hcju/proofwikips huggingface dataset:
Definition:Finite Path
Let $G$ be a graph. A '''finite path''' in $G$ is a path consisting of a finite number of edges.
Definition:Cantor Space
Let $\CC$ be the Cantor set. Let $\tau_d$ be the Euclidean topology on $\R$. Then since $\CC \subseteq \R$, we can endow $\CC$ with the subspace topology $\tau_\CC$. The topological space $\struct {\CC, \tau_\CC}$ is referred to as the '''Cantor space'''.
Definition:Hyperbolic Sine
The '''hyperbolic sine''' function is defined on the complex numbers as: :$\sinh: \C \to \C$: :$\forall z \in \C: \sinh z := \dfrac {e^z - e^{-z} } 2$
Definition:Hyperbolic Cosine
The '''hyperbolic cosine''' function is defined on the complex numbers as: :$\cosh: \C \to \C$: :$\forall z \in \C: \cosh z := \dfrac {e^z + e^{-z} } 2$
Definition:Norm
A '''norm''' is a measure which describes a sense of the '''size''' or '''length''' of a mathematical object. In its various contexts: === Ring === {{:Definition:Norm/Ring}} === Division Ring === {{:Definition:Norm/Division Ring}} === Vector Space === {{:Definition:Norm/Vector Space}} === Algebra === {{:Definition:Norm/Algebra}} === Unital Algebra === Let $R$ be a division ring with norm $\norm {\,\cdot\,}_R$. {{:Definition:Norm/Unital Algebra}}
Definition:Hyperbolic Tangent
{{:Definition:Hyperbolic Tangent/Definition 1}}
Definition:Domain (Category Theory)
Let $f: X \to Y$ be a morphism. Then the '''domain''' of $f$ is defined to be the object $X$. This is usually denoted $X = \Dom f$ or $X = \map D f$.
Definition:Codomain (Category Theory)
Let $f: X \to Y$ be a morphism. Then the '''codomain''' of $f$ is defined to be the object $Y$. This is usually denoted $Y = \Cdm f$.
Definition:Codomain (Set Theory)/Relation
The '''codomain''' of a relation $\RR \subseteq S \times T$ is the set $T$. It can be denoted $\Cdm \RR$.
Definition:Codomain (Set Theory)/Mapping
Let $f: S \to T$ be a mapping. The '''codomain''' of $f$ is the set $T$. It is denoted on  by $\Cdm f$.
Definition:Domain (Set Theory)/Relation
Let $\RR \subseteq S \times T$ be a relation. The '''domain''' of $\RR$ is defined and denoted as: :$\Dom \RR := \set {s \in S: \exists t \in T: \tuple {s, t} \in \RR}$ That is, it is the same as what is defined here as the preimage of $\RR$.
Definition:Domain (Set Theory)/Mapping
Let $f: S \to T$ be a mapping. The '''domain''' of $f$ is $S$, and can be denoted $\Dom f$. In the context of mappings, the '''domain''' and the preimage of a mapping are the same set.
Definition:Domain (Set Theory)/Binary Operation
Let $\circ: S \times S \to T$ be a binary operation. The '''domain''' of $\circ$ is the set $S$ and can be denoted $\Dom \circ$.
Definition:Finite Sequence
A '''finite sequence''' is a sequence whose domain is finite. === Length of Sequence === {{transclude:Definition:Length of Sequence|increase=1|section=tc}}
Definition:Operation/Binary Operation
A binary operation is the special case of an operation where the operation has exactly two operands. A '''binary operation''' is a mapping $\circ$ from the Cartesian product of two sets $S \times T$ to a universe $\mathbb U$: :$\circ: S \times T \to \mathbb U: \map \circ {s, t} = y \in \mathbb U$ If $S = T$, then $\circ$ can be referred to as a '''binary operation on''' $S$.
Definition:Locus
A '''locus''' is a set of points which satisfy a particular condition. Such points usually form a curve or a surface, depending on the context.
Definition:Vertical Tangent Line
Let $P = \left({c, f \left({c}\right)}\right)$ be a point on the graph of a real function $f$. The vertical line $x = c$ is a vertical tangent line to the graph of $f$ at $P$  any of the following hold: : $(1): \quad f$ is right continuous at $c$ and $\displaystyle \lim_{x \mathop \to c ^+} f' \left({x}\right) = +\infty$ : $(2): \quad f$ is right continuous at $c$ and $\displaystyle \lim_{x \mathop \to c ^+} f' \left({x}\right) = -\infty$ : $(3): \quad f$ is left continuous at $c$ and $\displaystyle \lim_{x \mathop \to c ^-} f' \left({x}\right) = +\infty$ : $(4): \quad f$ is left continuous at $c$ and $\displaystyle \lim_{x \mathop \to c ^-} f' \left({x}\right) = -\infty$
Definition:Left Zero
An element $z_L \in S$ is called a '''left zero element''' (or just '''left zero''') : :$\forall x \in S: z_L \circ x = z_L$
Definition:Right Zero
An element $z_R \in S$ is called a '''right zero element''' (or just '''right zero''') : :$\forall x \in S: x \circ z_R = z_R$
Definition:Divergent Improper Integral
An improper integral of a real function $f$ is said to '''diverge''' if any of the following hold: : $(1): \quad f$ is continuous on $\left[{a \,.\,.\, +\infty}\right)$ and the limit $\displaystyle \lim_{b \mathop \to +\infty} \int_a^b f \left({x}\right) \ \mathrm d x$ does not exist : $(2): \quad f$ is continuous on $\left({-\infty  \,.\,.\, b}\right]$ and the limit $\displaystyle \lim_{a \mathop \to -\infty} \int_a^b f \left({x}\right) \ \mathrm d x$ does not exist : $(3): \quad f$ is continuous on $\left[{a \,.\,.\, b}\right)$, has an infinite discontinuity at $b$, and the limit $\displaystyle \lim_{c \mathop \to b^-} \int_a^c f \left({x}\right) \ \mathrm dx$ does not exist : $(4): \quad f$ is continuous on $\left({a \,.\,.\, b}\right]$, has an infinite discontinuity at $a$, and the limit $\displaystyle \lim_{c \mathop \to a^+} \int_c^b f \left({x}\right) \ \mathrm dx$ does not exist.
Definition:Definite Integral/Limits of Integration
In the expression $\displaystyle \int_a^b \map f x \rd x$, the values $a$ and $b$ are called the '''limits of integration'''. If there is no danger of confusing the concept with limit of a function or of a sequence, just '''limits'''. Thus $\displaystyle \int_a^b \map f x \rd x$ can be voiced: : '''The integral of (the function) $f$ of $x$ {{WRT|Integration}} $x$ (evaluated) between the limits (of integration) $a$ and $b$.''' More compactly (and usually), it is voiced: : '''The integral of $f$ of $x$ {{WRT|Integration}} $x$ between $a$ and $b$''' or: : '''The integral of $f$ of $x$ dee $x$ from $a$ to $b$''' === Lower Limit === {{:Definition:Definite Integral/Limits of Integration/Lower Limit}} === Upper Limit === {{:Definition:Definite Integral/Limits of Integration/Upper Limit}} == Also known as == The interval defined by the '''limits of integration''' can be referred to as the '''range of integration'''. Some sources refer to it as the '''interval of integration'''. == Also see == From the Fundamental Theorem of Calculus, we have that: :$\displaystyle \int_a^b \map f x \rd x = \map F b - \map F a$ where $F$ is a primitive of $f$, that is: :$\map f x = \dfrac \d {\d x} \map F x$ Then $\map F b - \map F a$ is usually written: :$\bigintlimits {\map F x} {x \mathop = a} {x \mathop = b} := \map F b - \map F a$ or, when there is no chance of ambiguity as to the independent variable: :$\bigintlimits {\map F x} a b := \map F b - \map F a$ Some sources use: :$\Big.{\map F x}\Big|_a^b := \map F b - \map F a$ but this is not recommended, as it is not so clear exactly where the expression being evaluated actually starts.
Definition:Cofinal Subset
Let $\left({S, \mathcal R}\right)$ be a relational structure, that is, a set $S$ endowed with a binary relation $\mathcal R$. Let $T \subseteq S$ be a subset of $S$. Then $T$ is a '''cofinal subset of $S$ with respect to $\mathcal R$''' : :$\forall x \in S: \exists t \in T: x \mathop {\mathcal R} t$
Definition:Net (Preordered Set)
Let $X$ be a nonempty set. Let $\left({\Lambda, \precsim}\right)$ be a preordered set. Let $F: \Lambda \to X$ be a mapping. Then $f$ is referred to as a '''net'''.
Definition:Inner Limit
Let $\left({\mathcal X, \tau}\right)$ be a Hausdorff topological space. Let $\left \langle {C_n}\right \rangle_{n \in \N}$ be a sequence of sets in $\mathcal X$. The '''inner limit''' of $\left \langle {C_n}\right \rangle_{n \in \N}$ is defined as: : $\displaystyle \liminf_{n \to\infty} \ C_n := \left\{{x : \exists N \text{ cofinite set of }\N, \exists x_v \in C_v \left({v \in N}\right) \text{ such that } x_v \to x}\right\}$ where $x_v \to x$ denotes convergence in the topology $\tau$.
Definition:Horizontal Asymptote
The horizontal line $y = L$ is a '''horizontal asymptote''' of the graph of a real function $f$ if either of the following limits exist: :$\displaystyle \lim_{x \mathop \to +\infty} \map f x = L_1$ :$\displaystyle \lim_{x \mathop \to -\infty} \map f x = L_2$
Definition:Strict Well-Ordering
=== Definition 1 === {{:Definition:Strict Well-Ordering/Definition 1}} === Definition 2 === {{:Definition:Strict Well-Ordering/Definition 2}}
Definition:Foundational Relation
Let $\left({A, \mathcal R}\right)$ be a relational structure where $A$ is either a proper class or a set. Then $\mathcal R$ is a '''foundational relation on $A$'''  every non-empty subset of $A$ has an $\mathcal R$-minimal element.
Definition:Coset/Right Coset
The '''right coset of $y$ modulo $H$''', or '''right coset of $H$ by $y$''', is: :$H y = \set {x \in G: \exists h \in H: x = h y}$ This is the equivalence class defined by right congruence modulo $H$. That is, it is the subset product with singleton: :$H y = H \set y$
Definition:Coset Space/Right Coset Space
The '''right coset space (of $G$ modulo $H$)''' is the quotient set of $G$ by right congruence modulo $H$, denoted $G / H^r$. It is the set of all the right cosets of $H$ in $G$.
Definition:Coset/Left Coset
The '''left coset of $x$ modulo $H$''', or '''left coset of $H$ by $x$''', is: :$x H = \set {y \in G: \exists h \in H: y = x h}$ This is the equivalence class defined by left congruence modulo $H$. That is, it is the subset product with singleton: :$x H = \set x H$
Definition:Coset Space/Left Coset Space
The '''left coset space (of $G$ modulo $H$)''' is the quotient set of $G$ by left congruence modulo $H$, denoted $G / H^l$. It is the set of all the left cosets of $H$ in $G$.
Definition:Congruence Modulo Subgroup/Right Congruence
:$\mathcal R^r_H = \set {\tuple {x, y} \in G \times G: x y^{-1} \in H}$ This is called '''right congruence modulo $H$'''.
Definition:Congruence Modulo Subgroup/Left Congruence
:$\mathcal R^l_H := \set {\tuple {x, y} \in G \times G: x^{-1} y \in H}$ This is called '''left congruence modulo $H$'''.
Definition:Transversal (Group Theory)/Left Transversal
$S$ is a '''left transversal for $H$ in $G$'''  every left coset of $H$ contains '''exactly one''' element of $S$.
Definition:Transversal (Group Theory)/Right Transversal
$S$ is a '''right transversal for $H$ in $G$'''  every right coset of $H$ contains '''exactly one''' element of $S$.
Definition:Transitive Class
Let $A$ denote a class, which can be either a set or a proper class. Then $A$ is '''transitive'''  every element of $A$ is also a subclass of $A$. That is, $A$ is '''transitive''' : :$x \in A \implies x \subseteq A$ or: :$\forall x: \forall y: \paren {x \in y \land y \in A \implies x \in A}$
Definition:Epsilon Relation
In the language of set theory $\in$, the membership primitive, is neither a class nor a set, but a primitive predicate. To simplify formulations, it is useful to introduce a class which behaves identically to the standard membership relation $\in$ for sets. This class, denoted $\Epsilon$, will be referred to as the '''epsilon relation'''. In class-builder notation: :$\Epsilon := \left\{{ \left({ x , y }\right) : x \in y }\right\}$ Thus, explicitly, $\Epsilon$ is a relation, taking arguments from ordered pairs of sets $x$ and $y$. It consists of precisely those ordered pairs $\left({ x , y }\right)$ satisfying $x \in y$. The behavior is thus seen to be identical to regular membership with sets. It is '''not''' the same as class membership, because $x$ and $y$ must be set variables. === Restriction of Epsilon Relation === {{:Definition:Epsilon Relation/Restriction}}
Definition:Point-to-Set Distance
Let $\mathcal X$ be a normed space. The '''point-to-set distance''' on $\mathcal X$ is a mapping: :$d: \mathcal X \times \mathcal P \left({\mathcal X}\right) \to \left[{0 \,.\,.\, \infty}\right)$ where $P \left({\mathcal X}\right)$ is the power set of $\mathcal X$. This mapping is defined as: :$d \left({x, C}\right) := \inf \left\{{\left\|{x - y}\right\|: y \in C}\right\}$ with the convention that for any $x \in \mathcal X$: :$d \left({x, \varnothing}\right) = +\infty$ or, to be more mathematically rigorous: :$\forall C \in \mathcal P \left({\mathcal X}\right) \setminus \left\{{\varnothing}\right\}: d \left({x, \varnothing}\right) > d \left({x, C}\right)$
Definition:Minkowski Sum
Let $V$ be a vector space. Let $A, B$ be two subsets of $V$. Then the '''Minkowski sum''' of $A$ and $B$, denoted as $A + B$, is defined as: :$A + B := \set {a + b: a \in A, b \in B}$ where the operation $+$ is vector addition. The '''Minkowski sum''' is therefore a relation in the power set of $V$ in the sense that it is a mapping: :$+: \powerset V^2 \to \powerset V$ {{NamedforDef|Hermann Minkowski|cat = Minkowski}}
Definition:Well-Defined/Mapping
Let $f: S \to T$ be a mapping. Let $\RR$ be an equivalence relation on $S$. Let $S / \RR$ be the quotient set determined by $\RR$. Let $\phi: S / \RR \to T$ be a mapping such that: :$\map \phi {\eqclass x \RR} = \map f x$ Then $\phi: S / \RR \to T$ is '''well-defined''' : :$\forall \tuple {x, y} \in \RR: \map f x = \map f y$
Definition:Well-Defined/Operation
Let $\struct {S, \circ}$ be an algebraic structure. Let $\RR$ be a congruence for $\circ$. Let $\circ_\RR$ be the operation induced on $S / \RR$ by $\circ$. Let $\struct {S / \RR, \circ_\RR}$ be the quotient structure defined by $\RR$, where $\circ_\RR$ is defined as: :$\eqclass x \RR \circ_\RR \eqclass y \RR = \eqclass {x \circ y} \RR$ Then $\circ_\RR$ is '''well-defined (on $S / \RR$)''' : :$x, x' \in \eqclass x \RR, y, y' \in \eqclass y \RR \implies x \circ y = x' \circ y'$
Definition:Topological Vector Space
Let $V$ be a vector space over a topological field $K$. Let $\tau$ be a topology on $V$. Then $\left({V, \tau}\right)$ is called a '''topological vector space''' : {{begin-axiom}} {{axiom | n = 1         | t = $\tau$ is a Hausdorff topology }} {{axiom | n = 2         | t = $+: V \times V \to V$ is continuous with respect to $\tau$ }} {{axiom | n = 3         | t = $\cdot: K \times V \to V$ is continuous with respect to $\tau$ }} {{end-axiom}}
Definition:Star Shaped Set
Let $V$ be a vector space over a field $K$. Let $W \subseteq V$ be a subset of $V$. Then $W$ is called a '''star shaped set''' : :$\forall x \in W: -x \in W$ where $-x$ is the negative of $x$.
Definition:Absorbent Set
Let $V$ be a vector space over a field $K$. Let $W \subseteq V$ be a subset of $V$. Let $a \in K$. Let the set $a \cdot W$ be defined as: : $a \cdot W := \left\{{a \cdot y: y \in W} \right\}$ Then $W$ is an '''absorbent set in $V$''' : : $\displaystyle \bigcup_{a \mathop \in K} a \cdot W = V$ which symbolically can be represented as: : $K \cdot W = V$
Definition:Closure (Abstract Algebra)/Algebraic Structure
Let $\struct {S, \circ}$ be an algebraic structure. Then $S$ has the property of '''closure under $\circ$''' : :$\forall \tuple {x, y} \in S \times S: x \circ y \in S$ $S$ is said to be '''closed under $\circ$''', or just that '''$\struct {S, \circ}$ is closed'''.
Definition:Closure (Abstract Algebra)/Scalar Product
Let $\struct {S, \circ}_R$ be an $R$-algebraic structure over a ring $R$. Let $T \subseteq S$ such that $\forall \lambda \in R: \forall x \in T: \lambda \circ x \in T$. Then $T$ is '''closed for scalar product'''. If $T$ is also closed for operations on $S$, then it is called a '''closed subset''' of $S$.
Definition:Directed Set
Let $\struct {S, \precsim}$ be a preordered set. Then $\struct {S, \precsim}$ is a '''directed set'''  every pair of elements of $S$ has an upper bound in $S$: :$\forall x, y \in S: \exists z \in S: x \precsim z$ and $y \precsim z$
Definition:Epigraph
Let $f: S \to \overline \R$ be an extended real-valued function. The '''epigraph''' of $f$ is the set: :$\map {\operatorname {epi} } f := \set {\tuple {x, \alpha} \in S \times \R: \map f x \le \alpha}$
Definition:Lower Level Set
Let $f: S \to \overline \R$ be an extended real-valued function. Let $\alpha \in \R$. The '''$\alpha$-lower level set of $f$''' is the set: :$\displaystyle \operatorname{lev} \limits_{\mathop \le \alpha} f := \left\{ {x \in S: f \left({x}\right) \le \alpha}\right\}$
Definition:Lower Semicontinuous
Let $f: S \to \R \cup \left\{{-\infty, \infty}\right\}$ be an extended real valued function. Let $S$ be endowed with a topology $\tau$.  Then $f$ is '''lower semicontinuous''' at $\bar x \in S$ : : $\displaystyle \liminf_{x \mathop \to \bar x} f \left({x}\right) = f \left({\bar x}\right)$ where $\displaystyle \liminf_{x \mathop \to \bar x} f \left({x}\right)$ stands for the lower limit of $f$ at $\bar x$. === Lower Semicontinuous on Subset === {{:Definition:Lower Semicontinuous/Subset}}
Definition:Ordinal Class
The '''Ordinal Class''' is defined as the class of all ordinals: :$\operatorname{On} = \{ x : x$ is an ordinal $\}$
Definition:Trivial Norm
{{#switch: 3|1 = =|2 = ==|3 = ===|4 = ====|5 = =====|6 = ======|#default = }} Division Ring {{#switch: 3|1 = =|2 = ==|3 = ===|4 = ====|5 = =====|6 = ======|#default = }} {{:Definition:Trivial Norm/Division Ring}} {{#switch: 3|1 = =|2 = ==|3 = ===|4 = ====|5 = =====|6 = ======|#default = }} Vector Space {{#switch: 3|1 = =|2 = ==|3 = ===|4 = ====|5 = =====|6 = ======|#default = }} {{:Definition:Trivial Norm/Vector Space}}
Definition:Lower Limit (Topological Space)
Let $\struct {S, \tau}$ be a topological space. Let $f: S \to \R \cup \set {-\infty, \infty}$ be an extended real-valued function. The '''lower limit''' of $f$ at some $x_0 \in S$ is defined as: :$\displaystyle \liminf_{x \mathop \to x_0} \map f x := \sup_{V \mathop \in \map \mho {x_0} } \set {\inf_{x \mathop \in V} \map f x}$ where $\map \mho {x_0}$ stands for the set of open neighborhoods of $x_0$.
Definition:Outer Limit
Let $\left({\mathcal X, \tau}\right)$ be a Hausdorff topological space. Let $\left \langle {C_n}\right \rangle_{n \in \N}$ be a sequence of sets in $\mathcal X$. The '''outer limit''' of $\left \langle {C_n}\right \rangle_{n \in \N}$ is defined as: : $\displaystyle \limsup_{n \to\infty} \ C_n := \left\{{x : \exists N \text{ cofinal set of }\N, \exists x_v \in C_v \left({v \in N}\right) \text{ such that } x_v \to x}\right\}$ where $x_v \to x$ denotes convergence in the topology $\tau$.
Definition:Painlevé-Kuratowski Convergence
Let $T = \struct {S, \tau}$ be a Hausdorff topological space. Let $\sequence {C_n}_{n \mathop \in \N}$ be a sequence of sets in $T$. Let $\sequence {C_n}_{n \mathop \in \N}$ be such that: :$\displaystyle \liminf_n C_n = \limsup_n C_n = C$ where: :$\displaystyle \liminf_n C_n$ denotes the inner limit of $\sequence {C_n}_{n \mathop \in \N}$ :$\displaystyle \limsup_n C_n$ denotes the outer limit of $\sequence {C_n}_{n \mathop \in \N}$ Then $\sequence {C_n}_{n \mathop \in \N}$ is said to be '''convergent in the sense of Painlevé-Kuratowski'''. It can be denoted as: :$C_n \overset K \to C$ or: :$\operatorname {K-lim} \limits_{n \mathop \to \infty} C_n = C$ or simply: :$\displaystyle \lim_n C_n = C$ {{NamedforDef|Paul Painlevé|name2 = Kazimierz Kuratowski|cat = Painlevé|cat2 = Kuratowski}}
Definition:Ambiguity
An '''ambiguity''' is a statement which has more than one distinct meaning. Thus a statement is '''ambiguous''' if, without extraneous clarification, it can be interpreted in more than one way.
Definition:Classical Probability Model
Let $\EE$ be an experiment The '''classical probability model''' on $\EE$ is a mathematical model that defines the probability space $\struct {\Omega, \Sigma, \Pr}$ of $\EE$ as follows: :$(1) \quad$ All outcomes of $\EE$ are equally likely :$(2) \quad$ There are a finite number of outcomes. Then: :$\map \Pr {\text {event occurring} } := \dfrac {\paren {\text {number of outcomes favorable to event} } } {\paren {\text {total number of outcomes possible} } }$ or formally: :$\map \Pr \omega := \dfrac {\card \Sigma} {\card \Omega}$ where: :$\card {\, \cdot \,}$ denotes the cardinality of a set :$\omega$ denotes an event: $\omega \in \Omega$ :$\Sigma$ denotes the event space: $\Sigma \subseteq \Omega$ :$\Omega$ denotes the sample space.
Definition:Equivocation
An '''equivocation''' is a shift in meaning of an ambiguous term. If an argument's persuasive force depends on utilizing an '''equivocation''' it is fallacious. To make such an argument is to commit the '''fallacy of equivocation'''.
Definition:Relative Frequency Model
The '''relative frequency model''' is a mathematical model that defines the probability of an event occurring as follows: :$\map \Pr {\text {event occurring} } := \dfrac {\paren {\text {observed number of times event has occurred in the past} } } {\paren {\text {observed number of times event has occurred or not occurred} } }$ That is, the probability of an event happening is defined as the '''relative frequency''' of a finite number of events of a particular type in some finite reference class of events.  Symbolically: :$\map \Pr \omega := \dfrac {f_{\omega} } n$ where: :$\omega$ is an elementary event :$f_{\omega}$ is how many times $\omega$ occurred :$n$ is the number of trials observed.
Definition:Relative Frequency
Let $S$ be a sample or a finite population. Let $\omega$ be a qualitative variable, or a class of a quantitative variable. The '''relative frequency''' of $\omega$ is defined as: :$\map {\operatorname {RF} } \omega := \dfrac {f_\omega} n$ where: :$f_\omega$ is the (absolute) frequency of $\omega$ :$n$ is the number of individuals in $S$.
Definition:Sorgenfrey Line
Let $\R$ be the set of real numbers Let $\BB$ be the set: :$\BB = \set {\hointr a b: a, b \in \R}$ where $\hointr a b$ is the half-open interval $\set {x \in \R: a \le x < b}$. Then $\BB$ is the basis for a topology $\tau$ on $\R$. The topological space $T = \struct {\R, \tau}$ is referred to as the '''Sorgenfrey line'''.
Definition:Sequence of Distinct Terms
A '''sequence of distinct terms of $S$''' is an injection from a subset of $\N$ into $S$. Thus a sequence $\sequence {a_k}_{k \mathop \in A}$ is a '''sequence of distinct terms''' : :$\forall j, k \in A: j \ne k \implies a_j \ne a_k$
Definition:Categorical Statement
Let $S$ and $P$ be predicates. A '''categorical statement''' is a statement that can be expressed in one of the following ways in natural language: {{begin-axiom}} {{axiom | n  = A         | lc = Universal Affirmative:         | t  = Every $S$ is $P$ }} {{axiom | n  = E         | lc = Universal Negative:         | t  = No $S$ is $P$ }} {{axiom | n  = I         | lc = Particular Affirmative:         | t  = Some $S$ is $P$ }} {{axiom | n  = O         | lc = Particular Negative:         | t  = Some $S$ is not $P$ }} {{end-axiom}}
Definition:Topological Group
Let $\left({G, \odot}\right)$ be a group. On its underlying set $G$, let $\left({G, \tau}\right)$ be a topological space. === Definition 1 === {{:Definition:Topological Group/Definition 1}} === Definition 2 === {{:Definition:Topological Group/Definition 2}}
Definition:Inversion Mapping/Topology
Let $T = \left({G, \circ, \tau}\right)$ be a topological group. Let $\phi: G \to G$ be the mapping defined as: : $\forall x \in G: \phi \left({x}\right) = x^{-1}$ Then $\phi$ is the '''inversion mapping''' of $T$.