{> 0}$ be a strictly positive integer. Let $S$ be the set of square matrices of order $n$ of real numbers whose determinant is either $1$ or $-1$. Let $\struct {S, \times}$ denote the algebraic structure formed by $S$ whose operation is (conventional) matrix multiplication. Then $\struct {S, \times}$ is a group.
Rational Number Space is not Weakly Sigma-Locally Compact
Let $\struct {\Q, \tau_d}$ be the rational number space under the Euclidean topology $\tau_d$. Then $\struct {\Q, \tau_d}$ is not weakly $\sigma$-locally compact.
Group of Order 3 is Unique
There exists exactly $1$ group of order $3$, up to isomorphism: :$C_3$, the cyclic group of order $3$.
Integer Reciprocal Space is Topological Space
Let $\struct {\R, \tau_d}$ be the real number line $\R$ under the usual (Euclidean) topology $\tau_d$. Let $A \subseteq \R$ be the set of all points on $\R$ defined as: :$A := \set {\dfrac 1 n: n \in \Z_{>0} }$ Then the integer reciprocal space $\struct {A, \tau_d}$ is a topological space.
Integer Reciprocal Space with Zero is not Extremally Disconnected
Let $A \subseteq \R$ be the set of all points on $\R$ defined as: :$A := \set 0 \cup \set {\dfrac 1 n : n \in \Z_{>0} }$ Let $\struct {A, \tau_d}$ be the integer reciprocal space with zero under the usual (Euclidean) topology. Then $A$ is not extremally disconnected.
Components of Integer Reciprocal Space with Zero are Single Points
Let $A \subseteq \R$ be the set of all points on $\R$ defined as: :$A := \set 0 \cup \set {\dfrac 1 n : n \in \Z_{>0} }$ Let $\struct {A, \tau_d}$ be the integer reciprocal space with zero under the usual (Euclidean) topology. Then the components of $A$ are singletons.
Non-Homeomorphic Sets may be Homeomorphic to Subsets of Each Other
Let $T_1 = \struct {S_1, \tau_1}$ and $T_2 = \struct {S_2, \tau_2}$ be topological spaces. Let $H_1 \subseteq S_1$ and $H_2 \subseteq S_2$. Then it is possible for: :$(1): \quad T_1$ to be homeomorphic to $H_2$ :$(2): \quad T_2$ to be homeomorphic to $H_1$ but: :$(3): \quad T_1$ and $T_2$ to not be homeomorphic.
Superspace of Homeomorphic Subspaces may not have Homeomorphism to Itself containing Subspace Homeomorphism
Let $T_1 = \struct {S_1, \tau_1}$ and $T_2 = \struct {S_2, \tau_2}$ be topological spaces. Let $H_1 \subseteq S_1$ and $H_2 \subseteq S_2$. Let $H_1$ and $H_2$ be a homeomorphic. Then it may be the case that there does not exist a homeomorphism $g: T_1 \to T_2$ such that: :$g \restriction_{H_1} = f$ where: :$g \restriction_{H_1}$ is the restriction of $g$ to $H_1$ :$f: H_1 \to H_2$ is a homeomorphism.
Finite Group is p-Group iff Order is Power of p
Let $p$ be a prime number. Let $G$ be a finite group. Then $G$ is a $p$-group  the order of $G$ is a power of $p$.
Trivial Ring from Abelian Group
Any abelian group $\struct {G, +}$ may be turned into a trivial ring by defining the ring product to be: :$\forall x, y \in G: x \circ y = e_G$
Product with Inverse on Homomorphic Image is Group Homomorphism
Let $G$ be a group. Let $H$ be an abelian group. Let $\theta: G \to H$ be a (group) homomorphism. Let $\phi: G \times G \to H$ be the mapping defined as: :$\forall \tuple {g_1, g_2} \in G \times G: \map \phi {g_1, g_2} = \map \theta {g_1} \map \theta {g_2}^{-1}$ Then $\phi$ is a homomorphism.
Divisor of Product may not be Divisor of Factors
Let $a, b, c \in \Z_{>0}$ be (strictly) positive integers. Let: :$c \divides a b$ where $\divides$ expresses the relation of divisibility. Then it is not necessarily the case that either $c \divides a$ or $c \divides b$.
Equality of Complex Numbers
Let $z_1 := a_1 + i b_1$ and $z_2 := a_2 + i b_2$ be complex numbers. Then $z_1 = z_2$  $a_1 = a_2$ and $b_1 = b_2$.
Divisor of Unit is Unit
Let $\struct {D, +, \circ}$ be an integral domain whose unity is $1_D$. Let $\struct {U_D, \circ}$ be the group of units of $\struct {D, +, \circ}$. Then: :$x \in D, u \in U_D: x \divides u \implies x \in U_D$ That is, if $x$ is a divisor of a unit, $x$ must itself be a unit.
Associatehood is Equivalence Relation
Let $\struct {D, +, \circ}$ be an integral domain whose zero is $0_D$ and whose unity is $1_D$. Let $\sim$ be the relation defined on $D$ as: $\forall x, y \in D: x \sim y$  $x$ is an associate of $y$ Then $\sim$ is an equivalence relation.
Field is Subfield of Itself
Let $\struct {F, +, \circ}$ be a field. Then $\struct {F, +, \circ}$ is a subfield of $\struct {F, +, \circ}$.
Tangent Exponential Formulation/Formulation 1
:$\tan z = i \dfrac {1 - e^{2 i z} } {1 + e^{2 i z} }$
Tangent Exponential Formulation/Formulation 3
:$\tan z = -i \paren {\dfrac {e^{i z} - e^{-i z} } {e^{i z} + e^{-i z} } }$
Roots of Complex Number/Exponential Form
:$z^{1 / n} = \set {r^{1 / n} e^{i \paren {\theta + 2 \pi k} / n}: k \in \set {0, 1, 2, \ldots, n - 1} }$
Normalizer of Center is Group
Let $G$ be a group. Let $\map Z G$ be the center of $G$. Let $x \in G$. Let $\map {N_G} x$ be the normalizer of $x$ in $G$. Then: :$\map Z G = \set {x \in G: \map {N_G} x = G}$ That is, the center of a group $G$ is the set of elements $x$ of $G$ such that the normalizer of $x$ is the whole of $G$. Thus: :$x \in \map Z G \iff \map {N_G} x = G$ and so: :$\index G {\map {N_G} x} = 1$ where $\index G {\map {N_G} x}$ is the index of $\map {N_G} x$ in $G$.
Characterisation of Non-Archimedean Division Ring Norms/Corollary 4
Let $R$ have characteristic $p > 0$. Then $\norm {\,\cdot\,}$ is a non_Archimedean norm on $R$.