hcju/proofwikips huggingface dataset:
Definition:Sigma-Algebra Generated by Collection of Mappings
Let $\left({X_i, \Sigma_i}\right)$ be measurable spaces, with $i \in I$ for some index set $I$. Let $X$ be a set, and let, for $i \in I$, $f_i: X \to X_i$ be a mapping. Then the '''$\sigma$-algebra generated by $\left({f_i}\right)_{i \in I}$''', $\sigma \left({f_i: i \in I}\right)$, is the smallest $\sigma$-algebra on $X$ such that every $f_i$ is $\sigma \left({f_i: i \in I}\right) \, / \, \Sigma_i$-measurable. That is, $\sigma \left({f_i: i \in I}\right)$ is subject to: :$(1):\quad \forall i \in I: \forall E_i \in \Sigma_i: f_i^{-1} \left({E_i}\right) \in \sigma \left({f_i: i \in I}\right)$ :$(2):\quad \sigma \left({f_i: i \in I}\right) \subseteq \Sigma$ for all $\sigma$-algebras $\Sigma$ on $X$ satisfying $(1)$ In fact, $\sigma \left({f_i: i \in I}\right)$ always exists, and is unique, as proved on Existence and Uniqueness of Sigma-Algebra Generated by Collection of Mappings.
Definition:Gradient Operator/Real Cartesian Space
Let $\R^n$ denote the real Cartesian space of $n$ dimensions. Let $\map f {x_1, x_2, \ldots, x_n}$ denote a real-valued function on $\R^n$. Let $\tuple {\mathbf e_1, \mathbf e_2, \ldots, \mathbf e_n}$ be the standard ordered basis on $\R^n$. Let $\mathbf u = u_1 \mathbf e_1 + u_2 \mathbf e_2 + \cdots + u_n \mathbf e_n = \displaystyle \sum_{k \mathop = 1}^n u_k \mathbf e_k$ be a vector in $\R^n$. Let the partial derivative of $f$ with respect to $u_k$ exist for all $u_k$. The '''gradient of $f$''' (at $\mathbf u$) is defined as: {{begin-eqn}} {{eqn | l = \grad f       | o = :=       | r = \nabla f }} {{eqn | r = \paren {\sum_{k \mathop = 1}^n \mathbf e_k \dfrac \partial {\partial x_k} } \map f {\mathbf u}       | c = {{Defof|Del Operator}} }} {{eqn | r = \sum_{k \mathop = 1}^n \dfrac {\map {\partial f} {\mathbf u} } {\partial x_k} \mathbf e_k       | c =  }} {{end-eqn}}
Definition:Directional Derivative
Let: :$f: \R^n \to \R, \mathbf x \mapsto \map f {\mathbf x}$  be a real-valued function such that the gradient: :$\nabla \map f {\mathbf x}$ exists. Let: :$\mathbf u$ be a unit vector in $\R^n$. The '''directional derivative of $f$ in the direction of $\mathbf u$''' is defined as: {{begin-eqn}} {{eqn | l = D_{\mathbf u} \map f {\mathbf x}       | r = \nabla \map f {\mathbf x} \cdot \mathbf u }} {{end-eqn}} where $\cdot$ denotes the dot product. {{expand|Geometric Interpretation}}
Definition:Pushforward Measure
Let $\struct {X, \Sigma}$ and $\struct {X', \Sigma'}$ be measurable spaces. Let $\mu$ be a measure on $\struct {X, \Sigma}$. Let $f: X \to X'$ be a $\Sigma \, / \, \Sigma'$-measurable mapping. Then the '''pushforward of $\mu$ under $f$''' is the mapping $f_* \mu: \Sigma' \to \overline \R$ defined by: :$\forall E' \in \Sigma': \map {f_* \mu} {E'} := \map \mu {\map {f^{-1} } {E'} }$ where $\overline \R$ denotes the extended real numbers.
Definition:Orthogonal Group
Let $k$ be a field. The '''($n$th) orthogonal group (on $k$)''', denoted $\map {\mathrm O} {n, k}$, is the following subset of the general linear group $\GL {n, k}$: :$\map {\mathrm O} {n, k} := \set {M \in \GL {n, k}: M^\intercal = M^{-1} }$ where $M^\intercal$ denotes the transpose of $M$.
Definition:Stieltjes Function
Let $f: \R \to \overline \R$ be a real function, where $\overline \R$ denotes the extended real numbers. Then $f$ is said to be a '''Stieltjes function''' : :$(1): \quad f$ is increasing :$(2): \quad f$ is left-continuous.
Definition:Stieltjes Function of Measure on Real Numbers
Let $\mu$ be a measure on $\R$ with the Borel $\sigma$-algebra $\mathcal B \left({\R}\right)$. The '''Stieltjes function of $\mu$''' is the mapping $F_\mu: \R \to \overline \R$ defined by: :$F_\mu \left({x}\right) := \begin{cases} \mu \left({ \left[{0 \,.\,.\, x}\right) \, }\right) & \text{if } x > 0\\ 0 & \text{if } x = 0\\ - \mu \left({ \left[{x \,.\,.\, 0}\right) \, }\right) & \text{if } x < 0 \end{cases}$ where $\overline \R$ denotes the extended real numbers.
Definition:Pre-Measure of Finite Stieltjes Function
Let $\mathcal J_{ho}$ denote the collection of half-open intervals in $\R$. Let $f: \R \to \R$ be a finite Stieltjes function. The '''pre-measure of $f$''' is the mapping $\mu_f: \mathcal J_{ho} \to \overline \R_{\ge 0}$ defined by: :$\mu_f \left({ \left[{a \,.\,.\, b}\right) \, }\right) := \begin{cases} f \left({b}\right) - f\left({a}\right) & \text{if } b \ge a \\ 0 & \text{otherwise} \end{cases}$ where $\overline \R_{\ge 0}$ denotes the set of positive extended real numbers.
Definition:Measure of Finite Stieltjes Function
Let $f: \R \to \R$ be a finite Stieltjes function. Let $\mu_f$ be the pre-measure of $f$. Let $\mu$ be the unique measure extending $\mu_f$ provided on Pre-Measure of Finite Stieltjes Function Extends to Unique Measure. Then $\mu$ is called the '''measure of $f$'''. This definition makes $\mu$ a measure on $\mathcal B \left({\R}\right)$, the Borel $\sigma$-algebra of $\R$.
Definition:Extended Real-Valued Function
Let $S$ be a set, and let $\overline \R$ denote the extended real numbers. A mapping $f: S \to \overline \R$ is said to be an '''extended real-valued function'''.
Definition:Extended Real Sigma-Algebra
Let $\left({\overline{\R}, \tau}\right)$ be the extended real number space. The '''extended real $\sigma$-algebra''' $\overline{\mathcal B}$ is the Borel $\sigma$-algebra $\mathcal B \left({\overline{\R}, \tau}\right)$.
Definition:Space of Measurable Functions
Let $\left({X, \Sigma}\right)$ be a measurable space. Then the '''space of $\Sigma$-measurable, real-valued functions''' $\mathcal M \left({\Sigma}\right)$ is the collection of all $\Sigma$-measurable, real-valued functions: :$\mathcal M \left({\Sigma}\right) := \left\{{f: X \to \R: f \text{ is $\Sigma$-measurable}}\right\}$ Similarly, the '''space of $\Sigma$-measurable, extended real-valued functions''' $\mathcal{M}_{\overline{\R}} \left({\Sigma}\right)$ is the collection of all $\Sigma$-measurable, extended real-valued functions: :$\mathcal{M}_{\overline{\R}} \left({\Sigma}\right) := \left\{{f: X \to \overline{\R}: f \text{ is $\Sigma$-measurable}}\right\}$ === Space of Positive Measurable Functions === The '''space of $\Sigma$-measurable, positive real-valued functions''' $\mathcal{M}^+ \left({\Sigma}\right)$ is the subset of positive $\Sigma$-measurable functions in $\mathcal M \left({\Sigma}\right)$: :$\mathcal{M}^+ \left({\Sigma}\right) := \left\{{f: X \to \R: f \text{ is positive $\Sigma$-measurable}}\right\}$ Analogously, the '''space of $\Sigma$-measurable, positive extended real-valued functions''' $\mathcal{M}_{\overline{\R}}^+ \left({\Sigma}\right)$ is defined as: :$\mathcal{M}_{\overline{\R}}^+ \left({\Sigma}\right) := \left\{{f: X \to \overline{\R}: f \text{ is positive $\Sigma$-measurable}}\right\}$
Definition:Standard Representation of Simple Function
Let $\left({X, \Sigma}\right)$ be a measurable space. Let $f: X \to \R$ be a simple function. A '''standard representation of $f$''' consists of: :a finite sequence $a_1, \ldots, a_n$ of real numbers :a partition $E_0, E_1, \ldots, E_n$ of $\Sigma$-measurable sets subject to: :$f = \displaystyle \sum_{j \mathop = 0}^n a_j \chi_{E_j}$ where $a_0 := 0$, and $\chi$ denotes characteristic function.
Definition:Space of Simple Functions
Let $\left({X, \Sigma}\right)$ be a measurable space. Then the '''space of simple functions on $\left({X, \Sigma}\right)$''', denoted $\mathcal E \left({\Sigma}\right)$, is the collection of all simple functions $f: X \to \R$: :$\mathcal E \left({\Sigma}\right) := \left\{{f: X \to \R: f \text{ is a simple function}}\right\}$ === Space of Positive Simple Functions === The '''space of positive simple functions on $\left({X, \Sigma}\right)$''', denoted $\mathcal{E}^+ \left({\Sigma}\right)$ is the subset of positive simple functions in $\mathcal E \left({\Sigma}\right)$: :$\mathcal{E}^+ \left({\Sigma}\right) := \left\{{f: X \to \R: f \text{ is a positive simple function}}\right\}$
Definition:Positive Part
Let $X$ be a set. Let $f: X \to \overline \R$ be an extended real-valued function. Then the '''positive part of $f$''', $f^+: X \to \overline \R$, is the extended real-valued function defined by: :$\forall x \in X: \map {f^+} x := \max \set {0, \map f x}$ where the maximum is taken with respect to the extended real ordering. That is: :$\forall x \in X: \map {f^+} x := \begin {cases} \map f x & : \map f x \ge 0 \\ 0 & : \map f x < 0 \end {cases}$
Definition:Negative Part
Let $X$ be a set, and let $f: X \to \overline \R$ be an extended real-valued function. Then the '''negative part of $f$''', $f^-: X \to \overline \R$, is the extended real-valued function defined by: :$\forall x \in X: \map {f^-} x := -\min \set {0, \map f x}$ where the minimum is taken with respect to the extended real ordering. That is: :$\forall x \in X: \map {f^-} x := \begin {cases} \map f x & : \map f x \le 0 \\ 0 & : \map f x > 0 \end {cases}$
Definition:Dedekind-Infinite
A set is said to be '''Dedekind-infinite'''  it is equivalent to (at least) one of its proper subsets.
Definition:Continuous Real Function/Left-Continuous/Point
Let $x_0 \in A$.  Then $f$ is said to be '''left-continuous at $x_0$'''  the limit from the left of $\map f x$ as $x \to x_0$ exists and: :$\displaystyle \lim_{\substack {x \mathop \to x_0^- \\ x_0 \mathop \in A} } \map f x = \map f {x_0}$ where $\displaystyle \lim_{x \mathop \to x_0^-}$ is a limit from the left.
Definition:Continuous Real Function/Right-Continuous/Point
Let $x_0 \in S$.  Then $f$ is said to be '''right-continuous at $x_0$'''  the limit from the right of $f \left({x}\right)$ as $x \to x_0$ exists and: :$\displaystyle \lim_{\substack{x \mathop \to x_0^+ \\ x_0 \mathop \in A}} f \left({x}\right) = f \left({x_0}\right)$ where $\displaystyle \lim_{x \mathop \to x_0^+}$ is a limit from the right.
Definition:Pointwise Operation/Real-Valued Functions
Let $\R^S$ be the set of all mappings $f: S \to \R$, where $\R$ is the set of real numbers. Let $\oplus$ be a binary operation on $\R$. Define $\oplus: \R^S \times \R^S \to \R^S$, called '''pointwise $\oplus$''', by: :$\forall f, g \in \R^S: \forall s \in S: \map {\paren {f \oplus g} } s := \map f s \oplus \map g s$ In the above expression, the operator on the  is the given $\oplus$ on the real numbers.
Definition:Pointwise Scalar Multiplication of Mappings/Real-Valued Functions
Let $S$ be a non-empty set, and let $\R^S$ be the set of all mappings $f: S \to \R$. Then '''pointwise ($\R$-)scalar multiplication''' on $\R^S$ is the binary operation $\cdot: \R \times \R^S \to \R^S$ defined by: :$\forall \lambda \in \R: \forall f \in \R^S: \forall s \in S: \map {\paren {\lambda \cdot f} } s := \lambda \cdot \map f s$ where the $\cdot$ on the right is real multiplication.
Definition:Pointwise Addition of Real-Valued Functions
Let $f, g: S \to \R$ be real-valued functions. Then the '''pointwise sum of $f$ and $g$''' is defined as: :$f + g: S \to \R:$ ::$\forall s \in S: \map {\paren {f + g} } s := \map f s + \map g s$ where the $+$ on the  is real-number addition.
Definition:Pointwise Multiplication
The (binary) operation of '''pointwise multiplication''' is defined on $\mathbb F^S$ as: :$\times: \mathbb F^S \times \mathbb F^S \to \mathbb F^S: \forall f, g \in \mathbb F^S:$ ::$\forall s \in S: \map {\paren {f \times g} } s := \map f s \times \map g s$ where the $\times$ on the  is conventional arithmetic multiplication.
Definition:Pointwise Limit
Let $S$ be a set. Let $\sequence {f_n}_{n \mathop \in \N}$, $f_n: S \to \R$ be a sequence of real-valued functions. Suppose that for all $s \in S$, the limit: :$\displaystyle \lim_{n \mathop \to \infty} \map {f_n} s$ exists. {{definition wanted|Incorporate infinite limits}} Then the '''pointwise limit of $\sequence {f_n}_{n \mathop \in \N}$''', denoted $\displaystyle \lim_{n \mathop \to \infty} f_n: S \to \R$, is defined as: :$\forall s \in S: \displaystyle \map {\paren {\lim_{n \mathop \to \infty} f_n} } s := \lim_{n \mathop \to \infty} \map {f_n} s$ '''Pointwise limit''' thence is an instance of a pointwise operation on real-valued functions. {{definition wanted|of course, makes sense in arbitrary metric/topological space. cover this}}
Definition:Absolute Value of Mapping/Real-Valued Function
Let $S$ be a set. Let $f: S \to \R$ be a real-valued function. Then the '''absolute value of $f$''', denoted $\size f: S \to \R$, is defined as: :$\forall s \in S: \map {\size f} s := \size {\map f s}$ where $\size {\map f s}$ denotes the absolute value function on $\R$. '''Absolute value''' thence is an instance of a pointwise operation on real-valued functions.
Definition:Extended Absolute Value
Let $\overline \R$ denote the extended real numbers. Extend the absolute value $\left\vert{\cdot}\right\vert$ on $\R$ to $\overline \R = \R \cup \left\{{+\infty, -\infty}\right\}$ by defining: :$\left\vert{-\infty}\right\vert = \left\vert{+\infty}\right\vert = +\infty$ Thus, the '''extended absolute value''' is a mapping $\left\vert{\cdot}\right\vert: \overline \R \to \overline \R$.
Definition:Absolute Value of Mapping/Extended Real-Valued Function
Let $S$ be a set, and let $f: S \to \overline \R$ be an extended real-valued function. Then the '''absolute value of $f$''', denoted $\size f: S \to \overline \R$, is defined as: :$\forall s \in S: \map {\size f} s := \size {\map f s}$ where $\size {\map f s}$ denotes the extended absolute value function on $\overline \R$. '''Absolute value''' thence is an instance of a pointwise operation on extended real-valued functions. Since extended absolute value coincides on $\R$ with the standard ordering, this definition incorporates the definition for real-valued functions.
Definition:Absolute Value of Mapping
Let $D$ be an ordered integral domain. Let $\size {\, \cdot \,}_D$ denote the absolute value function on $D$. Let $S$ be a set. Let $f: S \to D$ be a mapping. Then the '''absolute value of $f$''', denoted $\size f_D: S \to D$, is defined as: :$\forall s \in S: \map {\size f_D} s := \size {\map f s}_D$ '''Absolute value''' thence is an instance of a pointwise operation on a mapping.
Definition:Pointwise Operation on Extended Real-Valued Functions
Let $S$ be a set, and let $f, g : S \to \overline \R$ be extended real-valued functions. Let $\lambda \in \R$. Then extended real-valued functions can be formed by defining (for all $s \in S$): :$\lambda \cdot f: S \to \R, \map {\paren {\lambda \cdot f} } s := \lambda \cdot \map f s$ :$f + g: S \to \R, \map {\paren {f + g} } s := \map f s + \map g s$ :$f \cdot g: S \to \R, \map {\paren {f \cdot g} } s := \map f s \cdot \map g s$ as is done on Pointwise Scalar Multiplication, Pointwise Addition and Pointwise Multiplication, respectively. More generally, let $\oplus$ be a binary operation on $\overline \R$. Define $f \oplus g: S \to \overline \R$, called '''pointwise $\oplus$''', by: :$\map {\paren {f \oplus g} } s := \map f s \oplus \map g s$ In above expressions, the subscript ${\overline \R}$ of an operator expresses that its operands are extended real numbers. Next, let $\family {f_i}_{i \mathop \in I}, f_i: S \to \overline \R$, be any $I$-indexed collection of extended real-valued functions, where $I$ is some index set. Suppose that $\oplus^I$ is an $I$-ary operation on $\overline \R$. Then define $\oplus^I \family {f_i}_{i \mathop \in I}: S \to \overline \R$, called '''pointwise $\oplus^I$''', by: :$\map {\oplus^I \family {f_i}_{i \mathop \in I} } s := \oplus^I \family {\map {f_i} s}_{i \mathop \in I}$
Definition:Ordered Subsemigroup
Let $\left({S, \circ, \preceq}\right)$ be an ordered structure. Let $T \subseteq S$ be a subset of $S$ such that: :$\left({T, \circ_T, \preceq_T}\right)$ is an ordered semigroup where: :$\circ_T$ is the operation induced on $H$ by $\circ$ :$\preceq_t$ is the restriction of $\preceq$ to $T \times T$. Then $\left({T, \circ_T, \preceq_T}\right)$ is an '''ordered subsemigroup''' of $\left({S, \circ, \preceq}\right)$.
Definition:Ordered Subgroup
An '''ordered subgroup''' $\struct {T, \circ, \preceq}$ of an ordered structure $\struct {S, \circ, \preceq}$ is an ordered group such that the group $\struct {T, \circ}$ is a subgroup of $\struct {S, \circ}$.
Definition:Totally Ordered Structure
When the ordering $\preceq$ is a total ordering, the structure $\left({S, \circ, \preceq}\right)$ is then a '''totally ordered structure'''.
Definition:Integral of Positive Simple Function
Let $\left({X, \Sigma, \mu}\right)$ be a measure space. Let $\mathcal E^+$ denote the space of positive simple functions Let $f: X \to \R, f \in \mathcal E^+$ be a positive simple function. Suppose that $f$ admits the following standard representation: :$\displaystyle f = \sum_{i \mathop = 0}^n a_i \chi_{E_i}$ where $a_0 := 0$, and $\chi$ denotes characteristic function. Then the '''$\mu$-integral of $f$''', denoted $I_\mu \left({f}\right)$, is defined by: :$I_\mu \left({f}\right) := \displaystyle \sum_{i \mathop = 0}^n a_i \mu \left({E_i}\right)$
Definition:Ordered Semigroup Isomorphism
Let $\left({S, \circ, \preceq}\right)$ and $\left({T, *, \preccurlyeq}\right)$ be ordered semigroups. An '''ordered semigroup isomorphism''' from $\left({S, \circ, \preceq}\right)$ to $\left({T, *, \preccurlyeq}\right)$ is a mapping $\phi: S \to T$ that is both: :$(1): \quad$ A semigroup isomorphism from the semigroup $\left({S, \circ}\right)$ to the semigroup $\left({T, *}\right)$ :$(2): \quad$ An order isomorphism from the ordered set $\left({S, \preceq}\right)$ to the ordered set $\left({T, \preccurlyeq}\right)$.
Definition:Isomorphism (Abstract Algebra)/Semigroup Isomorphism
Let $\struct {S, \circ}$ and $\struct {T, *}$ be semigroups. Let $\phi: S \to T$ be a (semigroup) homomorphism. Then $\phi$ is a semigroup isomorphism  $\phi$ is a bijection.
Definition:Semigroup Homomorphism
Let $\left({S, \circ}\right)$ and $\left({T, *}\right)$ be semigroups. Let $\phi: S \to T$ be a mapping such that $\circ$ has the morphism property under $\phi$. That is, $\forall a, b \in S$: :$\phi \left({a \circ b}\right) = \phi \left({a}\right) * \phi \left({b}\right)$ Then $\phi: \left({S, \circ}\right) \to \left({T, *}\right)$ is a semigroup homomorphism.
Definition:Semigroup Epimorphism
Let $\left({S, \circ}\right)$ and $\left({T, *}\right)$ be semigroups. Let $\phi: S \to T$ be a (semigroup) homomorphism. Then $\phi$ is a semigroup epimorphism  $\phi$ is a surjection.
Definition:Semigroup Monomorphism
Let $\struct {S, \circ}$ and $\struct {T, *}$ be semigroups. Let $\phi: S \to T$ be a (semigroup) homomorphism. Then $\phi$ is a semigroup monomorphism  $\phi$ is an injection.
Definition:Integral of Positive Measurable Function
Let $\struct {X, \Sigma, \mu}$ be a measure space. Define the '''$\mu$-integral of positive measurable functions''', denoted $\displaystyle \int \cdot \rd \mu: \MM_{\overline \R}^+ \to \overline \R_{\ge 0}$, as: :$\forall f \in \MM_{\overline \R}^+: \displaystyle \int f \rd \mu := \sup \set {\map {I_\mu} g: g \le f, g \in \EE^+}$ where: :$\MM_{\overline \R}^+$ denotes the space of positive $\Sigma$-measurable functions :$\overline \R_{\ge 0}$ denotes the positive extended real numbers :$\sup$ is a supremum in the extended real ordering :$\map {I_\mu} g$ denotes the $\mu$-integral of the positive simple function $g$ :$g \le f$ denotes pointwise inequality :$\EE^+$ denotes the space of positive simple functions === Integral Sign === {{:Definition:Integral Sign}}
Definition:Semigroup Endomorphism
Let $\left({S, \circ}\right)$ be a semigroups. Let $\phi: S \to S$ be a (semigroup) homomorphism from $S$ to itself. Then $\phi$ is a semigroup endomorphism.
Definition:Semigroup Automorphism
Let $\left({S, \circ}\right)$ be a semigroup. Let $\phi: S \to S$ be a (semigroup) isomorphism from $S$ to itself. Then $\phi$ is a semigroup automorphism.
Definition:Ordered Semigroup Automorphism
Let $\left({S, \circ, \preceq}\right)$ be an ordered semigroup. An '''ordered semigroup automorphism''' from $\left({S, \circ, \preceq}\right)$ to itself is a mapping $\phi: S \to S$ that is both: :$(1): \quad$ A semigroup automorphism, that is, a semigroup isomorphism from the semigroup $\left({S, \circ}\right)$ to itself :$(2): \quad$ An order isomorphism from the ordered set $\left({S, \preceq}\right)$ to itself.
Definition:Totally Ordered Semigroup
A '''totally ordered semigroup''' is a totally ordered structure $\left({S, \circ, \preceq}\right)$ such that $\left({S, \circ}\right)$ is a semigroup.
Definition:Ordered Semigroup Monomorphism
Let $\left({S, \circ, \preceq}\right)$ and $\left({T, *, \preccurlyeq}\right)$ be ordered semigroups. An '''ordered semigroup monomorphism''' from $\left({S, \circ, \preceq}\right)$ to $\left({T, *, \preccurlyeq}\right)$ is a mapping $\phi: S \to T$ that is both: :$(1): \quad$ A (semigroup) monomorphism from the semigroup $\left({S, \circ}\right)$ to the semigroup $\left({T, *}\right)$ :$(2): \quad$ An order embedding from the ordered set $\left({S, \preceq}\right)$ to the ordered set $\left({T, \preccurlyeq}\right)$.
Definition:Zero (Number)/Naturally Ordered Semigroup
Let $\left({S, \circ, \preceq}\right)$ be a naturally ordered semigroup. Then from axiom $(NO 1)$, $\left({S, \circ, \preceq}\right)$ has a smallest element. This smallest element of $\left({S, \circ, \preceq}\right)$ is called '''zero''' and has the symbol $0$. That is: :$\forall n \in S: 0 \preceq n$
Definition:Closed Interval/Integer Interval
The '''integer interval between $m$ and $n$''' is denoted and defined as: : $\closedint m n = \begin{cases} \set {x \in S: m \le x \le n} & : m \le n \\ \O & : n < m \end{cases}$ where $\O$ is the empty set.
Definition:Restriction of Ordering
Let $\left({S, \preceq}\right)$ be an ordered set. Let $T \subseteq S$ be a subset of $S$. Then the '''restriction of $\preceq$ to $T$''', denoted $\preceq \restriction_T$, is defined as: :${\preceq \restriction_T} := {\preceq} \cap \left({T \times T}\right)$ viewing ${\preceq} \subseteq S \times S$ as a relation on $S$. Here, $\times$ denotes Cartesian product. Thence the '''restriction of $\preceq$ to $T$''' is an instance of a restriction of a relation. {{MissingLinks|Undoubtedly there are similar things to this, which need to be put in 'Also see'}}
Definition:Multiplication/Natural Numbers
Let $\N$ be the natural numbers. '''Multiplication''' on $\N$ is the basic operation $\times$ everyone is familiar with. For example: :$3 \times 4 = 12$ :$13 \times 7 = 91$
Definition:Distributive Operation/Distributand
Let $\circ$ be distributive over $*$. Then $*$ is a '''distributand''' of $\circ$.
Definition:Distributive Operation/Distributor
Let $\circ$ be distributive over $*$. Then $\circ$ is a '''distributor''' of $*$.
Definition:Increasing Sequence of Mappings
Let $S$ be a set, and let $\left({T, \preceq}\right)$ be an ordered set. Let $\left({f_n}\right)_{n \mathop \in \N}, f_n: S \to T$ be a sequence of mappings. Then $\left({f_n}\right)_{n \mathop \in \N}$ is said to be an '''increasing sequence (of mappings)''' : :$\forall s \in S: \forall m, n \in \N: m \le n \implies f_m \left({s}\right) \preceq f_n \left({s}\right)$ That is,  $m \le n \implies f_m \preceq f_n$, where $\preceq$ denotes pointwise inequality.
Definition:Pointwise Inequality
Let $S$ be a set, and let $\struct {T, \preceq}$ be an ordered set. Let $f, g: S \to T$ be mappings. Then '''$f$ pointwise precedes $g$''', denoted $f \preceq g$, : :$\forall s \in S: \map f s \preceq \map g s$ Thence it can be seen that '''pointwise precedence''' is an instance of an induced relation on mappings. {{wtd|Not entirely happy with the naming}}
Definition:Pointwise Inequality of Real-Valued Functions
Let $S$ be a set. Let $f, g: S \to \R$ be real-valued functions. Then '''pointwise inequality of $f$ and $g$''', denoted $f \le g$, is defined to hold : :$\forall s \in S: f \left({s}\right) \le g \left({s}\right)$ where $\le$ denotes the usual ordering on the real numbers $\R$. Thence '''pointwise inequality''' of real-valued functions is an instance of an induced relation on mappings.