hcju/proofwikips huggingface dataset:
Composite of Homomorphisms is Homomorphism/Algebraic Structure
Let: : $\left({S_1, \circ_1, \circ_2, \ldots, \circ_n}\right)$ : $\left({S_2, *_1, *_2, \ldots, *_n}\right)$ : $\left({S_3, \oplus_1, \oplus_2, \ldots, \oplus_n}\right)$ be algebraic structures. Let: : $\phi: \left({S_1, \circ_1, \circ_2, \ldots, \circ_n}\right) \to \left({S_2, *_1, *_2, \ldots, *_n}\right)$ : $\psi: \left({S_2, *_1, *_2, \ldots, *_n}\right) \to \left({S_3, \oplus_1, \oplus_2, \ldots, \oplus_n}\right)$ be homomorphisms. Then the composite of $\phi$ and $\psi$ is also a homomorphism.
Composite of Homomorphisms is Homomorphism/R-Algebraic Structure
Let: : $\left({S_1, *_1}\right)_R$ : $\left({S_2, *_2}\right)_R$ : $\left({S_3, *_3}\right)_R$ be $R$-algebraic structures with the same number of operations. Let: : $\phi: \left({S_1, *_1}\right)_R \to \left({S_2, *_2}\right)_R$ : $\psi: \left({S_2, *_2}\right)_R \to \left({S_3, *_3}\right)_R$ be homomorphisms. Then the composite of $\phi$ and $\psi$ is also a homomorphism.
Subgroup is Subgroup of Normalizer
Let $G$ be a group. A subgroup $H \le G$ is a subgroup of its normalizer: :$H \le G \implies H \le \map {N_G} H$
Inclusion Mappings to Topological Sum from Components
Let $\struct {X, \tau_1}$ and $\struct {Y, \tau_2}$ be topological spaces. Let $\struct {Z, \tau_3}$ be the topological sum of $X$ and $Y$ where $\tau_3$ is the topology generated by $\tau_1$ and $\tau_2$. Then $\tau_3$ is the finest topology on $Z$ in which the inclusion mappings from $\struct {X, \tau_1}$ and $\struct {Y, \tau_2}$ to $\struct {Z, \tau_3}$ are continuous.
Principal Ultrafilter is All Sets Containing Cluster Point
Let $S$ be a set. Let $\powerset S$ denote the power set of $S$. Let $\FF \subset \powerset S$ be a principal ultrafilter on $S$. Let its cluster point be $x$. Then $\FF$ is the set of all subsets of $S$ which contain $x$.
Equivalence of Definitions of T4 Space
{{TFAE|def = T4 Space|view = $T_4$ space}} Let $T = \struct {S, \tau}$ be a topological space.
Equivalence of Definitions of T5 Space
{{TFAE|def = T5 Space|view = $T_5$ space}} Let $T = \struct {S, \tau}$ be a topological space.
Product Space is T2 iff Factor Spaces are T2
Let $\mathbb S = \family{\struct{S_\alpha, \tau_\alpha}}_{\alpha \mathop \in I}$ be an indexed family of topological spaces for $\alpha$ in some indexing set $I$. Let $\displaystyle T = \struct{S, \tau} = \displaystyle \prod_{\alpha \mathop \in I} \struct{S_\alpha, \tau_\alpha}$ be the product space of $\mathbb S$. Then $T$ is a $T_2$ (Hausdorff) space  each of $\struct{S_\alpha, \tau_\alpha}$ is a $T_2$ (Hausdorff) space.
Product Space is Completely Hausdorff iff Factor Spaces are Completely Hausdorff
Let $\mathbb S = \family {\struct {S_\alpha, \tau_\alpha} }_{\alpha \mathop \in I}$ be an indexed family of topological spaces for $\alpha$ in some indexing set $I$. Let $\displaystyle T = \struct{S, \tau} = \displaystyle \prod_{\alpha \mathop \in I} \struct {S_\alpha, \tau_\alpha}$ be the product space of $\mathbb S$. Then $T$ is a $T_{2 \frac 1 2}$ (completely Hausdorff) space  each of $\struct{S_\alpha, \tau_\alpha}$ is a $T_{2 \frac 1 2}$ (completely Hausdorff) space.
Product Space is T3 iff Factor Spaces are T3
Let $\mathbb S = \family {\struct {S_\alpha, \tau_\alpha} }_{\alpha \mathop \in I}$ be an indexed family of non-empty topological spaces for $\alpha$ in some indexing set $I$. Let $\displaystyle T = \struct{S, \tau} = \displaystyle \prod_{\alpha \mathop \in I} \struct{S_\alpha, \tau_\alpha}$ be the product space of $\mathbb S$. Then $T$ is a $T_3$ space  each of $\struct {S_\alpha, \tau_\alpha}$ is a $T_3$ space.
Product Space is T3 1/2 iff Factor Spaces are T3 1/2
Let $\mathbb S = \family {\struct {S_\alpha, \tau_\alpha} }_{\alpha \mathop \in I}$ be an indexed family of topological spaces for $\alpha$ in some indexing set $I$ with $S_\alpha \ne \O$ for every $\alpha \in I$. Let $\displaystyle T = \struct {S, \tau} = \displaystyle \prod_{\alpha \mathop \in I} \struct {S_\alpha, \tau_\alpha}$ be the product space of $\mathbb S$. Then $T$ is a $T_{3 \frac 1 2}$ space  each of $\struct {S_\alpha, \tau_\alpha}$ is a $T_{3 \frac 1 2}$ space.
Factor Spaces are T4 if Product Space is T4
Let $\mathbb S = \family {\struct{S_\alpha, \tau_\alpha} }_{\alpha \mathop \in I}$ be an indexed family of non-empty topological spaces for $\alpha$ in some indexing set $I$. Let $\displaystyle T = \struct {S, \tau} = \displaystyle \prod_{\alpha \mathop \in I} \struct {S_\alpha, \tau_\alpha}$ be the product space of $\mathbb S$. Let $T$ be a $T_4$ space. Then each of $\struct {S_\alpha, \tau_\alpha}$ is a $T_4$ space.
Factor Spaces are T5 if Product Space is T5
Let $\mathbb S = \family {\struct {S_\alpha, \tau_\alpha} }_{\alpha \mathop \in I}$ be an indexed family of topological spaces for $\alpha$ in some indexing set $I$. Let $\displaystyle T = \struct {S, \tau} = \displaystyle \prod_{\alpha \mathop \in I} \struct {S_\alpha, \tau_\alpha}$ be the product space of $\mathbb S$. Let $T$ be a $T_5$ space. Then each of $\struct {S_\alpha, \tau_\alpha}$ is a $T_5$ space.
T4 Property is not Hereditary
Let $T = \struct {S, \tau}$ be a topological space which is a $T_4$ space. Let $T_H = \struct {H, \tau_H}$, where $\O \subset H \subseteq S$, be a subspace of $T$. Then it does not necessarily follow that $T_H$ is a $T_4$ space.
Derivative of Arcsecant Function
:$\dfrac {\map \d {\arcsec x} } {\d x} = \dfrac 1 {\size x \sqrt {x^2 - 1} } = \begin {cases} \dfrac {+1} {x \sqrt {x^2 - 1} } & : 0 < \arcsec x < \dfrac \pi 2 \ (\text {that is: $x > 1$}) \\ \dfrac {-1} {x \sqrt {x^2 - 1} } & : \dfrac \pi 2 < \arcsec x < \pi \ (\text {that is: $x < -1$})  \\ \end{cases}$
Paracompactness is Preserved under Projections
Let $I$ be an indexing set. Let $\family {\struct {S_\alpha, \tau_\alpha} }_{\alpha \mathop \in I}$ be a family of topological spaces indexed by $I$. Let $\displaystyle \struct {S, \tau} = \prod_{\alpha \mathop \in I} \struct {S_\alpha, \tau_\alpha}$ be the product space of $\family {\struct {S_\alpha, \tau_\alpha} }_{\alpha \mathop \in I}$. Let $\pr_\alpha: \struct {S, \tau} \to \struct {S_\alpha, \tau_\alpha}$ be the projection on the $\alpha$ coordinate. If $\struct {S, \tau}$ is paracompact, then each of $\struct {S_\alpha, \tau_\alpha}$ is also paracompact.
Connected Space is Connected Between Two Points
Let $T$ be a topological space which is connected. Then $T$ is connected between two points.
Equality is Reflexive
:$\forall a: a = a$
Closed Set in Metric Space is G-Delta
Let $\left({X, d}\right)$ be a metric space, and let $F \subset X$ be a closed set.  Then $F$ is a $G_\delta$ set of X.
Totally Disconnected Space is Totally Pathwise Disconnected
Let $T = \struct {S, \tau}$ be a topological space which is totally disconnected. Then $T$ is a totally pathwise disconnected space.
Second-Countable T3 Space is T5
Let $T = \struct {S, \tau}$ be a $T_3$ space which is also second-countable. Then $T$ is a $T_5$ space.
T3 Space is Fully T4 iff Paracompact
Let $T = \struct {S, \tau}$ be a $T_3$ space. Then $T$ is fully $T_4$  $T$ is paracompact.
Mean Value Theorem for Integrals
Let $f$ be a continuous real function on the closed interval $\closedint a b$. Then there exists a real number $k \in \closedint a b$ such that: :$\displaystyle \int_a^b \map f x \rd x = \map f k \paren {b - a}$
Cover of Doubletons of Infinite Particular Point Space has no Locally Finite Refinement
Let $T = \struct {S, \tau_p}$ be an infinite particular point space. Let $\CC$ be the open cover of $T$ defined as: :$\CC = \set {\set {x, p}: x \in S, x \ne p}$ Then $\CC$ has no open refinement which is locally finite.
Path-Connected iff Path-Connected to Point
Let $T = \left({S, \tau}\right)$ be a topological space. Let $U \subseteq S$ be a non-empty subset of $T$. Then $U$ is path-connected : :$\exists p \in U: \forall q \in U: \exists f: \left[{0 \,.\,.\, 1}\right] \to U: f$ continuous, $f \left({0}\right) = p$ and $f \left({1}\right) = q$
Uncountable Fort Space is not Separable
Let $T = \struct {S, \tau_p}$ be a Fort space on an uncountable set $S$. Then $T$ is not a separable space.
Countable Fort Space is Separable
Let $T = \struct {S, \tau_p}$ be a Fort space on a countably infinite set $S$. Then $T$ is a separable space.
Countable Fort Space is Second-Countable
Let $T = \struct {S, \tau_p}$ be a Fort space on a countably infinite set $S$. Then $T$ is a second-countable space.
Fort Space is Regular
Let $T = \struct {S, \tau_p}$ be a Fort space. Then $T$ is a regular space.
Countable Fort Space is Metrizable
Let $T = \struct {S, \tau_p}$ be a Fort space on a countably infinite set $S$. Then $T$ is a metrizable space.
Fortissimo Space is not Pseudocompact
Let $T = \struct {S, \tau_p}$ be a Fortissimo space. Then $T$ is not a pseudocompact space.
Double Pointed Fortissimo Space is not Sigma-Compact
Let $T = \struct {S, \tau}$ be a Fortissimo space. Let $T \times D$ be the double pointed topology on $T$. Then $T \times D$ is not $\sigma$-compact.
Neighborhood of Origin in Arens-Fort Space is not Compact
Let $T = \struct {S, \tau}$ be the Arens-Fort space. Let $N_p$ be a neighborhood of the point $p = \tuple {0, 0}$. Then $N_p$ is not compact.
Arens-Fort Space is not Extremally Disconnected
Let $T = \struct {S, \tau}$ be the Arens-Fort space. Then $T$ is not extremally disconnected.
Modified Fort Space is Totally Disconnected
Let $T = \struct {S, \tau_{a, b} }$ be a modified Fort space. Then $T$ is totally disconnected.
Sets in Modified Fort Space are Disconnected
Let $T = \struct {S, \tau_{a, b}}$ be a modified Fort space. Let $H$ be a subset of $S$ with more than one point. Then $H$ is disconnected.
Modified Fort Space is not Totally Separated
Let $T = \struct {S, \tau_{a, b} }$ be a modified Fort space. Then $T$ is not totally separated.
Modified Fort Space is Scattered
Let $T = \struct {S, \tau_{a, b} }$ be a modified Fort space. Then $T$ is scattered.
Isolated Points in Subsets of Modified Fort Space
Let $T = \struct {S, \tau_{a, b} }$ be a modified Fort space. Let $H \subseteq S$ contain more than two points. Then $H$ contains an isolated point.
Modified Fort Space is Sequentially Compact
Let $T = \struct {S, \tau_{a, b} }$ be a modified Fort space. Then $T$ is sequentially compact.
Tangent Function is Odd
:$\map \tan {-x} = -\tan x$ That is, the tangent function is odd.
Tangent of Sum
:$\map \tan {a + b} = \dfrac {\tan a + \tan b} {1 - \tan a \tan b}$
Natural Logarithm Function is Differentiable
The (real) natural logarithm function is differentiable.
Countable Basis of Real Number Line
Let $\struct {\R, \tau_d}$ be the real number line with the usual (Euclidean) topology. Let $\BB$ be the set of subsets of $\R$ defined as: :$\BB = \set {\openint a b: a, b \in \Q,\ a < b}$ That is, $\BB$ is the set of open intervals of $\R$ whose endpoints are rational numbers. Then $\BB$ forms a countable basis of $\struct {\R, \tau_d}$
Second-Countable Space is Separable
Let $T = \struct {S, \tau}$ be a second-countable topological space. Then $T$ is also a separable space.
Real Number Line is Second-Countable
Let $\struct {\R, \tau_d}$ be the real number line with the usual (Euclidean) topology. Then $\struct {\R, \tau_d}$ is second-countable.
Real Number Line is not Countably Compact
Let $\struct {\R, \tau_d}$ be the real number line with the usual (Euclidean) topology. Then $\struct {\R, \tau_d}$ is not countably compact.
Real Number Line is Sigma-Compact
Let $\struct {\R, \tau_d}$ be the real number line with the usual (Euclidean) topology. Then $\struct {\R, \tau_d}$ is $\sigma$- compact.
Closed Subset of Real Number Line is G-Delta
Let $\struct {\R, \tau_d}$ be the real number line with the usual (Euclidean) topology. Let $H \subseteq \R$ be a closed subset of $\R$. Then $H$ is a $G_\delta$ set.
Real Number Line is Paracompact
Let $\struct {\R, \tau_d}$ be the real number line with the usual (Euclidean) topology. Then $\struct {\R, \tau_d}$ is paracompact.
Real Number Line with Off-Center Distance Function is Quasimetric Space
Let $\struct {\R, \tau_d}$ be the real number line with the usual (Euclidean) topology. Then $\tau_d$ can be given by a quasimetric defined as: :$\map d {x, y} = \begin {cases} y - x & : y \ge x \\ 2 \paren {x - y} & : y < x \end {cases}$ Thus $\struct {\R, \tau_d}$ is a quasimetric space.
Sub-basis for Uniformity on Real Number Line
Let $\struct {\R, \tau_d}$ be the real number line with the usual (Euclidean) topology. Let $a, b \in \R$ such that $a < b$. Let $S_{a b}$ be the set of subsets of $\R$ defined as: :$S_{a b} = \set {\tuple {x, y}: x, y < b \text{ or } x, y > a}$ Then $S_{ab}$ is a basis for a uniformity $U$ which generates the usual topology on $\R$. Note that $U$ is clearly not the usual metric uniformity. {{MissingLinks|Lots of assumed definitions here which might not be properly specified.}}
NAND with Equal Arguments
:$p \uparrow p \dashv \vdash \neg p$ That is, the NAND of a proposition with itself corresponds to the negation operation.
NAND is Commutative
:$p \uparrow q \dashv \vdash q \uparrow p$
NAND is not Associative
:$p \uparrow \left({q \uparrow r}\right) \not \vdash \left({p \uparrow q}\right) \uparrow r$
NOR with Equal Arguments
:$p \downarrow p \dashv \vdash \neg p$ That is, the NOR of a proposition with itself corresponds to the negation operator.
NOR is Commutative
:$p \downarrow q \dashv \vdash q \downarrow p$
NOR is not Associative
:$p \downarrow \left({q \downarrow r}\right) \not \vdash \left({p \downarrow q}\right) \downarrow r$
Representation of Ternary Expansions
Let $x \in \R$ be a real number. Let $x$ be represented in base $3$ notation. While it may be possible for $x$ to have two different such representations, for example: :$\dfrac 1 3 = 0.100000 \ldots_3 = 0.022222 \ldots_3$ it is not possible for $x$ be written in more than one way without using the digit $1$.
Equivalence of Definitions of Cantor Set
{{TFAE|def = Cantor Set|view = the Cantor Set $\CC$}}:
Cantor Set is Closed in Real Number Space
Let $\CC$ be the Cantor set. Let $\struct {\R, \tau_d}$ be the real number space $\R$ under the Euclidean topology $\tau_d$. Then $\CC$ is a closed subset of $\struct {\R, \tau_d}$.
Word Metric is Metric
Let $\left({G, \circ}\right)$ be a group. Let $S$ be a generating set for $G$ which is closed under inverses (that is, $x^{-1} \in S \iff x\in S$). Let $d_S$ be the associated word metric. Then $d_S$ is a metric on $G$.
Cantor Space is Compact
Let $\CC$ be the Cantor set. Let $\struct {\R, \tau_d}$ be the real number space $\R$ under the Euclidean topology $\tau_d$. Then $\CC$ is a compact subset of $\struct {\R, \tau_d}$.
Cantor Space is Complete Metric Space
Let $T = \struct {\CC, \tau_d}$ be the Cantor space. Then $T$ is a complete metric space.
Cantor Space satisfies all Separation Axioms
Let $T = \struct {\CC, \tau_d}$ be the Cantor space. Then $T$ satisfies all the separation axioms.
Cantor Space is Second-Countable
Let $T = \struct {\CC, \tau_d}$ be the Cantor space. Then $T$ is a second-countable space.
Cantor Space is Dense-in-itself
Let $T = \struct {\CC, \tau_d}$ be the Cantor space. Then $T$ is dense-in-itself.
Cantor Space is not Scattered
Let $T = \struct {\CC, \tau_d}$ be the Cantor space. Then $T$ is not scattered.
Cantor Space is Perfect
Let $T = \struct {\CC, \tau_d}$ be the Cantor space. Then $\CC$ is a perfect set of the real number space $\R$ under the usual (Euclidean) topology $\tau_d$.
Cantor Space is Nowhere Dense
Let $T = \left({\mathcal C, \tau_d}\right)$ be the Cantor space. Then $T$ is nowhere dense in $\left[{0 \,.\,.\, 1}\right]$.
Cantor Space is Meager in Closed Unit Interval
Let $T = \struct {\CC, \tau_d}$ be the Cantor space. Then $T$ is meager in $\closedint 0 1$.
Cantor Set is Uncountable
The Cantor set $\mathcal C$ is uncountable.
Equality is Symmetric
:$\forall a, b: a = b \implies b = a$
Biconditional is Commutative/Formulation 1/Proof 2
: $p \iff q \dashv \vdash q \iff p$
Derivative of Hyperbolic Cosine Function
:$\map {\dfrac \d {\d x} } {\cosh u} = \sinh u \dfrac {\d u} {\d x}$
Cantor Space is Totally Separated
Let $T = \struct {\CC, \tau_d}$ be the Cantor space. Then $T$ is totally separated.
Cantor Space is not Extremally Disconnected
Let $T = \struct {\CC, \tau_d}$ be the Cantor space. Then $T$ is not extremally disconnected.
Cantor Space as Countably Infinite Product
Let $A_n = \struct {\set {0, 2}, \tau_n}$ be the discrete space of the two points $0$ and $2$. Let $\displaystyle A = \prod_{n \mathop = 1}^\infty A_n$. Let $\struct {A, \tau}$ be the product space where $\tau$ is the Tychonoff topology on $A$. Then $A$ is homeomorphic to the Cantor space.
Cantor Space is not Locally Connected
Let $T = \struct {\CC, \tau_d}$ be the Cantor space. Then $T$ is not locally connected.
Local Connectedness is not Preserved under Infinite Product
The property of local connectedness is not preserved under the operation of forming an infinite product space.
Difference of Squares of Hyperbolic Cosine and Sine
:$\cosh^2 x - \sinh^2 x = 1$ where $\cosh$ and $\sinh$ are hyperbolic cosine and hyperbolic sine.
Derivative of Arccosecant Function
:$\dfrac {\map \d {\arccsc x} } {\d x} = \dfrac {-1} {\size x \sqrt {x^2 - 1} } = \begin {cases} \dfrac {-1} {x \sqrt {x^2 - 1} } & : 0 < \arccsc x < \dfrac \pi 2 \ (\text {that is: $x > 1$}) \\ \dfrac {+1} {x \sqrt {x^2 - 1} } & : -\dfrac \pi 2 < \arccsc x < 0 \ (\text {that is: $x < -1$})  \\ \end{cases}$
Power of Sum Modulo Prime
Let $p$ be a prime number. Then: :$\paren {a + b}^p \equiv a^p + b^p \pmod p$
Volume of Right Circular Cone
The volume $V$ of a right circular cone is given by: :$V = \dfrac 1 3 \pi r^2 h$ where: : $r$ is the radius of the base : $h$ is the height of the cone, that is, the distance between the apex and the center of the base.
Intersection of Topologies is Topology
Let $\left({\tau_i}\right)_{i \in I}$ be an arbitrary indexed set of topologies for a set $S$. Then $\tau := \displaystyle \bigcap_{i \mathop \in I} {\tau_i}$ is also a topology for $S$.
Power Set is Closed under Set Difference
Let $S$ be a set. Let $\powerset S$ be the power set of $S$. Then: :$\forall A, B \in \powerset S: A \setminus B \in \powerset S$ where $A \setminus B$ denotes the set difference of $A$ and $B$.
Powers of Semigroup Element Commute
Let $\struct {S, \odot}$ be a semigroup. Let $a \in S$. Let $m, n \in \Z_{>0}$. Then: :$\forall m, n \in \Z_{>0}: a^n \odot a^m = a^m \odot a^n$
Equivalence of Definitions of Derivative
{{TFAE|def = Derivative of Real Function at Point}} Let $I$ be an open real interval. Let $f: I \to \R$ be a real function defined on $I$. Let $\xi \in I$ be a point in $I$.
Derivative of x to the x
:$\dfrac \d {\d x} x^x = x^x \paren {\ln x + 1}$
Commutativity of Powers in Semigroup
:$\forall m, n \in \N_{>0}: a \circ b = b \circ a \implies a^m \circ b^n = b^n \circ a^m$ but it is not necessarily the case that: :$\forall m, n \in \N_{>0}: a^m \circ b^n = b^n \circ a^m \implies a \circ b = b \circ a$
Power of Product of Commutative Elements in Semigroup
:$\forall n \in \N_{>1}: \paren {x \circ y}^n = x^n \circ y^n \iff x \circ y = y \circ x$
Sum of Arcsecant and Arccosecant
Let $x \in \R$ be a real number such that $\left|{x}\right| \ge 1$. Then: : $\operatorname{arcsec} x + \operatorname{arccsc} x = \dfrac \pi 2$ where $\operatorname{arcsec}$ and $\operatorname{arccsc}$ denote arcsecant and arccosecant respectively.
Equation of Straight Line Tangent to Circle
Let $\tuple {a, b}$ be the center of a circle $\mathcal C$. Let $P_n = \tuple {x_n, y_n}$ be any point on $\mathcal C$. The equation of a non-vertical tangent line $\mathcal T$ to $\mathcal C$ is given by: :$y - y_n = \dfrac {a - x_n} {y_n - b} \paren {x - x_n}$ The equations of the vertical tangent lines to $\mathcal C$ are: :$x = r - a$ for $P = \tuple {r - a, b}$ :$x = a - r$ for $P = \tuple {a - r, b}$
Index Laws for Monoids/Negative Index
:$\forall n \in \Z: \paren {a^n}^{-1} = a^{-n} = \paren {a^{-1} }^n$
Infinite Limit Theorem
Let $f$ be a real function of $x$ of the form  :$\map f x = \dfrac {\map g x} {\map h x}$ Further, let $g$ and $h$ be continuous on some open interval $\mathbb I$, where $c$ is a constant in $\mathbb I$. If: :$(1): \quad \map g c \ne 0$ :$(2): \quad \map h c = 0$ :$(3): \quad \forall x \in \mathbb I: x \ne c \implies \map h x \ne 0$ then the limit of $f$ as $x \to c$ will not exist, and: :$\displaystyle \lim_{x \mathop \to c^+} \map f x = +\infty$ or $-\infty$ :$\displaystyle \lim_{x \mathop \to c^-} \map f x = +\infty$ or $-\infty$
Index Laws for Monoids/Product of Indices
:$\forall m, n \in \Z: a^{n m} = \paren {a^m}^n = \paren {a^n}^m$
Index Laws for Monoids/Sum of Indices
:$\forall m, n \in \Z: a^{n + m} = a^n \circ a^m$
Powers of Group Elements/Negative Index
:$\forall n \in \Z: \paren {g^n}^{-1} = g^{-n} = \paren {g^{-1} }^n$
Powers of Group Elements/Sum of Indices
:$\forall m, n \in \Z: g^m \circ g^n = g^{m + n}$
Powers of Group Elements/Product of Indices
:$\forall m, n \in \Z: \paren {g^m}^n = g^{m n} = \paren {g^n}^m$
Group Element Commutes with Inverse
Let $\struct {G, \circ}$ be a group whose identity is $e$. Let $x \in G$. Then: :$x \circ x^{-1} = x^{-1} \circ x$ That is, $x$ commutes with its inverse $x^{-1}$.
Commutativity of Powers in Monoid
:$\forall m, n \in \Z: a^m \circ b^n = b^n \circ a^m$
Power of Product of Commutative Elements in Monoid
:$\forall n \in \Z: \paren {a \circ b}^n = a^n \circ b^n$
Commutativity of Powers in Group
: $\forall m, n \in \Z: a^m \circ b^n = b^n \circ a^m$
Power of Product of Commutative Elements in Group
:$a \circ b = b \circ a \iff \forall n \in \Z: \paren {a \circ b}^n = a^n \circ b^n$
Primitive of Tangent Function/Cosine Form
:$\displaystyle \int \tan x \rd x = -\ln \size {\cos x} + C$ where $\cos x \ne 0$.
Primitive of Reciprocal
:$\ds \int \frac {\d x} x = \ln \size x + C$ for $x \ne 0$.
Powers of Commuting Elements of Semigroup Commute
:$\forall m, n \in \N_{>0}: \paren {\circ^m a} \circ \paren {\circ^n b} = \paren {\circ^n b} \circ \paren {\circ^m a}$
Right Operation All Elements Right Zeroes
Let $\struct {S, \rightarrow}$ be an algebraic structure in which the operation $\rightarrow$ is the right operation. Then no matter what $S$ is, $\struct {S, \rightarrow}$ is a semigroup all of whose elements are right zeroes. Thus it can be seen that any right zero in a semigroup is not necessarily unique.
Left Operation All Elements Left Zeroes
Let $\struct {S, \leftarrow}$ be an algebraic structure in which the operation $\leftarrow$ is the left operation. Then no matter what $S$ is, $\struct {S, \leftarrow}$ is a semigroup all of whose elements are left zeroes. Thus it can be seen that any left zero in a semigroup is not necessarily unique.
More than One Right Zero then No Left Zero
Let $\struct {S, \circ}$ be an algebraic structure. If $\struct {S, \circ}$ has more than one left zero, then it has no right zero. Likewise, if $\struct {S, \circ}$ has more than one right zero, then it has no left zero.
Left and Right Zero are the Same
Let $\struct {S, \circ}$ be an algebraic structure. Let $z_L \in S$ be a left zero, and $z_R \in S$ be a right zero. Then $z_L = z_R$, that is, both the left and right zero are the same, and are therefore a zero $z$. Furthermore, $z$ is the ''only'' left and right zero for $\circ$.
Primitive of Cotangent Function
:$\displaystyle \int \cot x \rd x = \ln \size {\sin x} + C$ where $\sin x \ne 0$.
Inner Limit in Hausdorff Space by Open Neighborhoods
Let $\left \langle{C_n}\right \rangle_{n \in \N}$ be a sequence of sets in a Hausdorff topological space $\left({\mathcal X, \tau}\right)$. Let $x \in \mathcal X$. Let $\mho \left({x}\right) := \left\{ {V \in \tau:\ x \in V}\right\}$ denote the set of open neighborhoods of $x$. Let $\mathcal N_\infty$ denote the set of cofinite subsets of $\N$: : $\mathcal N_\infty := \left\{ {N \subset \N: \N \setminus N \text{ is finite} }\right\}$ Then the inner limit of $\left \langle{C_n}\right \rangle_{n \in \N}$ is: :$\displaystyle \liminf_n C_n = \left\{ {x \in \mathcal X: \forall V \in \mho \left({x}\right): \exists N \in \mathcal N_\infty: \forall n \in N: C_n \cap V \ne \varnothing}\right\}$ or equivalently: : $\displaystyle \liminf_n C_n = \left\{{x \in \mathcal X: \forall V \in \mho \left({x}\right): \exists N_0 \in \N: \forall n \ge N_0: C_n \cap V \ne \varnothing}\right\}$