hcju/proofwikips huggingface dataset:
Definition:Smooth Path/Complex
Let $\left[{a \,.\,.\, b}\right]$ be a closed real interval. Let $\gamma: \left[{a \,.\,.\, b}\right] \to \C$ be a path in $\C$. That is, $\gamma$ is a continuous complex-valued function from $\left[{a \,.\,.\, b}\right]$ to $\C$. Define the real function $x : \left[{a \,.\,.\, b}\right] \to \R$ by: :$\forall t \in \left[{a \,.\,.\, b}\right]: x \left({t}\right) = \operatorname{Re} \left({\gamma \left({t}\right)}\right)$ Define the real function $y: \left[{a \,.\,.\, b}\right] \to \R$ by: :$\forall t \in \left[{a \,.\,.\, b}\right]: y \left({t}\right) = \operatorname{Im} \left({\gamma \left({t}\right)}\right)$ where: : $\operatorname{Re} \left({\gamma \left({t}\right)}\right)$ denotes the real part of the complex number $\gamma \left({t}\right)$ : $\operatorname{Im} \left({\gamma \left({t}\right)}\right)$ denotes the imaginary part of $\gamma \left({t}\right)$. Then $\gamma$ is a '''smooth path (in $\C$)''' : :$(1): \quad$ Both $x$ and $y$ are continuously differentiable :$(2): \quad$ For all $t \in \left[{a \,.\,.\, b}\right]$, either $x' \left({t}\right) \ne 0$ or $y' \left({t}\right) \ne 0$.
Definition:Contour/Complex Plane
Let $C_1, \ldots, C_n$ be directed smooth curves in the complex plane $\C$. For each $i \in \left\{ {1, \ldots, n}\right\}$, let $C_i$ be parameterized by the smooth path $\gamma_i: \left[{a_i \,.\,.\, b_i}\right] \to \C$. For each $i \in \left\{ {1, \ldots, n - 1}\right\}$, let the endpoint of $\gamma_i$ equal the start point of $\gamma_{i + 1}$: :$\gamma_i \left({b_i}\right) = \gamma_{i + 1} \left({a_{i + 1} }\right)$ Then the finite sequence $\left\langle{C_1, \ldots, C_n}\right\rangle$ is a '''contour'''.
Definition:Connected (Topology)/Set/Definition 1
$H$ is a '''connected set of $T$'''  it is not the union of any two non-empty separated sets of $T$.
Definition:Connected (Topology)/Set/Definition 2
$H$ is a '''connected set of $T$'''  it is not '''disconnected in $T$'''.
Definition:Connected (Topology)/Set/Definition 3
$H$ is a '''connected set of $T$''' : :the topological subspace $\struct {H, \tau_H}$ of $T$ is a connected topological space.
Definition:Contour/Closed/Complex Plane
$C$ is a '''closed contour'''  the start point of $C$ is equal to the end point of $C$: :$\gamma_1 \left({a_1}\right) = \gamma_n \left({b_n}\right)$
Definition:Complex Riemann Integral
Let $\mathbb I := \closedint a b$ be a closed real interval. Let $f: \mathbb I \to \C$ be a bounded complex function. Define the real function $x: \mathbb I \to \R$ by: :$\forall t \in \mathbb I: \map x t = \map \Re {\map f t}$ Define the real function $y: \mathbb I \to \R$ by: :$\forall t \in \mathbb I: \map y t = \map \Im {\map f t}$ Here: :$\map \Re {\map f t}$ denotes the real part of the complex number $\map f t$ :$\map \Im {\map f t}$ denotes the imaginary part of $\map f t$. Suppose that both $x$ and $y$ are Riemann integrable over $\mathbb I$. Then the '''complex (Riemann) integral of $f$ over $\mathbb I$''' is defined as: :$\displaystyle \int_a^b \map f t \rd t = \int_a^b \map \Re {\map f t} \rd t + i \int_a^b \map \Im {\map f t} \rd t$ $f$ is formally defined as '''(properly) complex integrable over $\mathbb I$ in the sense of Riemann''', or '''(properly) complex Riemann integrable over $\mathbb I$'''. More usually (and informally), we say: :'''$f$ is (Riemann) complex integrable over $\mathbb I$.'''
Definition:Path (Topology)/Initial Point
The '''initial point''' of $\gamma$ is $\map \gamma a$. That is, the path '''starts''' (or '''begins''') at $\map \gamma a$.
Definition:Compatible Atlases
Let $M$ be a topological space. Let $\mathscr F, \mathscr G$ be $d$-dimensional atlases of class $C^k$ on $M$. === Definition 1 === {{:Definition:Compatible Atlases/Definition 1}} === Definition 2 === {{:Definition:Compatible Atlases/Definition 2}}
Definition:Path (Topology)/Final Point
The '''final point''' of $\gamma$ is $\map \gamma b$. That is, the path '''ends''' (or '''finishes''') at $\map \gamma b$.
Definition:Coordinate System/Coordinate Function
Let $\left \langle {a_n} \right \rangle$ be a coordinate system of a unitary $R$-module $G$. For each $x \in G$ let $x_1, x_2, \ldots, x_n$ be the coordinates of $x$ relative to $\left \langle {a_n} \right \rangle$. Then for $i = 1, \ldots, n$ the mapping $f_i : G \to R$ defined by $f_i \left({x}\right) = x_i$ is called the '''$i$-th coordinate function on $G$ relative to $\left \langle {a_n} \right \rangle$'''.
Definition:Component (Topology)/Definition 1
From Connectedness of Points is Equivalence Relation, $\sim$ is an equivalence relation. From the Fundamental Theorem on Equivalence Relations, the points in $T$ can be partitioned into equivalence classes. These equivalence classes are called the '''(connected) components''' of $T$. If $x \in S$, then the '''component of $T$ containing $x$''' (that is, the set of points $y \in S$ with $x \sim y$) is denoted by $\map {\operatorname {Comp}_x} T$.
Definition:Component (Topology)/Definition 2
The '''component of $T$ containing $x$''' is defined as: : $\displaystyle \operatorname{Comp}_x \left({T}\right) = \bigcup \left\{{A \subseteq S: x \in A \land A}\right.$ is connected $\left.\right\}$
Definition:Component (Topology)/Definition 3
The '''component of $T$ containing $x$''' is defined as: : the maximal connected set of $T$ that contains $x$.
Definition:Closed Ball/Metric Space/Radius
The value $\epsilon$ is referred to as the '''radius''' of $\map { {B_\epsilon}^-} a$.
Definition:Continuous Mapping (Metric Space)/Point/Definition 1
'''$f$ is continuous at (the point) $a$ (with respect to the metrics $d_1$ and $d_2$)''' : :$\forall \epsilon \in \R_{>0}: \exists \delta \in \R_{>0}: \forall x \in A_1: \map {d_1} {x, a} < \delta \implies \map {d_2} {\map f x, \map f a} < \epsilon$ where $\R_{>0}$ denotes the set of all strictly positive real numbers.
Definition:Continuous Mapping (Metric Space)/Point/Definition 2
$f$ is '''continuous at (the point) $a$ (with respect to the metrics $d_1$ and $d_2$)''' : :$(1): \quad$ The limit of $f \left({x}\right)$ as $x \to a$ exists :$(2): \quad \displaystyle \lim_{x \to a} f \left({x}\right) = f \left({a}\right)$.
Definition:Continuous Mapping (Metric Space)/Point/Definition 3
$f$ is '''continuous at (the point) $a$ (with respect to the metrics $d_1$ and $d_2$)''' : :$\forall \epsilon \in \R_{>0}: \exists \delta \in \R_{>0}: f \left[{B_\delta \left({a; d_1}\right)}\right] \subseteq B_\epsilon \left({f \left({a}\right); d_2}\right)$ where $B_\epsilon \left({f \left({a}\right); d_2}\right)$ denotes the open $\epsilon$-ball of $f \left({a}\right)$ with respect to the metric $d_2$, and similarly for $B_\delta \left({a; d_1}\right)$.
Definition:Continuous Mapping (Metric Space)/Space/Definition 1
$f$ is '''continuous from $\left({A_1, d_1}\right)$ to $\left({A_2, d_2}\right)$'''  it is continuous at every point $x \in A_1$.
Definition:Limit of Function (Metric Space)/Epsilon-Delta Condition
:$\forall \epsilon \in \R_{>0}: \exists \delta \in \R_{>0}: 0 < d_1 \left({x, c}\right) < \delta \implies d_2 \left({f \left({x}\right), L}\right) < \epsilon$ That is, for every real positive $\epsilon$ there exists a real positive $\delta$ such that ''every'' point in the domain of $f$ within $\delta$ of $c$ has an image within $\epsilon$ of some point $L$ in the codomain of $f$.
Definition:Limit of Function (Metric Space)/Epsilon-Ball Condition
:$\forall \epsilon \in \R_{>0}: \exists \delta \in \R_{>0}: f \left({B_\delta \left({c; d_1}\right) \setminus \left\{{c}\right\}}\right) \subseteq B_\epsilon \left({L; d_2}\right)$. where: : $B_\delta \left({c; d_1}\right) \setminus \left\{{c}\right\}$ is the deleted $\delta $-neighborhood of $c$ in $M_1$ : $B_\epsilon \left({L; d_2}\right)$ is the open $\epsilon$-ball of $L$ in $M_2$. That is, for every open $\epsilon$-ball of $L$ in $M_2$, there exists a deleted $\delta $-neighborhood of $c$ in $M_1$ whose image is a subset of that open $\epsilon$-ball.
Definition:Totally Bounded Metric Space/Definition 1
A metric space $M = \struct {A, d}$ is '''totally bounded''' : :for every $\epsilon \in \R_{>0}$ there exists a finite $\epsilon$-net for $M$. That is, $M$ is '''totally bounded''' : :for every $\epsilon \in \R_{>0}$ there exists a finite set of points $x_1, \ldots, x_n \in A$ such that: ::$\displaystyle A = \bigcup_{i \mathop = 1}^n \map {B_\epsilon} {x_i}$ :where $\map {B_\epsilon} {x_i}$ denotes the open $\epsilon$-ball of $x_i$. That is: $M$ is '''totally bounded''' , given any $\epsilon \in \R_{>0}$, one can find a finite number of open $\epsilon$-balls which cover $A$.
Definition:Totally Bounded Metric Space/Definition 2
A metric space $M = \struct {A, d}$ is '''totally bounded''' : :for every $\epsilon \in \R_{>0}$ there exist finitely many points $x_0, \dots, x_n \in A$ such that: ::$\displaystyle \inf_{0 \mathop \le i \mathop \le n} \map d {x_i, x} \le \epsilon$ :for all $x \in A$.
Definition:Uniform Convergence/Metric Space
Let $S$ be a set. Let $M = \left({A, d}\right)$ be a metric space. Let $\left \langle {f_n} \right \rangle$ be a sequence of mappings $f_n: S \to A$. Let: : $\forall \epsilon \in \R_{>0}: \exists N \in \R: \forall n \ge N, \forall x \in S: d \left({f_n \left({x}\right), f \left({x}\right)}\right) < \epsilon$ Then '''$\left \langle {f_n} \right \rangle$ converges to $f$ uniformly on $S$ as $n \to \infty$'''.
Definition:Uniform Convergence/Real Numbers
Let $\sequence {f_n}$ be a sequence of real functions defined on $D \subseteq \R$. Let: :$\forall \epsilon \in \R_{>0}: \exists N \in \R: \forall n \ge N, \forall x \in D: \size {\map {f_n} x - \map f x} < \epsilon$ That is:  :$\displaystyle \forall \epsilon \in \R_{>0}: \exists N \in \R: \forall n \ge N: \sup_{x \mathop \in D} \size {\map {f_n} x - \map f x} < \epsilon$ Then '''$\sequence {f_n}$ converges to $f$ uniformly on $D$ as $n \to \infty$'''.
Definition:Real Function/Domain
Let $f: S \to \R$ be a real function. The domain of $f$ is the set $S$. It is frequently the case that $S$ is not explicitly specified. If this is so, then it is understood that the domain is to consist of ''all'' the values in $\R$ for which the function is defined. This often needs to be determined as a separate exercise in itself, by investigating the nature of the function in question.
Definition:Modulo Operation/Modulo Zero
:$\forall x \in \R: x \bmod 0 = x$ This can be considered as a '''special case''' of the modulo operation, but it is interesting to note that most of the results concerning the modulo operation still hold.
Definition:Modulo Operation/Modulo One
:$x \bmod 1 = x - \left \lfloor {x}\right \rfloor$ from which it follows directly that: :$x = \left \lfloor {x}\right \rfloor + \left({x \bmod 1}\right)$
Definition:Symmetric Difference/Definition 1
:$S * T := \paren {S \setminus T} \cup \paren {T \setminus S}$
Definition:Symmetric Difference/Definition 2
:$S * T = \paren {S \cup T} \setminus \paren {S \cap T}$
Definition:Symmetric Difference/Definition 3
: $S * T = \left({S \cap \overline T}\right) \cup \left({\overline S \cap T}\right)$
Definition:Symmetric Difference/Definition 4
:$S * T = \paren {S \cup T}\cap \paren {\overline S \cup \overline T}$
Definition:Standard Deviation
Let $X$ be a random variable. Then the '''standard deviation of $X$''', written $\sigma_X$ or $\sigma$, is defined as the principal square root of the variance of $X$: :$\sigma_X := \sqrt {\var X}$
Definition:Transfinite Ordinal
Let $\alpha$ be an ordinal. Then $\alpha$ is said to be '''transfinite'''  it is an infinite set.
Definition:Characteristic Function of Random Variable
Let $X$ be a random variable on a probability space $\struct {\Omega, \Sigma, \Pr}$. The '''characteristic function of $X$''' is the mapping $\phi: \R \to \C$ defined by: :$\map \phi t = \expect {e^{i t X} }$ where: :$i$ is the imaginary unit :$\expect \cdot$ denotes expectation.
Definition:Transfinite Sequence
Let $\alpha$ be an infinite ordinal. Let $\left({x_\beta}\right)_{\beta \mathop \in \alpha}$ be an $\alpha$-indexed family. Then $\left({x_\beta}\right)_{\beta \mathop \in \alpha}$ is called a '''transfinite sequence'''.
Definition:Nest
Let $S$ be a set. Let $\powerset S$ be its power set. Let $N \subseteq \powerset S$ be a subset of $\powerset S$. Then $N$ is a '''nest''' : :$\forall X, Y \in N: X \subseteq Y$ or $Y \subseteq X$ === Class Theory === {{:Definition:Nest/Class Theory}}
Definition:Ideal (Order Theory)
Let $\left({S, \preceq}\right)$ be an ordered set. Let $I \subseteq S$ be a non-empty subset of $S$. Then $I$ is an '''ideal''' of $S$  it is both a lower set and a directed set. That is, $I$ is an '''ideal''' : :$\forall x \in I, y \in P: y \preceq x \implies y \in I$ :$\forall x, y \in I: \exists z \in I: x \preceq z$ and $y \preceq z$
Definition:Characteristic Function (Set Theory)/Set/Support
Let $S$ be a set Let $E \subseteq S$ be a subset. Let $\chi_E: S \to \set {0, 1}$ be the characteristic function of $E$. The '''support''' of $\chi_E$, denoted $\operatorname{supp} \chi_E$, is the set $E$. That is: :$\operatorname{supp} \chi_E = \set {x \in S: \map {\chi_E} x = 1}$
Definition:Contour/Length/Complex Plane
Let $C$ be a '''contour''' in $C$ defined by the (finite) sequence $\left\langle{C_1, \ldots, C_n}\right\rangle$ of directed smooth curves in $\C$. Let $C_i$ be parameterized by the smooth path $\gamma_i: \left[{a_i\,.\,.\,b_i}\right] \to \C$ for all $i \in \left\{ {1, \ldots, n}\right\}$. The '''length''' of $C$ is defined as: :$\displaystyle L \left({C}\right) := \sum_{i \mathop = 1}^n \int_{a_i}^{b_i} \left\vert{\gamma_i' \left({t}\right) }\right\vert \rd t$
Definition:Two Ring
Let $\struct {\Z_2, +_2, \times_2}$ be the ring of integers modulo $2$. Then $\struct {\Z_2, +_2, \times_2}$ is referred to as the '''two ring'''.
Definition:Idempotent Ring
Let $\struct {R, +, \circ}$ be a ring. $R$ is an '''idempotent ring'''  its ring product $\circ$ is an idempotent operation.
Definition:Boolean Ring Axioms
{{begin-axiom}} {{axiom | n = \text A 0         | lc= Closure under addition         | q = \forall a, b \in R         | m = a * b \in R }} {{axiom | n = \text A 1         | q = \forall a, b, c \in R         | m = \paren {a * b} * c = a * \paren {b * c}         | lc= Associativity of addition }} {{axiom | n = \text A 2         | q = \forall a, b \in R         | m = a * b = b * a         | lc= Commutativity of addition }} {{axiom | n = \text A 3         | q = \exists 0_R \in R: \forall a \in R         | m = a * 0_R = a = 0_R * a         | lc= Identity element for addition: the zero }} {{axiom | n = \text {AC} 2         | q = \forall a \in R         | m = a * a = 0_R         | lc= Characteristic 2 for addition: }} {{axiom | n = \text M 0         | q = \forall a, b \in R         | m = a \circ b \in R         | lc= Closure under product }} {{axiom | n = \text M 1         | q = \forall a, b, c \in R         | m = \paren {a \circ b} \circ c = a \circ \paren {b \circ c}         | lc= Associativity of product }} {{axiom | n = \text M 2         | q = \exists 1_R \in R: \forall a \in R         | m = 1_R \circ a = a = a \circ 1_R         | lc= Identity element for product: the unity }} {{axiom | n = \text {MI}         | q = \forall a \in R         | m = a \circ a = a         | lc= Idempotence of product }} {{axiom | n = \text D         | q = \forall a, b, c \in R         | m = a \circ \paren {b * c} = \paren {a \circ b} * \paren {a \circ c},         | lc= Product is distributive over addition }} {{axiom | m = \paren {a * b} \circ c = \paren {a \circ c} * \paren {b \circ c} }} {{end-axiom}} These criteria are called the '''Boolean ring axioms'''.
Definition:Two-Valued Function
Let $2$ be the two ring. Let $f: X \to 2$ be a mapping with codomain $2$. Then $f$ is called a '''$2$-valued function'''.
Definition:Ring of Idempotents
Let $\left({R, +, \circ}\right)$ be a commutative ring. Let $A$ be the set of all idempotent elements of $R$ under $\circ$: :$A = \left\{{x \in R: x \circ x = x}\right\}$ Define a binary operation $\circ$ on $A$ by: :$\forall x, y \in A: x \oplus y := x + y - 2 x \circ y$ Denote also with $\circ$ the restriction of $\circ$ to $A$. The algebraic structure $\left({A, \oplus, \circ}\right)$ is called the '''ring of idempotents''' of $R$. It is an idempotent ring, as shown on Ring of Idempotents is Idempotent Ring.
Definition:Algebraic Element of Ring Extension
Let $\struct {R, +, \circ}$ be a commutative ring with unity whose zero is $0_R$ and whose unity is $1_R$. Let $\struct {D, +, \circ}$ be an integral subdomain of $R$. Let $x \in R$. Then $x$ is '''algebraic over $D$''' : :$\exists \map f x$ over $D$ such that $\map f x = 0_R$ where $\map f x$ is a non-null polynomial in $x$ over $D$.
Definition:Algebraic Element of Field Extension
Let $E / F$ be a field extension. Let $\alpha \in E$. === Definition 1 === {{:Definition:Algebraic Element of Field Extension/Definition 1}} === Definition 2 === {{:Definition:Algebraic Element of Field Extension/Definition 2}}
Definition:Algebraic Field Extension
A field extension $E / F$ is said to be '''algebraic''' : : $\forall \alpha \in E: \alpha$ is algebraic over $F$
Definition:Prime Ideal (Order Theory)
Let $I$ be an ideal in an ordered set $S$. Then $I$ is a '''prime ideal''' in $S$  $S \setminus I$ is a filter.
Definition:Distributive Lattice
Let $\left({S, \vee, \wedge, \preceq}\right)$ be a lattice. Then $\left({S, \vee, \wedge, \preceq}\right)$ is '''distributive'''  one (hence all) of the following equivalent statements holds: {{begin-axiom}} {{axiom | n = 1         | q = \forall x, y, z \in S         | m = x \wedge \left({y \vee z}\right) = \left({x \wedge y}\right) \vee \left({x \wedge z}\right) }} {{axiom | n = 1'         | q = \forall x, y, z \in S         | m = \left({x \vee y}\right) \wedge z = \left({x \wedge z}\right) \vee \left({y \wedge z}\right) }} {{axiom | n = 2         | q = \forall x, y, z \in S         | m = x \vee \left({y \wedge z}\right) = \left({x \vee y}\right) \wedge \left({x \vee z}\right) }} {{axiom | n = 2'         | q = \forall x, y, z \in S         | m = \left({x \wedge y}\right) \vee z = \left({x \vee z}\right) \wedge \left({y \vee z}\right) }} {{end-axiom}}
Definition:Stone Space
Let $B$ be a Boolean algebra. The '''Stone space of $B$''' is the topological space: :$\map S B = \struct {U, \tau}$ where: :$U$ is the set of ultrafilters in $B$ :$\tau$ is the topology generated by the basis consisting of all sets of the form: :$\set {x \in \map S B: b \in x}$ :for some $b \in B$.
Definition:Contour Integral/Complex
Let $C$ be a contour defined by a finite sequence $C_1, \ldots, C_n$ of directed smooth curves. Let $C_i$ be parameterized by the smooth path $\gamma_i: \closedint {a_i} {b_i} \to \C$ for all $i \in \set {1, \ldots, n}$. Let $f: \Img C \to \C$ be a continuous complex function, where $\Img C$ denotes the image of $C$. The '''contour integral of $f$ along $C$''' is defined by: :$\displaystyle \int_C \map f z \rd z = \sum_{i \mathop = 1}^n \int_{a_i}^{b_i} \map f {\map {\gamma_i} t} \map {\gamma_i'} t \rd t$ From Contour Integral is Well-Defined, it follows that the complex Riemann integral on the right side is defined and is independent of the parameterizations of $C_1, \ldots, C_n$. === Contour Integral along Closed Contour === {{:Definition:Contour Integral/Complex/Closed}}
Definition:Smooth Path/Closed/Complex Plane
Let $\gamma$ be a smooth path in $\C$. $\gamma$ is a '''closed smooth path'''  $\gamma$ is a closed path. That is,  $\gamma \left({a}\right) = \gamma \left({b}\right)$.
Definition:Ordered Sum/General Definition
Let $S_1, S_2, \ldots, S_n$ be tosets. Then we define $T_n$ as the ordered sum of $S_1, S_2, \ldots, S_n$ as: :$\forall n \in \N_{>0}: T_n = \begin{cases} S_1 & : n = 1 \\ T_{n-1} + S_n & : n > 1 \end{cases}$
Definition:Moore-Smith Sequence
Let $\left({S, \preceq}\right)$ be a directed set. Let $T$ be a set. Let $f: S \to T$ be a mapping. Then $f$ is a '''Moore-Smith sequence''' in $T$.
Definition:Ordered Square
Let: : $S = \left[{0 \,.\,.\, 1}\right] \times \left[{0 \,.\,.\, 1}\right]$ where $\left[{0 \,.\,.\, 1}\right]$ is the closed real interval from $0$ to $1$. Let $\preceq$ be the lexicographic ordering on $S$ induced by the usual ordering of $\left[{0 \,.\,.\, 1}\right]$. Let $\tau$ be the order topology on $S$ induced by $\preceq$. Then $\left({S, \tau}\right)$ is the '''ordered square'''.
Definition:Cartesian Product of Relations
Let $\left\langle{S_i}\right\rangle_{i \mathop \in I}$ and $\left\langle{T_i}\right\rangle_{i \mathop \in I}$ be families of sets indexed by $I$. For each $i \in I$, let $\mathcal R_i \subseteq S_i \times T_i$ be a relation from $S_i$ to $T_i$. Let $S$ and $T$ be the Cartesian product of $\left\langle{S_i}\right\rangle_{i \mathop \in I}$ and $\left\langle{T_i}\right\rangle_{i \mathop \in I}$ respectively: :$\displaystyle S = \prod_{i \mathop \in I} S_i$ :$\displaystyle T = \prod_{i \mathop \in I} T_i$ Then the '''product''' of the relations $\mathcal R_i$ is defined as the relation $\mathcal R \subseteq S \times T$ such that: :$x \mathrel{\mathcal R} y \iff \forall i \in I: x_i \mathrel{\mathcal R_i} y_i$
Definition:Time-Constructible Function
=== Definition 1 === {{:Definition:Time-Constructible Function/Definition 1}} === Definition 2 === {{:Definition:Time-Constructible Function/Definition 2}}
Definition:Boolean Algebra
=== Definition 1 === {{:Definition:Boolean Algebra/Definition 1}} === Definition 2 === {{:Definition:Boolean Algebra/Definition 2}} === Definition 3 === {{:Definition:Boolean Algebra/Definition 3}} The operations $\vee$ and $\wedge$ are called '''join''' and '''meet''', respectively. The identities $\bot$ and $\top$ are called '''bottom''' and '''top''', respectively. Also, $\neg a$ is called the '''complement''' of $a$. The operation $\neg$ is called '''complementation'''.
Definition:Concatenation of Contours
Let $\R^n$ be a real cartesian space of $n$ dimensions. Let $C$ and $D$ be contours in $\R^n$. Thus: :$C$ is a (finite) sequence of directed smooth curves $\sequence {C_1, \ldots, C_n}$ :$D$ is a (finite) sequence of directed smooth curves $\sequence {D_1, \ldots, D_m}$. Let $C_i$ be parameterized by the smooth path: :$\gamma_i: \closedint {a_i} {b_i} \to \R^n$ for all $i \in \set {1, \ldots, n}$ Let $D_i$ be parameterized by the smooth path: :$\sigma_i: \closedint {c_i} {d_i} \to \R^n$ for all $i \in \set {1, \ldots, m}$ Let $\map {\gamma_n} {b_n} = \map {\sigma_1} {c_1}$. Then the '''concatenation of the contours''' $C$ and $D$, denoted $C \cup D$, is the contour defined by the (finite) sequence: :$\sequence {C_1, \ldots, C_n, D_1, \ldots, D_m}$
Definition:Probability Density Function
Let $\struct {\Omega, \Sigma, \Pr}$ be a probability space. Let $X: \Omega \to \R$ be a continuous random variable on $\struct {\Omega, \Sigma, \Pr}$. Let $\Omega_X = \Img X$, the image of $X$. Then the '''probability density function''' of $X$ is the mapping $f_X: \R \to \closedint 0 1$ defined as: :$\forall x \in \R: \map {f_X} x = \begin {cases} \displaystyle \lim_{\epsilon \mathop \to 0^+} \frac {\map \Pr {x - \frac \epsilon 2 \le X \le x + \frac \epsilon 2} } \epsilon & : x \in \Omega_X \\ 0 & : x \notin \Omega_X \end {cases}$
Definition:Finite State Machine
A '''finite state machine''' is an ordered tuple: : $F = \left({ S, A, I, \Sigma, T }\right)$ where: : $S$ is the (finite) set of states : $A \subseteq S$ is the set of ''accepting'' states : $I \in S$ is the ''initial state'' : $\Sigma$ is the ''alphabet'' of symbols that can be fed into the machine : $T : \left({ S \times \Sigma }\right) \rightarrow S$ is the ''transition function''. A finite state machine operates as follows: :$(1): \quad$ At the beginning, the ''current state'' $s$ of the finite state machine is $I$. :$(2): \quad$ One by one, the ''input'' (a sequence of symbols from $\Sigma$) is fed into the machine. :$(3): \quad$ After each input symbol $\sigma$, the current state $s$ is set to the result of $T\left({s, \sigma}\right)$. If, at the end of processing an input word $w$, $s \in A$, the finite state machine is said to '''accept''' $w$, otherwise to '''reject''' it. The set of words $w$ ''accepted'' by the machine $F$ is called the '''accepted language''' $L\left({F}\right)$.