hcju/proofwikips huggingface dataset:
Definition:Numerators and Denominators of Continued Fraction
Let $F$ be a field. Let $n \in \N \cup \set \infty$ be an extended natural number. Let $C = \sqbrk {a_0, a_1, a_2, \ldots}$ be a continued fraction in $F$ of length $n$. === Definition 1: recursive definition === The '''sequence of numerators''' of $C$ is the sequence $\sequence {p_k}_{0 \mathop \le k \mathop \le n}$ that is recursively defined by: :$p_k = \begin {cases}   a_0 & : k = 0 \\   a_1 a_0 + 1 & : k = 1 \\   a_k p_{k - 1} + p_{k - 2} & : k \ge 2 \end {cases}$ The '''sequence of denominators''' of $C$ is the sequence $\sequence {q_k}_{0 \mathop \le k \mathop \le n}$ that is recursively defined by: :$q_k = \begin {cases}   1 & : k = 0 \\   a_1 & : k = 1 \\   a_k q_{k - 1} + q_{k - 2} & : k \ge 2 \end {cases}$ === Definition 2: using matrix products === Let $k \ge 0$, and let the indexed matrix product: :$\displaystyle \prod_{i \mathop = 0}^k \begin {pmatrix}   a_i & 1 \\   1 & 0 \end {pmatrix} = \begin {pmatrix}   x_{1 1}^{\paren k} & x_{1 2}^{\paren k} \\   x_{2 1}^{\paren k} & x_{2 2}^{\paren k} \end {pmatrix}$ The $k$th '''numerator''' is $x_{1 1}^{\paren k}$ and the $k$th '''denominator''' is $x_{2 1}^{\paren k}$.
Definition:Generalized Continued Fraction
Let $k$ be a field. Informally, a '''generalized continued fraction''' in $k$ is an expression of the form: :$b_0 + \cfrac {a_1} {b_1 + \cfrac {a_2} {b_2 + \cfrac {a_3} {\ddots \cfrac {} {b_{n-1} + \cfrac {a_n} {b_n + \cfrac {a_{n+1}} {\ddots}}} }}}$ Formally, a '''generalized continued fraction''' in $k$ is a pair of sequences $((b_n)_{n\geq 0}, (a_n)_{n\geq 1})$ in $k$, called '''sequence of partial denominators''' and '''sequence of partial numerators''' respectively.
Definition:Sheffer Operator
A '''Sheffer operator''' is a truth function which forms a functionally complete singleton set.
Definition:Functionally Complete
Let $S$ be a set of truth functions. Then $S$ is '''functionally complete'''  all possible truth functions are definable from $S$.
Definition:Complement of Truth Function
Let $f: \Bbb B^k \to \Bbb B$ be a truth function. The '''complement''' of $f$ is the function $f'$ defined by: :$f': \Bbb B^k \to \Bbb B, f' \left({p}\right) = \neg \left({f \left({p}\right)}\right)$
Definition:Support of Continuous Mapping
=== Continuous Real-Valued Function in $\R^n$ === {{Definition:Support of Continuous Mapping/Real-Valued}} === General topological group === Let $X$ be a topological space. Let $G$ be a topological group with identity $e$. Let $f : X \to G$ be a continuous mapping. The '''support of $f$''' is the closure of the set of elements of $X$ that do not map to $e$ under $f$: :$\operatorname{supp} \left({f}\right) = \overline{\left\{{x \in X: f \left({x}\right) \ne e}\right\}}$
Definition:Propositional Function
A '''propositional function''' $\map P {x_1, x_2, \ldots}$ is an operation which acts on the objects denoted by the object variables $x_1, x_2, \ldots$ in a particular universe to return a truth value which depends on: :$(1): \quad$ The values of $x_1, x_2, \ldots$ :$(2): \quad$ The nature of $P$. === Satisfaction === {{:Definition:Propositional Function/Satisfaction}}
Definition:Boolean Interpretation/Formula
Let $v: \mathcal L_0 \to \left\{{T, F}\right\}$ be a (partial) boolean interpretation. Let $\phi$ be a WFF of propositional logic. Then $v$ is called a '''boolean interpretation for $\phi$''' iff $v$ is defined at $\phi$. Otherwise, $v$ is called a '''partial (boolean) interpretation for $\phi$'''.
Definition:Boolean Interpretation/Set of Formulas
Let $v: \mathcal L_0 \to \left\{{T, F}\right\}$ be a (partial) boolean interpretation. Let $\mathcal F$ be a set of WFFs of $\mathcal L_0$. Then $v$ is called a '''boolean interpretation for $\mathcal F$''' iff $v$ is defined on $\mathcal F$. Otherwise, $v$ is called a '''partial (boolean) interpretation for $\mathcal F$'''.
Definition:Boolean Interpretation/Truth Value
Let $\phi$ be a WFF of propositional logic. Let $v: \mathcal L_0 \to \left\{{T, F}\right\}$ be a boolean interpretation of $\phi$. The '''truth value of $\phi$ under $v$''' is $v (\phi)$.
Definition:Legendre Polynomial
The '''Legendre polynomials''' are the solutions to Legendre's differential equation.
Definition:Inverse Hyperbolic Sine/Complex
=== Definition 1 === {{:Definition:Inverse Hyperbolic Sine/Complex/Definition 1}} === Definition 2 === {{:Definition:Inverse Hyperbolic Sine/Complex/Definition 2}}
Definition:Variable/Propositional Logic
A '''statement variable''' is a variable which is used to stand for arbitrary and unspecified statements. For '''statement variables''', lowercase letters are usually used, e.g.: :$p, q, r, \ldots{}$, etc. or lowercase Greek letters, e.g.: :$\phi, \psi, \chi$ etc. The citing of a variable can be interpreted as an assertion that the statement represented by that symbol is true.  That is: :'''$p$''' means :'''$p \text { is true}$'''
Definition:Scope (Logic)/Connective
Let $\mathcal L_0$ be the language of propositional logic. Let $\circ$ be a connective of $\mathcal L_0$. Let $\mathbf W$ be a well-formed formula of $\mathcal L_0$. The '''scope''' of an occurrence of $\circ$ in $\mathbf W$ is the smallest well-formed part of $\mathbf W$ containing this occurrence of $\circ$.
Definition:Abbreviation of WFFs of Propositional Logic
The WFFs of propositional logic can be made more readable by allowing them to be abbreviated. The resulting strings are not actually WFFs as such, but can be translated back uniquely into full WFFs. === Rules for Abbreviation of WFFs === {{:Definition:Abbreviation of WFFs of Propositional Logic/Rules}}
Definition:Abbreviation of WFFs of Propositional Logic/Standard Abbreviation
The string obtained by applying as many of the rules for abbreviation of WFFs  to a WFF $\mathbf A$ as possible is known as the '''standard abbreviation of $\mathbf A$'''.
Definition:Binding Priority/Propositional Logic
The binding priority convention which is almost universally used for the connectives of propositional logic is: :$(1): \quad \neg$ binds more tightly than $\lor$ and $\land$ :$(2): \quad \lor$ and $\land$ bind more tightly than $\implies$ and $\impliedby$ :$(3): \quad \implies$ and $\impliedby$ bind more tightly than $\iff$ Note that there is no overall convention defining which of $\land$ and $\lor$ bears a higher binding priority, and therefore we consider them to have equal priority. Because of this fact, ''unless specifically defined'', expressions such as $p \land q \lor r$ can not be interpreted unambiguously, and parenthesis ''must'' be used to determine the exact priorities which are to be used to interpret particular statements which may otherwise be ambiguous. Most sources do not recognise the use of $\impliedby$ as a separate connective from $\implies$, so the priority of one over the other is rarely a question.
Definition:Main Connective/Propositional Logic/Definition 1
Let $\mathbf C$ be a WFF of propositional logic. Let $\circ$ be a binary connective. Then $\circ$ is the '''main connective''' iff the scope of $\circ$ is $\mathbf C$.
Definition:Main Connective/Propositional Logic/Definition 2
Let $\mathbf C$ be a WFF of propositional logic such that: : $\mathbf C = \left({\mathbf A \circ \mathbf B}\right)$ where both $\mathbf A$ and $\mathbf B$ are both WFFs and $\circ$ is a binary connective. Then $\circ$ is the '''main connective''' of $\mathbf C$. Alternatively, let $\mathbf A$ be a WFF of propositional logic such that: : $\mathbf A = \neg \mathbf B$ where $\mathbf B$ is a WFF. Then $\neg$ is the '''main connective''' of $\mathbf A$.
Definition:Main Connective/Propositional Logic/Definition 3
Let $T$ be a WFF of propositional logic in the labeled tree specification. Suppose $T$ has more than one node. Then the label of the root of $T$ is called the '''main connective''' of $T$.
Definition:Classical Propositional Logic
'''Classical propositional logic''' is the branch of propositional logic based upon Aristotelian logic, whose philosophical tenets state that: :$(1): \quad$ A statement must be either true or false, and there can be no middle value. :$(2): \quad$ A statement cannot be both true and not true at the same time, that is, it may not contradict itself.
Definition:Labeled Tree for Propositional Logic
A '''labeled tree for propositional logic''' is a system containing: * A rooted tree $T$; * A countable set $\mathbf H$ of WFFs of propositional logic; * A WFF $\Phi \left({t}\right)$ attached to each non-root node $t$ of $T$. Such a structure can be denoted $\left({T, \mathbf H, \Phi}\right)$. === Hypothesis Set === {{:Definition:Labeled Tree for Propositional Logic/Hypothesis Set}} === Attached === {{:Definition:Labeled Tree for Propositional Logic/Attached}} === Child WFF === {{:Definition:Labeled Tree for Propositional Logic/Child WFF}} === Ancestor WFF === {{:Definition:Labeled Tree for Propositional Logic/Ancestor WFF}} === Along a Branch === {{:Definition:Labeled Tree for Propositional Logic/Along a Branch}}
Definition:Labeled Tree for Propositional Logic/Attached
Let $t$ be a non-root node of $T$. Let $\mathbf A$ be a WFF. Then '''$\mathbf A$ is attached to $t$'''  $\mathbf A = \Phi \left({t}\right)$.
Definition:Labeled Tree for Propositional Logic/Child WFF
A WFF that is attached to a child of a node $t$ is called a '''child WFF of $t$'''.
Definition:Labeled Tree for Propositional Logic/Ancestor WFF
A WFF that is attached to an ancestor node of a node $t$ is called an '''ancestor WFF of $t$'''.
Definition:Labeled Tree for Propositional Logic/Along a Branch
Let $\Gamma$ be a branch of $T$. Let $\mathbf A$ be a WFF that is attached to a node $t \in \Gamma$. Then '''$\mathbf A$ occurs along $\Gamma$'''. This includes the case where $\mathbf A \in \mathbf H$, that is, $\mathbf A$ is attached to the root node.
Definition:Intuitionistic Propositional Logic
The '''intuitionist''' school of mathematics is one which adopts the following philosophical position: :"Although we may know that it is not the case that a statement $p$ is (provably) false, we don't necessarily know that it is (provably) true either." Thus the intuitionist school rejects the Law of the Excluded Middle. The classical school, by affirming that if a statement is not true it must be false, and if it is not false it must be true, accepts as an axiom that "not not-true" must mean "true". {{refactor|The below needs its own place (it's "Natural Deduction for Intuitionistic Propositional Logic")}} The system of '''intuitionistic propositional logic''' is, in consequence, based on the same axioms as that of classical propositional logic except for that disputed Law of the Excluded Middle.
Definition:Language of Propositional Logic/Labeled Tree
There are many formal languages expressing propositional logic. The formal language used on  is defined on Definition:Language of Propositional Logic. This page defines the formal language $\LL_0$ used in: * {{BookReference|Mathematical Logic for Computer Science|2012|M. Ben-Ari|ed=3rd|edpage=Third Edition}} Explanations are omitted as this is intended for reference use only. === Alphabet === ==== Letters ==== The letters used are an infinite set of symbols $\PP_0$. This is the same as the  definition. ==== Signs ==== ===== Connectives ===== The following '''connectives''' are used: {{begin-eqn}} {{eqn | ll= \bullet       | l = \neg       | o = :       | r = \)the negation sign\(       | c =  }} {{eqn | ll= \bullet       | l = \lor       | o = :       | r = \)the disjunction sign\(       | c =  }} {{eqn | ll= \bullet       | l = \land       | o = :       | r = \)the conjunction sign\(       | c =  }} {{eqn | ll= \bullet       | l = \implies       | o = :       | r = \)the conditional sign\(       | c =  }} {{eqn | ll= \bullet       | l = \iff       | o = :       | r = \)the biconditional sign\(       | c =  }} {{eqn | ll= \bullet       | l = \oplus       | o = :       | r = \)the exclusive or sign\(       | c =  }} {{eqn | ll= \bullet       | l = \downarrow       | o = :       | r = \)the nor sign\(       | c =  }} {{eqn | ll= \bullet       | l = \uparrow       | o = :       | r = \)the nand sign\(       | c =  }} {{end-eqn}} See the  definition. === Collation System === The collation system used is that of labeled trees and adding ancestors. {{explain|"adding ancestors" signifies combining labeled trees into a new one by creating a common ancestor for their roots}} See the  definition. === Formal Grammar === The following bottom-up formal grammar is used. Let $\PP_0$ be the vocabulary of $\LL_0$. The WFFs of $\LL_0$ are the smallest set $\FF$ of labeled trees such that: {{begin-axiom}} {{axiom | n = T1         | t = For each letter $p \in \PP_0$, the labeled tree with one node, whose label is $p$, is in $\FF$. }} {{axiom | n = T2         | t = For $T \in \FF$, the labeled tree obtained by adding an ancestor with label $\neg$ to the root node of $T$, is again in $\FF$. }} {{axiom | n = T3         | t = For $T_1 \in \FF$ and $T_2 \in \FF$ and a binary connective $\mathsf B$, the labeled tree obtained by adding a common ancestor labeled $\mathsf B$ of the root nodes of $T_1$ and $T_2$, is again in $\FF$. }} {{end-axiom}} Graphically, this means one has the following means to construct WFFs: ::$\begin{xy}\xymatrix{  p & &  \neg   \ar@{-}[d] & & &  \mathsf B   \ar@{-}[ld]   \ar@{-}[rd] \\ & &  <{\sf WFF}> & &   <{\sf WFF}> & &  <{\sf WFF}> }\end{xy}$ See the  definition.
Definition:Absolute Value/Ordered Integral Domain
Let $\struct {D, +, \times, \le}$ be an ordered integral domain whose zero is $0_D$. Then for all $a \in D$, the '''absolute value''' of $a$ is defined as: :$\size a = \begin{cases} a & : 0_D \le a \\ -a & : a < 0_D \end{cases}$
Definition:Truth Function
Let $\mathbb B$ be the set of truth values, and let $k$ be a natural number. A mapping $f: \mathbb B^k \to \mathbb B$ is called a '''truth function'''.
Definition:Logical NAND/Truth Function
The NAND connective defines the truth function $f^\uparrow$ as follows: {{begin-eqn}} {{eqn | l=f^\uparrow \left({F, F}\right)       | r=T }} {{eqn | l=f^\uparrow \left({F, T}\right)       | r=T }} {{eqn | l=f^\uparrow \left({T, F}\right)       | r=T }} {{eqn | l=f^\uparrow \left({T, T}\right)       | r=F }} {{end-eqn}}
Definition:Logical NOR/Truth Function
The NOR connective defines the truth function $f^\downarrow$ as follows: {{begin-eqn}} {{eqn | l=f^\downarrow \left({F, F}\right)       | r=T }} {{eqn | l=f^\downarrow \left({F, T}\right)       | r=F }} {{eqn | l=f^\downarrow \left({T, F}\right)       | r=F }} {{eqn | l=f^\downarrow \left({T, T}\right)       | r=F }} {{end-eqn}}
Definition:Exclusive Or/Truth Function
The exclusive or connective defines the truth function $f^\oplus$ as follows: {{begin-eqn}} {{eqn | l=f^\oplus \left({F, F}\right)       | r=F }} {{eqn | l=f^\oplus \left({F, T}\right)       | r=T }} {{eqn | l=f^\oplus \left({T, F}\right)       | r=T }} {{eqn | l=f^\oplus \left({T, T}\right)       | r=F }} {{end-eqn}}
Definition:Truth Table/Characteristic
Let $\circledcirc$ be a logical connective. The '''characteristic truth table''' for $\circledcirc$ is the truth table describing the truth function of $\circledcirc$: :$\begin{array}{|cc||c|} \hline p & q & p \circledcirc q \\ \hline F & F & x \\ F & T & x \\ T & F & x \\ T & T & x \\ \hline \end{array}$ where $x$ is replaced by either $F$ or $T$ as appropriate on each row. The characteristic truth tables of the various logical connectives are listed below. === Logical Negation === {{:Definition:Logical Not/Truth Table}} === Logical Conjunction === {{:Definition:Conjunction/Truth Table}} === Logical Disjunction === {{:Definition:Disjunction/Truth Table}} === Biconditional === {{:Definition:Biconditional/Truth Table}} === Exclusive Disjunction === {{:Definition:Exclusive Or/Truth Table}} === Conditional === {{:Definition:Conditional/Truth Table}} === Logical NAND === {{:Definition:Logical NAND/Truth Table}} === Logical NOR === {{:Definition:Logical NOR/Truth Table}}
Definition:Generator of Monoid
Let $\left({M, \circ}\right)$ be a monoid Let $S \subseteq M$. Let $H$ be the smallest submonoid of $M$ such that $S \subseteq H$. Then: * $S$ is a '''generator'''  of $\left({H, \circ}\right)$ * $S$ '''generates''' $\left({H, \circ}\right)$ * $\left({H, \circ}\right)$ is the '''submonoid of $\left({M, \circ}\right)$ generated by $S$'''. This is written $H = \left\langle {S} \right\rangle$.
Definition:Boolean Interpretation
Let $\LL_0$ be the language of propositional logic, with vocabulary $\PP_0$. A '''boolean interpretation''' for $\LL_0$ is a propositional function: :$v: \PP_0 \to \set {\T, \F}$
Definition:Isomorphism (Abstract Algebra)/Monoid Isomorphism
Let $\struct {S, \circ}$ and $\struct {T, *}$ be monoids. Let $\phi: S \to T$ be a (monoid) homomorphism. Then $\phi$ is a monoid isomorphism  $\phi$ is a bijection.
Definition:Monoid Automorphism
Let $\struct {S, \circ}$ be a monoid. Let $\phi: S \to S$ be a (monoid) isomorphism from $S$ to itself. Then $\phi$ is a monoid automorphism.
Definition:Additive Group of Real Numbers
The '''additive group of real numbers''' $\struct {\R, +}$ is the set of real numbers under the operation of addition.
Definition:Additive Group of Rational Numbers
The '''additive group of rational numbers''' $\left({\Q, +}\right)$ is the set of rational numbers under the operation of addition.
Definition:Multiplicative Group of Rational Numbers
The '''multiplicative group of rational numbers''' $\left({\Q_{\ne 0}, \times}\right)$ is the set of rational numbers without zero under the operation of multiplication.
Definition:Additive Group of Complex Numbers
The '''additive group of complex numbers''' $\struct {\C, +}$ is the set of complex numbers under the operation of addition.
Definition:Multiplicative Group of Complex Numbers
The '''multiplicative group of complex numbers''' $\struct {\C_{\ne 0}, \times}$ is the set of complex numbers without zero under the operation of multiplication.
Definition:Multiplicative Group of Real Numbers
The '''multiplicative group of real numbers''' $\struct {\R_{\ne 0}, \times}$ is the set of real numbers without zero under the operation of multiplication.
Definition:Field of Complex Numbers
The '''field of complex numbers''' $\struct {\C, +, \times}$ is the set of complex numbers under the two operations of addition and multiplication.
Definition:Field of Real Numbers
The '''field of real numbers''' $\struct {\R, + \times, \le}$ is the set of real numbers under the two operations of addition and multiplication, with an ordering $\le$ compatible with the ring structure of $\R$..
Definition:Field of Rational Numbers
The '''field of rational numbers''' $\struct {\Q, + \times, \le}$ is the set of rational numbers under the two operations of addition and multiplication, with an ordering $\le$ compatible with the ring structure of $\Q$.
Definition:Multiplicative Monoid of Integers Modulo m
Let $m \in \Z$ such that $m > 1$. The '''multiplicative monoid of integers modulo $m$''' $\struct {\Z_m, \times_m}$ is the set of integers modulo $m$ under the operation of multiplication modulo $m$.
Definition:Additive Group of Integers Modulo m
Let $m \in \Z$ such that $m > 1$. The '''additive group of integers modulo $m$''' $\struct {\Z_m, +_m}$ is the set of integers modulo $m$ under the operation of addition modulo $m$.
Definition:Radical Extension
Let $L / F$ be a field extension. Then $L$ is a '''radical extension of $F$''' iff there exist $\alpha_1, \ldots, \alpha_m \in F$ and $n_1, \ldots, n_2 \in \Z_{>0}$ such that: :$(1): \quad L = K \left[{\alpha_1, \ldots, \alpha_m}\right]$ :$(2): \quad \alpha_1^{n_1} \in F$ :$(3): \quad \forall i \in \N_m: \alpha_i^{n_i} \in F \left[{\alpha_1, \ldots, \alpha_{i-1}}\right]$ {{explain|link to definitions of $K \left[{\alpha_1, \ldots, \alpha_m}\right]$ and $F \left[{\alpha_1, \ldots, \alpha_{i-1} }\right]$}}
Definition:Additive Monoid of Natural Numbers
The '''additive monoid of natural numbers''' $\left({\N, +}\right)$ is the set of natural numbers under the operation of addition.
Definition:Language of Propositional Logic/Formal Grammar/WFF
Let $\mathbf A$ be approved of by the formal grammar of propositional logic. Then $\mathbf A$ is called a '''well-formed formula of propositional logic'''. Often, one abbreviates "well-formed formula", speaking of a '''WFF of propositional logic''' instead.
Definition:Statement Form/Substatement
A '''substatement''' of a statement form $\mathbf A$ is another statement form which occurs as a part of $\mathbf A$.
Definition:Finite Monoid
A '''finite monoid''' is a monoid whose underlying set is finite.
Definition:Additive Group of Integer Multiples
Let $n \in \Z_{>0}$. The '''additive group $\left({n \Z, +}\right)$ of integer multiples of $n$'''  is the set of integer multiples of $n$ under the operation of addition.
Definition:Root of Unity/Complex
Let $n \in \Z_{>0}$ be a (strictly) positive integer. The '''complex $n$th roots of unity''' are the elements of the set: :$U_n = \set {z \in \C: z^n = 1}$
Definition:Root of Unity/Primitive
A '''primitive $n$th root of unity of $F$''' is an element $\alpha \in U_n$ such that: :$U_n = \set {1, \alpha, \ldots, \alpha^{n - 1} }$
Definition:Affine Group of One Dimension
Let $S$ be the set of mappings $f_{a b}: \R \to \R$ defined as: :$S := \left\{{f_{a b}: x \mapsto a x + b: a \in \R_{\ne 0}, b \in \R}\right\}$ The algebraic structure $\left({S, \circ}\right)$, where $\circ$ denotes composition of mappings, is called the '''$1$-dimensional affine group on $\R$''' and can be denoted $\operatorname{Af}_1 \left({\R}\right)$.
Definition:Multiplicative Group of Positive Rational Numbers
The '''multiplicative group of positive rational numbers''' $\left({\Q_{> 0}, \times}\right)$ is the set of (strictly) positive rational numbers under the operation of multiplication.
Definition:Multiplicative Group of Positive Real Numbers
The '''multiplicative group of positive real numbers''' $\left({\R_{> 0}, \times}\right)$ is the set of (strictly) positive real numbers under the operation of multiplication.
Definition:Googol
A '''googol''' is defined to be $10^{100}$.
Definition:Googolplex
A '''googolplex''' is defined to be $10$ to the power of a googol, that is $10^{10^{100}}$ or $10^{\mathrm {googol}}$.
Definition:Hyperbola
=== Intersection with Cone === {{:Definition:Conic Section/Intersection with Cone}} {{:Definition:Conic Section/Intersection with Cone/Hyperbola}} === Focus-Directrix Property === {{:Definition:Conic Section/Focus-Directrix Property/Hyperbola}} === Equidistance Property === {{:Definition:Hyperbola/Equidistance}}
Definition:Parabola
=== Intersection with Cone === {{:Definition:Conic Section/Intersection with Cone}} {{:Definition:Conic Section/Intersection with Cone/Parabola}} === Focus-Directrix Property === {{:Definition:Conic Section/Focus-Directrix Property/Parabola}}
Definition:Centralizer/Group Subset
Let $\struct {G, \circ}$ be a group. Let $S \subseteq G$. The '''centralizer of $S$ (in $G$)''' is the set of elements of $G$ which commute with all $s \in S$: :$\map {C_G} S = \set {x \in G: \forall s \in S: x \circ s = s \circ x}$
Definition:Möbius Strip
A '''Möbius strip''' is a surface with boundary obtained by twisting a side of a rectangle by $180$ degrees, then joining the twisted side and the opposite side of it together. {{expand|An illustration of this process will aid understanding}}
Definition:Differentiable at Point
=== Real Function === {{:Definition:Differentiable Mapping/Real Function/Point}} === Complex Function === {{:Definition:Differentiable Mapping/Complex Function/Point}} === Real-Valued Function === {{:Definition:Differentiable Mapping/Real-Valued Function/Point}} === Between Manifolds === {{:Definition:Differentiable Mapping between Manifolds/Point}}
Definition:Faithful Group Action
Let $G$ be a group with identity $e$. Let $X$ be a set. Let $\phi: G \times X \to X$ be a group action. === Definition 1 === {{:Definition:Faithful Group Action/Definition 1}} === Definition 2 === {{:Definition:Faithful Group Action/Definition 2}}
Definition:Transitive Group Action
Let $G$ be a group. Let $S$ be a set. Let $*: G \times S \to S$ be a group action. The group action is '''transitive'''  for any $x, y \in S$ there exists $g \in G$ such that $g * x = y$. That is,  for all $x \in S$: :$\Orb x = S$ where $\Orb x$ denotes the orbit of $x \in S$ under $*$.