hcju/proofwikips huggingface dataset:
Definition:Greatest Common Divisor/Also known as
The '''greatest common divisor''' is often seen abbreviated as '''GCD''', '''gcd''' or '''g.c.d.''' Some sources write $\gcd \set {a, b}$ as $\tuple {a, b}$, but this notation can cause confusion with ordered pairs. The notation $\map \gcd {a, b}$ is frequently seen, but the set notation, although a little more cumbersome, can be argued to be preferable. The '''greatest common divisor''' is also known as the '''highest common factor''', or '''greatest common factor'''. '''Highest common factor''' when it occurs, is usually abbreviated as '''HCF''', '''hcf''' or '''h.c.f.''' It is written $\hcf \set {a, b}$ or $\map \hcf {a, b}$. The archaic term '''greatest common measure''' can also be found, mainly in such as .
Definition:Ring of Sequences
Let $\struct {R, +, \circ}$ be a ring. Given the natural numbers $\N$, the '''ring of sequences over $R$''' is the ring of mappings $\struct {R^\N, +', \circ'}$ where: :$(1): \quad R^\N$ is the set of sequences in $R$ :$(2): \quad +'$ and $\circ'$ are the (pointwise) operations induced by $+$ and $\circ$.
Definition:Ring of Cauchy Sequences
Let $\struct {R, +, \circ, \norm {\, \cdot \,} }$ be a normed division ring. Let $\struct {R^\N, +', \circ'}$ be the ring of sequences over $R$. Let $\CC$ be the set of Cauchy sequences on $R$. The '''ring of Cauchy sequences over $R$''' is the subring $\struct {\CC, +', \circ'}$ of $R^\N$ with unity. The (pointwise) ring operations on the '''ring of Cauchy sequences over $R$''' are defined as: :$\forall \sequence {x_n}, \sequence {y_n} \in R^\N: \sequence {x_n} +' \sequence {y_n} = \sequence {x_n + y_n}$ :$\forall \sequence {x_n}, \sequence {y_n} \in R^\N: \sequence {x_n} \circ' \sequence {y_n} = \sequence {x_n \circ y_n}$ The zero of the '''ring of Cauchy sequences''' is the sequence $\tuple {0, 0, 0, \dots}$, where $0$ is the zero in $R$. The unity of the '''ring of Cauchy sequences''' is the sequence $\tuple {1, 1, 1, \dots}$, where $1$ is the unity of $R$. By Corollary to Cauchy Sequences form Ring with Unity, if $R$ is a valued field then the '''ring of Cauchy sequences over $R$''' is a commutative ring with unity.
Definition:Image/Also denoted as
The notation $\Img f$ to denote the image of a mapping $f$ is specific to . The usual notation is $\map {\mathrm {Im} } f$ or a variant, but this is too easily confused with $\map \Im z$, the imaginary part of a complex number. Hence the non-standard usage $\Img f$. Some sources use $f \sqbrk S$, where $S$ is the domain of $f$. Others just use $\map f S$, but that notation is deprecated on  so as not to confuse it with the notation for the image of an element.
Definition:Inverse Image Mapping/Relation/Definition 1
The '''inverse image mapping''' of $\mathcal R$ is the mapping $\mathcal R^\gets: \powerset T \to \powerset S$ that sends a subset $Y \subseteq T$ to its preimage $\map {\mathcal R^{-1} } Y$ under $\mathcal R$: :$\forall Y \in \powerset T: \map {\mathcal R^\gets} Y = \begin {cases} \set {s \in S: \exists t \in Y: \tuple {t, s} \in \mathcal R^{-1} } & : \Img {\mathcal R} \cap Y \ne \O \\ \O & : \Img {\mathcal R} \cap Y = \O \end {cases}$
Definition:Inverse Image Mapping/Relation/Definition 2
The '''inverse image mapping''' of $\mathcal R$ is the direct image mapping of the inverse $\mathcal R^{-1}$ of $\mathcal R$: :$\mathcal R^\gets = \paren {\mathcal R^{-1} }^\to: \powerset T \to \powerset S$ That is: :$\forall Y \in \powerset T: \map {\mathcal R^\gets} Y = \set {s \in S: \exists t \in Y: \tuple {t, s} \in \mathcal R^{-1} }$
Definition:Inverse Image Mapping/Mapping/Definition 1
The '''inverse image mapping''' of $f$ is the mapping $f^\gets: \powerset T \to \powerset S$ that sends a subset $Y \subseteq T$ to its preimage $f^{-1} \paren T$ under $f$: :$\forall Y \in \powerset T: \map {f^\gets} Y = \begin {cases} \set {s \in S: \exists t \in Y: \map f s = t} & : \Img f \cap Y \ne \O \\ \O & : \Img f \cap Y = \O \end {cases}$
Definition:Inverse Image Mapping/Mapping/Definition 2
The '''inverse image mapping''' of $f$ is the direct image mapping of the inverse $f^{-1}$ of $f$: :$f^\gets = \paren {f^{-1} }^\to: \powerset T \to \powerset S$: That is: :$\forall Y \in \powerset T: \map {f^\gets} Y = \set {s \in S: \exists t \in Y: \map f s = t}$
Definition:Mapping/Also known as
Words which are often used to mean the same thing as '''mapping''' include: :'''transformation''' (particularly in the context of self-maps) :'''operator''' or '''operation''' :'''function''' (usually in the context of numbers) :'''map''' (but this term is discouraged, as the term is also used by some writers for '''planar graph'''). Some sources introduce the concept with informal words such as '''rule''' or '''idea''' or '''mathematical notion'''. Sources which define a '''mapping (function)''' to be only a many-to-one relation refer to a '''mapping (function)''' as a '''total mapping (total function)'''. Some use the term '''single-valued relation'''. Sources which go into analysis of multifunctions often refer to a conventional '''mapping''' as: :a '''single-valued mapping''' or '''single-valued function''' :a '''many-to-one mapping''', '''many-to-one transformation''', or '''many-to-one correspondence''', and so on. The wording can vary, for example: '''many-one''' can be seen for '''many-to-one'''. A '''mapping $f$ from $S$ to $T$''' is also described as a '''mapping on $S$ into $T$'''.
Definition:Image (Set Theory)/Mapping/Element/Also known as
The image of an element $s$ under a mapping $f$ is also called the '''functional value''', or '''value''', of $f$ at $s$. The terminology: :'''$f$ maps $s$ to $\map f s$''' :'''$f$ assigns the value $\map f s$ to $s$''' :'''$f$ carries $s$ into $\map f s$ can be found. The modifier '''by $f$''' can also be used for '''under $f$'''. Thus, for example, the '''image of $s$ by $f$''' means the same as the '''image of $s$ under $f$'''. In the context of computability theory, the following terms are frequently found: If $\tuple {x, y} \in f$, then $y$ is often called the '''output''' of $f$ for '''input''' $x$, or simply, the '''output of $f$ at $x$'''.
Definition:Surjection/Also known as
The phrase '''$f$ is surjective''' is often used for '''$f$ is a surjection'''. Authors who prefer to limit the jargon of mathematics tend to use the term '''an onto mapping''' for '''a surjection''', and '''onto''' for '''surjective'''. A mapping which is '''not surjective''' is thence described as '''into'''. A surjection $f$ from $S$ to $T$ is sometimes denoted: :$f: S \twoheadrightarrow T$ to emphasize surjectivity. {{LatexFor|for = f: S \twoheadrightarrow T}}
Definition:Constructed Semantics/Instance 2
Let $\mathcal L_0$ be the language of propositional logic. The constructed semantics $\mathscr C_2$ for $\mathcal L_0$ is used for the following results: * Hilbert Proof System Instance 2 Independence Results: Independence of $(A1)$ === Structures === Define the structures of $\mathscr C_2$ as mappings $v$ by the Principle of Recursive Definition, as follows. Let $\mathcal P_0$ be the vocabulary of $\mathcal L_0$. Let a mapping $v: \mathcal P_0 \to \{ 0, 1, 2 \}$ be given. Next, regard the following as definitional abbreviations: {{begin-axiom}} {{axiom | n  = 1         | lc = Conjunction         | ml = \mathbf A \land \mathbf B         | mo = {{=}}_{\text{def} }         | mr = \neg \left({ \neg \mathbf A \lor \neg \mathbf B }\right) }} {{axiom | n  = 2         | lc = Conditional         | ml = \mathbf A \implies \mathbf B         | mo = {{=}}_{\text{def} }         | mr = \neg \mathbf A \lor \mathbf B }} {{axiom | n  = 3         | lc = Biconditional         | ml = \mathbf A \iff \mathbf B         | mo = {{=}}_{\text{def} }         | mr = \left({\mathbf A \implies \mathbf B}\right) \land \left({\mathbf B \implies \mathbf A}\right) }} {{end-axiom}} It only remains to define $v \left({ \neg \phi }\right)$ and $v \left({ \phi \lor \psi}\right)$ recursively, by: {{begin-eqn}} {{eqn | l = v \left({\neg \phi }\right)       | o = :=       | r = \begin{cases} 1 & : \text{if $v \left({\phi}\right) = 0$} \\ 0 & : \text{if $v \left({\phi}\right) = 1$} \\ 2 & : \text{if $v \left({\phi}\right) = 2$}\end{cases} }} {{eqn | l = v \left({ \phi \lor \psi }\right)       | o = :=       | r = \begin{array}{c{{!}}ccc} \phi \lor \psi & 0 & 1 & 2 \\ \hline 0 & 0 & 0 & 0 \\ 1 & 0 & 1 & 2 \\ 2 & 0 & 2 & 0 \\ \end{array} }} {{end-eqn}} === Validity === Define validity in $\mathscr C_2$ by declaring: :$\models_{\mathscr C_2} \phi$  $v \left({\phi}\right) = 0$
Definition:Constructed Semantics/Instance 3
Let $\mathcal L_0$ be the language of propositional logic. The constructed semantics $\mathscr C_3$ for $\mathcal L_0$ is used for the following results: * Hilbert Proof System Instance 2 Independence Results: Independence of $(A2)$ === Structures === Define the structures of $\mathscr C_3$ as mappings $v$ by the Principle of Recursive Definition, as follows. Let $\mathcal P_0$ be the vocabulary of $\mathcal L_0$. Let a mapping $v: \mathcal P_0 \to \{ 0, 1, 2 \}$ be given. Next, regard the following as definitional abbreviations: {{begin-axiom}} {{axiom | n  = 1         | lc = Conjunction         | ml = \mathbf A \land \mathbf B         | mo = {{=}}_{\text{def} }         | mr = \neg \left({ \neg \mathbf A \lor \neg \mathbf B }\right) }} {{axiom | n  = 2         | lc = Conditional         | ml = \mathbf A \implies \mathbf B         | mo = {{=}}_{\text{def} }         | mr = \neg \mathbf A \lor \mathbf B }} {{axiom | n  = 3         | lc = Biconditional         | ml = \mathbf A \iff \mathbf B         | mo = {{=}}_{\text{def} }         | mr = \left({\mathbf A \implies \mathbf B}\right) \land \left({\mathbf B \implies \mathbf A}\right) }} {{end-axiom}} It only remains to define $v \left({ \neg \phi }\right)$ and $v \left({ \phi \lor \psi}\right)$ recursively, by: {{begin-eqn}} {{eqn | l = v \left({\neg \phi }\right)       | o = :=       | r = \begin{cases} 2 & : \text{if $v \left({\phi}\right) = 0$} \\ 1 & : \text{if $v \left({\phi}\right) = 1$} \\ 0 & : \text{if $v \left({\phi}\right) = 2$}\end{cases} }} {{eqn | l = v \left({ \phi \lor \psi }\right)       | o = :=       | r = \begin{array}{c{{!}}ccc} \phi \lor \psi & 0 & 1 & 2 \\ \hline 0 & 0 & 0 & 0 \\ 1 & 0 & 1 & 2 \\ 2 & 0 & 2 & 2 \\ \end{array} }} {{end-eqn}} === Validity === Define validity in $\mathscr C_3$ by declaring: :$\models_{\mathscr C_3} \phi$  $v \left({\phi}\right) \in \set{ 0, 1 }$
Definition:Constructed Semantics/Instance 4
Let $\mathcal L_0$ be the language of propositional logic. The constructed semantics $\mathscr C_4$ for $\mathcal L_0$ is used for the following results: * Hilbert Proof System Instance 2 Independence Results: Independence of $(A3)$ === Structures === Define the structures of $\mathscr C_4$ as mappings $v$ by the Principle of Recursive Definition, as follows. Let $\mathcal P_0$ be the vocabulary of $\mathcal L_0$. Let a mapping $v: \mathcal P_0 \to \{ 0, 1, 2, 3 \}$ be given. Next, regard the following as definitional abbreviations: {{begin-axiom}} {{axiom | n  = 1         | lc = Conjunction         | ml = \mathbf A \land \mathbf B         | mo = {{=}}_{\text{def} }         | mr = \neg \left({ \neg \mathbf A \lor \neg \mathbf B }\right) }} {{axiom | n  = 2         | lc = Conditional         | ml = \mathbf A \implies \mathbf B         | mo = {{=}}_{\text{def} }         | mr = \neg \mathbf A \lor \mathbf B }} {{axiom | n  = 3         | lc = Biconditional         | ml = \mathbf A \iff \mathbf B         | mo = {{=}}_{\text{def} }         | mr = \left({\mathbf A \implies \mathbf B}\right) \land \left({\mathbf B \implies \mathbf A}\right) }} {{end-axiom}} It only remains to define $v \left({ \neg \phi }\right)$ and $v \left({ \phi \lor \psi}\right)$ recursively, by: {{begin-eqn}} {{eqn | l = v \left({\neg \phi }\right)       | o = :=       | r = \begin{cases} 1 & : \text{if $v \left({\phi}\right) = 0$} \\ 0 & : \text{if $v \left({\phi}\right) = 1$} \\ 0 & : \text{if $v \left({\phi}\right) = 2$} \\ 2 & : \text{if $v \left({\phi}\right) = 3$}\end{cases} }} {{eqn | l = v \left({ \phi \lor \psi }\right)       | o = :=       | r = \begin{array}{c{{!}}cccc} \phi \lor \psi & 0 & 1 & 2 & 3\\ \hline 0 & 0 & 0 & 0 & 0 \\ 1 & 0 & 1 & 2 & 3 \\ 2 & 0 & 2 & 2 & 0 \\ 3 & 0 & 3 & 3 & 3 \end{array} }} {{end-eqn}} === Validity === Define validity in $\mathscr C_4$ by declaring: :$\models_{\mathscr C_4} \phi$  $v \left({\phi}\right) = 0$
Definition:Cauchy Functional Equation
Let $f: \R \to \R$ be a real function. Then the '''Cauchy functional equation''' is: :$\forall x, y \in \R: f \paren {x + y} = f \paren x + f \paren y$
Definition:Constructed Semantics/Instance 5
Let $\mathcal L_0$ be the language of propositional logic. The constructed semantics $\mathscr C_5$ for $\mathcal L_0$ is used for the following results: * Hilbert Proof System Instance 2 Independence Results: Independence of $(A4)$ === Structures === Define the structures of $\mathscr C_5$ as mappings $v$ by the Principle of Recursive Definition, as follows. Let $\mathcal P_0$ be the vocabulary of $\mathcal L_0$. Let a mapping $v: \mathcal P_0 \to \{ 0, 1, 2, 3 \}$ be given. Next, regard the following as definitional abbreviations: {{begin-axiom}} {{axiom | n  = 1         | lc = Conjunction         | ml = \mathbf A \land \mathbf B         | mo = {{=}}_{\text{def} }         | mr = \neg \left({ \neg \mathbf A \lor \neg \mathbf B }\right) }} {{axiom | n  = 2         | lc = Conditional         | ml = \mathbf A \implies \mathbf B         | mo = {{=}}_{\text{def} }         | mr = \neg \mathbf A \lor \mathbf B }} {{axiom | n  = 3         | lc = Biconditional         | ml = \mathbf A \iff \mathbf B         | mo = {{=}}_{\text{def} }         | mr = \left({\mathbf A \implies \mathbf B}\right) \land \left({\mathbf B \implies \mathbf A}\right) }} {{end-axiom}} It only remains to define $v \left({ \neg \phi }\right)$ and $v \left({ \phi \lor \psi}\right)$ recursively, by: {{begin-eqn}} {{eqn | l = v \left({\neg \phi }\right)       | o = :=       | r = \begin{cases} 1 & : \text{if $v \left({\phi}\right) = 0$} \\ 0 & : \text{if $v \left({\phi}\right) = 1$} \\ 0 & : \text{if $v \left({\phi}\right) = 2$} \\ 2 & : \text{if $v \left({\phi}\right) = 3$}\end{cases} }} {{eqn | l = v \left({ \phi \lor \psi }\right)       | o = :=       | r = \begin{array}{c{{!}}cccc} \phi \lor \psi & 0 & 1 & 2 & 3\\ \hline 0 & 0 & 0 & 0 & 0 \\ 1 & 0 & 1 & 2 & 3 \\ 2 & 0 & 2 & 2 & 0 \\ 3 & 0 & 3 & 0 & 3 \end{array} }} {{end-eqn}} === Validity === Define validity in $\mathscr C_5$ by declaring: :$\models_{\mathscr C_5} \phi$  $v \left({\phi}\right) = 0$
Definition:Subtraction/Real Numbers
Let $\struct {\R, +, \times}$ be the field of real numbers. The operation of '''subtraction''' is defined on $\R$ as: :$\forall a, b \in \R: a - b := a + \paren {-b}$ where $-b$ is the negative of $b$ in $\R$.
Definition:Subtraction/Rational Numbers
Let $\struct {\Q, +, \times}$ be the field of rational numbers. The operation of '''subtraction''' is defined on $\Q$ as: :$\forall a, b \in \Q: a - b := a + \paren {-b}$ where $-b$ is the negative of $b$ in $\Q$.
Definition:Self-Inverse Element/Definition 1
$x$ is a '''self-inverse element of $\struct {S, \circ}$'''  $x \circ x = e$.
Definition:Self-Inverse Element/Definition 2
$x$ is a '''self-inverse element of $\struct {S, \circ}$''' : :$x$ is invertible and: :$x = x^{-1}$, where $x^{-1}$ is the inverse of $x$.
Definition:Symmetric Group/n Letters
Let $S_n$ denote the set of permutations on $n$ letters. Let $\struct {S_n, \circ}$ denote the symmetric group on $S_n$. Then $\struct {S_n, \circ}$ is referred to as the '''symmetric group on $n$ letters'''.
Definition:Symmetric Group/Notation
In order not to make notation for operations on a '''symmetric group''' overly cumbersome, '''product notation''' is usually used for mapping composition. Thus $\pi \circ \rho$ is written $\pi \rho$. Also, for the same reason, rather than using $I_{S_n}$ for the identity mapping, the symbol $e$ is usually used.
Definition:Symmetric Group/Also known as
In view of the isomorphism between symmetric groups on sets of the same cardinality, the terminology '''symmetric group of degree $n$''' is often used when the nature of the underlying set is immaterial. Some sources use the term '''$n$th symmetric group'''. These terms will sometimes be used on . Some sources refer to the '''symmetric group''' on a set as the '''full symmetric group (on $S$)'''. Others use the term '''complete symmetric group'''. Similarly, the '''symmetric group on $n$ letters''' can be found referred to as the '''full symmetric group on $n$ letters'''. The term '''(full) symmetric group on $n$ objects''' can be found for both the general '''symmetric group''' and the '''symmetric group on $n$ letters''' Some sources use the notation $S \paren A$ to denote the set of permutations on a given set $A$, and thence $S \paren A$ to denote the '''symmetric group''' on $A$. In line with this, the notation $S \paren n$ is often used for $S_n$ to denote the '''symmetric group on $n$ letters'''. Others use $\mathcal S_n$ or some such variant. The notation $\operatorname {Sym} \paren n$ for $S_n$ can also be found. Some older sources denote the '''symmetric group on $A$''' as $\mathfrak S_A$. Such sources consequently denote the '''symmetric group on $n$ letters''' as $\mathfrak S_n$. However, this ''fraktur'' font is rarely used nowadays as it is cumbersome to reproduce and awkward to read. Be careful not to refer to $\struct {\Gamma \paren S, \circ}$ for $\card S = n$ or $S_n$ as the '''symmetric group of order $n$''', as the order of these groups is not $n$ but $n!$, from Order of Symmetric Group. === Isomorphism between Symmetric Groups === {{:Definition:Symmetric Group/Isomorphism}}
Definition:Rotation (Geometry)
A '''rotation''' in the context of Euclidean geometry is an isometry from a Euclidean Space $\R^n$ as follows. A '''rotation''' is defined usually for either: :$n = 2$, representing the plane or: :$n = 3$, representing ordinary space. === Rotation in the Plane === {{:Definition:Rotation (Geometry)/Plane}} === Rotation in Space === {{:Definition:Rotation (Geometry)/Space}}
Definition:Rotation (Geometry)/Plane
A '''rotation''' $r_\alpha$ in the plane is an isometry on the Euclidean Space $\Gamma = \R^2$ as follows. Let $O$ be a distinguished point in $\Gamma$, which has the property that: :$\map {r_\alpha} O = O$ That is, $O$ maps to itself. Let $P \in \Gamma$ such that $P \ne O$. Let $OP$ be joined by a straight line. Let a straight line $OP'$ be constructed such that: :$(1): \quad OP' = OP$ :$(2): \angle POP' = \alpha$ such that $OP \to OP'$ is in the anticlockwise direction: :300px Then: :$\map {r_\alpha} P = P'$ Thus $r_\alpha$ is a '''rotation (in the plane) of (angle) $\alpha$ about (the axis) $O$'''.
Definition:Rotation (Geometry)/Space
A '''rotation''' $r_\theta$ in space is an isometry on the Euclidean Space $\Gamma = \R^3$ as follows. Let $AB$ be a distinguished straight line in $\Gamma$, which has the property that: :$\forall P \in AB: \map {r_\theta} P = P$ That is, all points on $AB$ map to themselves. Let $P \in \Gamma$ such that $P \notin AB$. Let a straight line be constructed from $P$ to $O$ on $AB$ such that $OP$ is perpendicular to $AB$. Let a straight line $OP'$ be constructed perpendicular to $AB$ such that: :$(1): \quad OP' = OP$ :$(2): \quad \angle POP' = \theta$ such that $OP \to OP'$ is in the anticlockwise direction: :400px Then: :$\map {r_\theta} P = P'$ Thus $r_\theta$ is a '''rotation (in space) of (angle) $\theta$ about (the axis) $O$'''.
Definition:Rotation (Geometry)/Axis
Let $r_\theta$ be a rotation in the Euclidean Space $\Gamma = \R^n$. The set $A$ of points in $\Gamma$ such that: :$\forall P \in A: \map {r_\theta} P = P$ is called the '''axis of rotation''' of $r_\theta$.
Definition:Reflection (Geometry)
A '''reflection''' in the context of Euclidean geometry is an isometry from a Euclidean Space $\R^n$ as follows. A '''reflection''' is defined usually for either: :$n = 2$, representing the plane or: :$n = 3$, representing ordinary space. === Reflection in the Plane === {{:Definition:Reflection (Geometry)/Plane}} === Reflection in Space === {{:Definition:Reflection (Geometry)/Space}} === Point Reflection in Space === {{:Definition:Reflection (Geometry)/Point}}
Definition:Reflection (Geometry)/Plane
A '''reflection''' $\phi_{AB}$ in the plane is an isometry on the Euclidean Space $\Gamma = \R^2$ as follows. Let $AB$ be a distinguished straight line in $\Gamma$, which has the property that: :$\forall P \in AB: \map {\phi_{AB} } P = P$ That is, every point on $AB$ maps to itself. Let $P \in \Gamma$ such that $P \notin AB$. Let a straight line be constructed from $P$ to $O$ on $AB$ such that $OP$ is perpendicular to $AB$. Let $PO$ be produced to $P'$ such that $OP = OP'$. :400px Then: :$\map {\phi_{AB} } P = P'$ Thus $\phi_{AB}$ is a '''reflection (in the plane) in (the axis of reflection) $AB$'''.
Definition:Translation Mapping/Abelian Group
Let $\struct {G, +}$ be an abelian group. Let $g \in G$. Then '''translation by $g$''' is the mapping $\tau_g: G \to G$ defined by: :$\forall h \in G: \map {\tau_g} h = h + \paren {-g}$ where $-g$ is the inverse of $g$ with respect to $+$ in $G$.
Definition:Translation Mapping/Euclidean Space
A '''translation''' $\tau_\mathbf x$ is an isometry on the Euclidean Space $\Gamma = \R^n$ defined as: :$\forall \mathbf y \in \R^n: \map {\tau_\mathbf x} {\mathbf y} = \mathbf y - \mathbf x$ where $\mathbf x$ is a vector in $\R^n$.
Definition:Aleph Number
The '''aleph numbers''' are a sequence of numbers used to represent the cardinality (or size) of infinite sets that can be well-ordered. The cardinality of the natural numbers is $\aleph_0$. The next larger cardinality is $\aleph_1$. Then $\aleph_2$ and so on.
Definition:Order of Group Element/Also known as
Some sources refer to the '''order''' of an element of a group as its '''period'''.
Definition:Order of Group Element/Also denoted as
The '''order''' of an element $x$ in a group is sometimes seen as $\map o x$. Some sources render it as $\map {\operatorname {Ord} } x$.
Definition:Cyclic Group/Notation
A '''cyclic group''' with $n$ elements is often denoted $C_n$. Some sources use the notation $\sqbrk g$ or $\gen g$ to denote the '''cyclic group''' generated by $g$. From Integers Modulo m under Addition form Cyclic Group, $\struct {\Z_m, +_m}$ is a '''cyclic group'''. Thus $\struct {\Z_m, +_m}$ often taken as the archetypal example of a '''cyclic group''', and the notation $\Z_m$ is used. This is justified as, from Cyclic Groups of Same Order are Isomorphic, $\Z_m$ is isomorphic to $C_m$. In certain contexts $\Z_m$ is particularly useful, as it allows results about '''cyclic groups''' to be demonstrated using number theoretical techniques.
Definition:Cyclic Group/Definition 1
The group $G$ is '''cyclic'''  every element of $G$ can be expressed as the power of one element of $G$: :$\exists g \in G: \forall h \in G: h = g^n$ for some $n \in \Z$.
Definition:Cyclic Group/Definition 2
The group $G$ is '''cyclic'''  it is generated by one element $g \in G$: :$G = \gen g$
Definition:Generator of Group
Let $\struct {G, \circ}$ be a group. Let $S \subseteq G$. Then '''$S$ is a generator of $G$''', denoted $G = \gen S$,  $G$ is the subgroup generated by $S$.
Definition:Cyclic Group/Generator
Let $a \in G$ be an element of $G$ such that $\gen a = G$. Then $a$ is '''a generator of $G$'''.
Definition:Multiplicative Group of Reduced Residues
Let $m \in \Z_{> 0}$ be a (strictly) positive integer. Let $\Z'_m$ denote the reduced residue system modulo $m$. Consider the algebraic structure: :$\struct {\Z'_m, \times_m}$ where $\times_m$ denotes multiplication modulo $m$. Then $\struct {\Z'_m, \times_m}$ is referred to as '''the multiplicative group of reduced residues modulo $m$'''.