hcju/proofwikips huggingface dataset:
Definition:Stirling Numbers of the First Kind/Unsigned/Definition 1
'''Unsigned Stirling numbers of the first kind''' are defined recursively by: :$\displaystyle {n \brack k} := \begin{cases} \delta_{n k} & : k = 0 \text { or } n = 0 \\ & \\ \displaystyle {n - 1 \brack k - 1} + \paren {n - 1} {n - 1 \brack k} & : \text{otherwise} \\ \end{cases}$
Definition:Stirling Numbers of the Second Kind/Definition 1
'''Stirling numbers of the second kind''' are defined recursively by: :$\displaystyle {n \brace k} := \begin{cases} \delta_{n k} & : k = 0 \text{ or } n = 0 \\ & \\ \displaystyle {n - 1 \brace k - 1} + k {n - 1 \brace k} & : \text{otherwise} \\ \end{cases}$
Definition:Stirling Numbers of the Second Kind/Definition 2
'''Stirling numbers  of the second kind''' are defined as the coefficients $\displaystyle {n \brace k}$ which satisfy the equation: :$\displaystyle x^n = \sum_k {n \brace k} x^{\underline k}$ where $x^{\underline k}$ denotes the $k$th falling factorial of $x$.
Definition:Stirling Numbers of the First Kind/Signed/Definition 1
'''Signed Stirling numbers of the first kind''' are defined recursively by: :$\map s {n, k} := \begin{cases} \delta_{n k} & : k = 0 \text{ or } n = 0 \\ \map s {n - 1, k - 1} - \paren {n - 1} \map s {n - 1, k} & : \text{otherwise} \\ \end{cases}$
Definition:Stirling Numbers of the First Kind/Signed/Definition 2
'''Signed Stirling numbers of the first kind''' are defined as the polynomial coefficients $\map s {n, k}$ which satisfy the equation: :$\displaystyle x^{\underline n} = \sum_k \map s {n, k} x^k$ where $x^{\underline n}$ denotes the $n$th falling factorial of $x$.
Definition:Relative Semantic Equivalence
Let $\mathcal F$ be a theory in the language of predicate logic. === Relative Semantic Equivalence of WFFs === {{:Definition:Relative Semantic Equivalence/WFF}} === Relative Semantic Equivalence of Terms === {{:Definition:Relative Semantic Equivalence/Term}}
Definition:Relative Semantic Equivalence/WFF
Let $\mathbf A, \mathbf B$ be WFFs. Let $\mathbf C$ be the universal closure of $\mathbf A \iff \mathbf B$. Then $\mathbf A$ and $\mathbf B$ are '''semantically equivalent with respect to $\mathcal F$''' : :$\mathcal F \models_{\mathrm{PL}} \mathbf C$ That is, iff $\mathbf C$ is a semantic consequence of $\mathcal F$.
Definition:Relative Semantic Equivalence/Term
Let $\tau_1, \tau_2$ be terms. Then $\tau_1$ and $\tau_2$ are '''semantically equivalent with respect to $\mathcal F$''' : :$\mathop{ \operatorname{val}_{\mathcal A} \left({\tau_1}\right) } \left[{\sigma}\right] = \mathop{ \operatorname{val}_{\mathcal A} \left({\tau_2}\right) } \left[{\sigma}\right]$ for all models $\mathcal A$ of $\mathcal F$ and assignments $\sigma$ for $\tau_1,\tau_2$ in $\mathcal A$. Here $\mathop{ \operatorname{val}_{\mathcal A} \left({\tau_1}\right) } \left[{\sigma}\right]$ denotes the value of $\tau_1$ under $\sigma$.
Definition:Substitution (Formal Systems)/Term/In Term
Let $\beta, \tau$ be terms of predicate logic. Let $x \in \mathrm{VAR}$ be a variable. Let $\beta \left({x \gets \tau}\right)$ be the term resulting from $\beta$ by replacing all occurrences of $x$ by $\tau$. Then $\beta \left({x \gets \tau}\right)$ is called the '''substitution instance of $\beta$ substituting $\tau$ for $x$'''.
Definition:Substitution (Formal Systems)/Term/In WFF
Let $\mathbf A$ be a WFF of predicate logic. Let $\tau$ be a term of predicate logic. Let $x \in \mathrm{VAR}$ be a variable. Let $\mathbf A \left({x \gets \tau}\right)$ be the WFF resulting from $\mathbf A$ by replacing all free occurrences of $x$ by $\tau$. Then $\mathbf A \left({x \gets \tau}\right)$ is called the '''substitution instance of $\mathbf A$ substituting $\tau$ for $x$'''.
Definition:Additive Semiring/Axioms
An additive semiring is an algebraic structure $\left({R, *, \circ}\right)$, on which are defined two binary operations $\circ$ and $*$, which satisfy the following conditions: {{begin-axiom}} {{axiom | n = A0         | q = \forall a, b \in S         | m = a * b \in S         | rc= Closure under $*$ }} {{axiom |  n = A1         |  q = \forall a, b, c \in S         | m = \left({a * b}\right) * c = a * \left({b * c}\right)         | rc= Associativity of $*$ }} {{axiom | n = A2         | q = \forall a, b \in S         | m = a * b = b * a         | rc= Commutativity of $*$ }} {{axiom | n = M0         | q = \forall a, b \in S         | m = a \circ b \in S         | rc= Closure under $\circ$ }} {{axiom | n = M1         | q = \forall a, b, c \in S         | m = \left({a \circ b}\right) \circ c = a \circ \left({b \circ c}\right)         | rc= Associativity of $\circ$ }} {{axiom | n = D         | q = \forall a, b, c \in S         | m = a \circ \left({b * c}\right) = \left({a \circ b}\right) * \left({a \circ c}\right), \left({a * b}\right) \circ c = \left({a \circ c}\right) * \left({a \circ c}\right)         | rc= $\circ$ is distributive over $*$ }} {{end-eqn}} These criteria are called the '''additive semiring axioms'''.
Definition:Join Semilattice/Definition 1
Let $\struct {S, \preceq}$ be an ordered set. Suppose that for all $a, b \in S$: :$a \vee b \in S$ where $a \vee b$ is the join of $a$ and $b$ with respect to $\preceq$. Then the ordered structure $\struct {S, \vee, \preceq}$ is called a '''join semilattice'''.
Definition:Join Semilattice/Definition 2
Let $\struct {S, \vee}$ be a semilattice. Let $\preceq$ be the ordering on $S$ defined by: :$a \preceq b \iff \paren {a \vee b} = b$ Then the ordered structure $\struct {S, \vee, \preceq}$ is called a '''join semilattice'''.
Definition:Parenthesization/Equivalent
Two parenthesizations of $a_1, \ldots, a_n$ are '''equivalent'''  the product defined by them yields the same result.
Definition:Trivial Norm/Vector Space
Let $\struct {K, +, \circ}$ be a division ring endowed with the trivial norm. Let $V$ be a vector space over $K$, with zero $0_V$. Then the map $\norm {\cdot}: V \to \R_+ \cup \set 0$ given by: :$\norm x = \begin{cases}   0 & : \text {if $x = 0_V$} \\   1 & : \text {otherwise} \end{cases}$ defines a norm on $V$, called the '''trivial norm'''.
Definition:Trivial Norm/Division Ring
Let $\struct {R, +, \circ}$ be a division ring, and denote its zero by $0_R$. Then the map $\norm {\cdot}: R \to \R_{\ge 0}$ given by: :$\norm x = \begin{cases}   0 & : \text{if $x = 0_R$}\\   1 & : \text{otherwise} \end{cases}$ defines a norm on $R$, called the '''trivial norm'''.
Definition:Signature (Logic)/Predicate Logic
Let $\mathcal L_1$ be the language of predicate logic. Then a '''signature''' for $\mathcal L_1$ is an explicit choice of the alphabet of $\mathcal L_1$. That is to say, it amounts to choosing, for each $n \in \N$: :A collection $\mathcal F_n$ of $n$-ary function symbols :A collection $\mathcal P_n$ of $n$-ary relation symbols It is often conceptually enlightening to explicitly address the $0$-ary function symbols separately, as constant symbols.
Definition:Variable/Also known as
When it occurs in a mathematical equation, a '''variable''' is often referred to as an '''unknown'''. In the specific context of elementary algebra, the ugly misnomer '''pronumeral''' is frequently found in Australia. This was introduced by extension of the concept of a {{WP|Pronoun|pronoun}}: a symbol that '''stands in''' for a '''numeral''', by which the term number is actually meant. Thankfully the term appears not to have caught on in general.
Definition:Bipartite Graph/Partite Set
Let $G = \left({A \mid B, E}\right)$ be a bipartite graph. Then $A$ and $B$ are called the '''partite sets''' of the bipartite graph.
Definition:Conjugation (Abstract Algebra)/Conjugate
Let $a \in A$. Then $C \left({a}\right)$ is called the '''conjugate''' of $a$.
Definition:Gamma Function/Partial
Let $m \in \Z_{\ge 0}$. The '''partial Gamma function at $m$''' is defined as: :$\displaystyle \Gamma_m \left({z}\right) := \frac {m^z m!} {z \left({z + 1}\right) \left({z + 2}\right) \cdots \left({z + m}\right)}$ which is valid except for $z \in \left\{{0, -1, -2, \ldots, -m}\right\}$.
Definition:Neighborhood (Real Analysis)/Open Subset
Let $N_\alpha$ be a subset of $\R$ which contains (as a subset) an open real set which itself contains (as an element) $\alpha$. Then $N_\alpha$ is a '''neighborhood''' of $\alpha$.
Definition:Neighborhood (Real Analysis)
Let $\alpha \in \R$ be a real number. === Open Subset Neighborhood === {{:Definition:Neighborhood (Real Analysis)/Open Subset}} === Epsilon-Neighborhood === {{:Definition:Neighborhood (Real Analysis)/Epsilon}}
Definition:Oscillation/Real Space/Oscillation at Point/Epsilon
The '''oscillation of $f$ at $x$''' is defined as: :$\displaystyle \omega_f \left({x}\right) := \inf \left\{{\omega_f \left({\left({x - \epsilon \,.\,.\, x + \epsilon}\right) \cap X}\right): \epsilon \in \R_{>0}}\right\}$ where $\omega_f \left({\left({x - \epsilon \,.\,.\, x + \epsilon}\right) \cap X}\right)$ denotes the oscillation of $f$ on $\left({x - \epsilon \,.\,.\, x + \epsilon}\right) \cap X$.
Definition:Subsignature
Let $\mathcal L, \mathcal L'$ be signatures for the language of predicate logic. Then $\mathcal L$ is said to be a '''subsignature''' of $\mathcal L'$''', denoted $\mathcal L \subseteq \mathcal L'$, , for each $n \in \N$: :$\mathcal F_n \left({\mathcal L}\right) \subseteq \mathcal F_n \left({\mathcal L'}\right)$ :$\mathcal P_n \left({\mathcal L}\right) \subseteq \mathcal P_n \left({\mathcal L'}\right)$ where $\mathcal F_n$ denotes the collection of $n$-ary function symbols, and $\mathcal P_n$ denotes the collection of $n$-ary predicate symbols. === Supersignature === {{:Definition:Subsignature/Supersignature}}
Definition:Subsignature/Supersignature
Let $\mathcal L$ be a subsignature of $\mathcal L'$. Then $\mathcal L'$ is said to be a '''supersignature''' of $\mathcal L$, denoted: :$\mathcal L' \supseteq \mathcal L$
Definition:Lower Semicontinuous/Subset
Let $A \subseteq S$, and $A \ne \varnothing$. The function $f$ is said to be '''lower semicontinuous on''' $A$  $f$ is lower semicontinuous at every $a \in A$.
Definition:Reduct of Structure
Let $\mathcal L, \mathcal L'$ be signatures of the language of predicate logic. Let $\mathcal L$ be a subsignature of $\mathcal L'$. Let $\mathcal A, \mathcal A'$ be structures for $\mathcal L, \mathcal L'$, respectively. Then $\mathcal A$ is called the '''reduct of $\mathcal A'$ to $\mathcal L$''' : :For all function symbols $f$ of $\mathcal L$, one has $f_{\mathcal A'} = f_{\mathcal A}$ :For all predicate symbols $p$ of $\mathcal L$, one has $p_{\mathcal A'} = p_{\mathcal A}$ where $f_{\mathcal A'}$ is the interpretation of the function symbol $f$ in the structure $\mathcal A'$. Symbolically, one may write $\mathcal A = \mathcal A' \restriction_{\mathcal L}$.
Definition:Expansion of Structure
Let $\mathcal L, \mathcal L'$ be signatures of the language of predicate logic. Let $\mathcal L'$ be a supersignature of $\mathcal L$. Let $\mathcal A, \mathcal A'$ be structures for $\mathcal L, \mathcal L'$, respectively. Then $\mathcal A'$ is called an '''expansion of $\mathcal A$ to $\mathcal L'$''' : :For all function symbols $f$ of $\mathcal L$, one has $f_{\mathcal A'} = f_{\mathcal A}$ :For all predicate symbols $p$ of $\mathcal L$, one has $p_{\mathcal A'} = p_{\mathcal A}$ where $f_{\mathcal A'}$ is the interpretation of the function symbol $f$ in the structure $\mathcal A'$.
Definition:Meet Irreducible
Let $\left({S, \wedge, \preceq}\right)$ be a meet semilattice. Let $g \in S$. Then $g$ is '''meet irreducible''' : :$\forall x, y \in S: g = x \wedge y \implies g = x$ or $g = y$
Definition:Stencil
A quintuple $\tuple {X, S_X, \map S 0, \Delta, \delta}$ defines a '''stencil''' : {{begin-axiom}} {{axiom | n = 1         | lc= Index Domain:         | t = $X$ is discrete: $X \subseteq \Z^{\size X}$ and of finite dimension: $\size X \in \N$ }} {{axiom | n = 2         | lc= State Range:         | t = $S_X$ is a well-defined set }} {{axiom | n = 3         | lc= Initial State:         | t = $\map S 0$ maps from index space to state space $\map S 0: X \to S_X$ }} {{axiom | n = 4         | lc= Neighbourhood Delta:         | t = $\Delta$ is a vector of '''index offsets''': $\Delta \in \paren {\Z^{\size X} }^{\size \Delta}$ }} {{axiom | n = 5         | lc= Transition Combinator:         | t = $\delta$ is a mapping $\delta: S_X^{\size \Delta} \to S_X$ from neighbourhood states to states }} {{end-axiom}} {{refactor|Put update sweep and stencil evolution into their own pages}} By means of the '''update sweep''' $\forall \vec x \in X: \map {\map S {n + 1} } {\vec x} = \displaystyle \map \delta {\prod_{i \mathop = 1}^{\size \Delta} \map {\map S n} {\vec x + \Delta_i} }$ this induces a '''stencil evolution''' $\N \ni n \mapsto \map S n \in \paren {X \to S_X}$ from the '''initial state''' $\map S 0$.
Definition:Conway Life
A cellular evolution deriving from a '''stencil''' shaped as: :$(1): \quad X := \Z^2$ :$(2): \quad S_X := 2 := \set {0, 1}$ :$(3): \quad \map S 0 \in \paren {\Z^2 \to 2}$ :$(4): \quad \Delta := \tuple {\tuple {0, 0}, \tuple {0, 1}, \tuple {1, 1}, \tuple {1, 0}, \tuple {1, -1}, \tuple {0, -1}, \tuple {-1, -1}, \tuple {-1, 0}, \tuple {-1, 1} }$ :$(5): \quad \map \delta {m, n_1, n_2, n_3, n_4, n_5, n_6, n_7, n_8} := \tuple {m, \displaystyle \sum_{i \mathop = 1}^8 n_i} \in \set {\tuple {0, 3}, \tuple {1, 2}, \tuple {1, 3} } ? 1 : 0$ is called a '''Conway life'''.
Definition:Legendre Symbol/Definition 1
The '''Legendre symbol''' $\left({\dfrac a p}\right)$ is defined as: :$\left({\dfrac a p}\right) := \begin{cases} +1 & : a^{\frac{(p - 1)} 2} \bmod p = 1 \\ 0 & : a^{\frac{(p - 1)} 2} \bmod p = 0 \\ -1 & : a^{\frac{(p - 1)} 2} \bmod p = p - 1 \end{cases}$
Definition:Legendre Symbol/Definition 2
The '''Legendre symbol''' $\left({\dfrac a p}\right)$ is defined as: :$\left({\dfrac a p}\right) := a^{\frac{(p - 1)} 2} \bmod p$
Definition:Termial
Let $n \in \Z_{> 0}$ be a positive integer. The '''termial''' of $n$ is denoted $n?$ and defined as: :$\displaystyle n? = \sum_{k \mathop = 1}^n k = 1 + 2 + \cdots + n$
Definition:Way Below Open
Let $\left({S, \preceq}\right)$ a preordered set. Let $X$ be a subset of $S$. Then $X$ is '''way below open''' : :$\forall x \in X: \exists y \in X: y \ll x$ where $\ll$ denotes the way below relation.
Definition:Way Above Closure
Let $\left({S, \preceq}\right)$ be an ordered set. Let $x \in S$. The '''way above closure''' of $x$, denoted by $x^\gg$, is defined by: :$x^\gg := \left\{ {y \in S: x \ll y}\right\}$ where $\ll$ denotes the way below relation.
Definition:Termial/Real Numbers
Let $x \in \R$ be a real number. The '''termial''' of $x$ is denoted $x?$ and defined as: :$\displaystyle x? = \dfrac {x \left({x + 1}\right)} 2$
Definition:Operation/Binary Operation/Product/Right
Let $x$ and $y$ be elements which are operated on by a given operation $\circ$. The '''right-hand product of $x$ by $y$''' is the product $x \circ y$.
Definition:Operation/Binary Operation/Product/Left
Let $x$ and $y$ be elements which are operated on by a given operation $\circ$. The '''left-hand product of $x$ by $y$''' is the product $y \circ x$.
Definition:Special Orthogonal Group
Let $k$ be a field. The '''special ($n$th) orthogonal group (on $k$)''', denoted $\map {\mathrm {SO} } {n, k}$, is: :the set of all proper orthogonal order-$n$ square matrices over $k$ :under (conventional) matrix multiplication. That is: $\map {\mathrm {SO} } {n, k} = \map {\mathrm O} {n, k} \cap \SL {n, k}$
Definition:Finite Semigroup
A '''finite semigroup''' is a semigroup of finite order. That is, a semigroup $\struct {S, \circ}$ is a '''finite semigroup'''  its underlying set $G$ is finite. That is, a '''finite semigroup''' is a semigroup with a finite number of elements.
Definition:Inverse Row form of Cayley Table for Group
Let $\struct {G, \circ}$ be a finite group. The Cayley table for $\struct {G, \circ}$ can be presented in a form such that the rows are headed by the inverse elements of the elements which head the corresponding columns. This form is known as the '''inverse row form''' of the Cayley table for $\struct {G, \circ}$.
Definition:Group Generated by Reciprocal of z and 1 minus z
Let: :$S = \set {f_1, f_2, f_3, f_4, f_5, f_6}$ where $f_1, f_2, \ldots, f_6$ are complex functions defined for all $z \in \C \setminus \set {0, 1}$ as: {{begin-eqn}} {{eqn | l = \map {f_1} z       | r = z }} {{eqn | l = \map {f_2} z       | r = \dfrac 1 {1 - z} }} {{eqn | l = \map {f_3} z       | r = \dfrac {z - 1} z }} {{eqn | l = \map {f_4} z       | r = \dfrac 1 z }} {{eqn | l = \map {f_5} z       | r = 1 - z }} {{eqn | l = \map {f_6} z       | r = \dfrac z {z - 1} }} {{end-eqn}} Let $\circ$ denote composition of functions. Then $\struct {S, \circ}$ is the '''group generated by $\dfrac 1 z$ and $1 - z$'''.
Definition:Coprime Residue Class
Let $m \in \Z: m \ge 1$. Let $a \in \Z$ such that: :$a \perp m$ where $\perp$ denotes that $a$ is prime to $m$. Let $\eqclass a m$ be the residue class of $a$ (modulo $m$): :$\set {x \in \Z: \exists k \in \Z: x = a + k m}$ Then $\eqclass a m$ is referred to as a '''coprime residue class'''.
Definition:Reduced Residue System
The '''reduced residue system modulo $m$''', denoted $\Z'_m$, is the set of all residue classes of $k$ (modulo $m$) which are prime to $m$: :$\Z'_m = \set {\eqclass k m \in \Z_m: k \perp m}$ Thus $\Z'_m$ is the '''set of all coprime residue classes modulo $m$''': :$\Z'_m = \set {\eqclass {a_1} m, \eqclass {a_2} m, \ldots, \eqclass {a_{\map \phi m} } m}$ where: :$\forall k: a_k \perp m$ :$\map \phi m$ denotes the Euler phi function of $m$.
Definition:Set of Residue Classes/Real Modulus
The quotient set of congruence modulo $z$ denoted $\R_z$ is: :$\R_z = \dfrac \R {\mathcal R_z}$ Thus $\R_z$ is the '''set of all residue classes modulo $z$'''. It follows from the Fundamental Theorem on Equivalence Relations that the quotient set $\R_z$ of congruence modulo $z$ forms a partition of $\R$.
Definition:Order Generating
Let $\left({S, \preceq}\right)$ be a preordered set. Let $X$ be a subset of $S$. Then $X$ is '''order generating'''  :$\forall x \in S: x^\succeq \cap X$ admits an infimum and $x = \inf \left({x^\succeq \cap X}\right)$
Definition:Finite Group/Axioms
A finite group is an algebraic structure $\struct {G, \circ}$ which satisfies the following four conditions: {{begin-axiom}} {{axiom | n = \text {FG} 0         | lc= Closure         | q = \forall a, b \in G         | m = a \circ b \in G }} {{axiom | n = \text {FG} 1         | lc= Associativity         | q = \forall a, b, c \in G         | m = a \circ \paren {b \circ c} = \paren {a \circ b} \circ c }} {{axiom | n = \text {FG} 2         | lc= Finiteness         | q = \exists n \in \N         | m = \order G = n }} {{axiom | n = \text {FG} 3         | lc= Cancellability         | q = \forall a, b, c \in G         | m = c \circ a = c \circ b \implies a = b }} {{axiom | m = a \circ c = b \circ c \implies a = b }} {{end-axiom}} These four stipulations are called the '''finite group axioms'''.
Definition:Group of Rotation Matrices Order 4
Consider the algebraic structure $S$ of rotation matrices: :$R_4 = \set {\begin {bmatrix} 1 & 0 \\ 0 & 1 \end {bmatrix}, \begin {bmatrix} 0 & 1 \\ -1 & 0 \end {bmatrix}, \begin {bmatrix} -1 & 0 \\ 0 & -1 \end {bmatrix}, \begin {bmatrix} 0 & -1 \\ 1 & 0 \end {bmatrix} }$ under the operation of (conventional) matrix multiplication. $R_4$ is the '''group of rotation matrices of order $4$'''.
Definition:Additive Group of Integers Modulo 4
The '''additive group of integers modulo $4$''' $\struct {\Z_4, +_4}$ is the set of integers modulo $4$ under the operation of addition modulo $4$.
Definition:Dimension of Vector Space/Vector
Informally, an element of an $n$-dimensional vector space is often referred to as an '''$n$-dimensional vector'''.  It must be understood that this is no more than a convenient shorthand. It is not the ''vector'' which possesses the dimensionality, but the ''space in which it is embedded''.
Definition:Complex Root
Let $z \in \C$ be a complex number such that $z \ne 0$. Let $n \in \Z_{>0}$ be a (strictly) positive integer. Let $w \in \C$ such that: :$w^n = z$ Then $w$ is a '''(complex) $n$th root of $z$''', and we can write: :$w = z^{1 / n}$
Definition:Root of Unity/Complex/Primitive
A '''primitive (complex) $n$th root of unity''' is an element $\alpha \in U_n$ such that: :$U_n = \set {1, \alpha, \alpha^2, \ldots, \alpha^{n - 1} }$
Definition:Prime Element (Order Theory)
Let $\struct {S, \wedge, \preceq}$ be a meet semilattice. Let $p \in S$. Then $p$ is '''a prime element (of $\struct {S, \wedge, \preceq}$)''' : :$\forall x, y \in S: \paren {x \wedge y \preceq p \implies x \preceq p \text { or } y \preceq p}$
Definition:Relation/Relation as Subset of Cartesian Product
Most treatments of set theory and relation theory define a '''relation on $S \times T$''' to refer to just the truth set itself: :$\RR \subseteq S \times T$ where: :$S \times T$ is the Cartesian product of $S$ and $T$. Thus under this treatment, $\RR$ is a set of ordered pairs, the first coordinate from $S$ and the second coordinate from $T$. This approach leaves the precise nature of $S$ and $T$ undefined.
Definition:Airy's Equation
'''Airy's equation''' is the second order ODE: :$y'' = x y$
Definition:Game
In the context of game theory, a '''game''' is any social situation which involves more than one person. More specifically, a '''game''' is a problem of strategy between two or more parties in which all parties involved seek (usually) to maximise their outcome. A '''game''' $G$ can be specified by: :its players :the moves available to each player :the preference relation for each player: that is: :$G = \sequence {N, \sequence {A_i}, \sequence {\succsim_i} }$ $G$ can also be specified by: :its players :the moves available to each player :the payoff function for each player: that is: :$G = \sequence {N, \sequence {A_i}, \sequence {u_i} }$
Definition:Game/Player
Each of the parties involved in a '''game''' are called '''players'''.
Definition:Game/Strategy
A '''strategy''' is a complete plan of action that defines what a '''player''' will do under all circumstances in a  '''game'''.
Definition:Game/Payoff
The '''payoff''' of a '''game''' is the reward or punishment made to a '''player''' at the end of the '''game''' as a result of combination of the various strategies employed.
Definition:Game of Perfect Information
A '''game of perfect information''' is a game in which all players have complete information on the state of the game.
Definition:Two-Person Game
A '''two-person game''' is a game in which there are (exactly) two players.
Definition:Zero-Sum Game
A '''zero-sum game''' is a game in which the total of the payoffs to all players is zero.
Definition:Equilibrium Point
An '''equilibrium point''' is a stable outcome of a game associated with a particular set of strategies.
Definition:Zero-Sum Game/Examples
Examples of zero-sum game include: * [https://en.wikipedia.org/wiki/Poker Poker]: the total amount of money on the table is fixed -- what one person loses the other players win.
Definition:Stable Outcome
An '''stable outcome''' is an end point of a game such that if a players were to use a new strategy, that player would experience a lower payoff.
Definition:Two-Person Zero-Sum Game
A '''two-person zero-sum game''' is a two-person game with zero sum.
Definition:Equilibrium Strategies
A system of strategies, one for each player, is '''in equilibrium'''  they result in an equilibrium point. Such strategies are known as '''equilibrium strategies'''. In the singular, an '''equilibrium strategy''' is one that contributes to an equilibrium point.
Definition:Linear Programming
'''Linear programming''' is the branch of mathematical programming which studies optimization of mathematical models whose requirements are represented by linear relationships.
Definition:Matching Pennies
There are two players: $A$ and $B$. Each player puts down a coin, traditionally a [https://en.wikipedia.org/wiki/Penny penny], either head or tail up, without showing it to the other player. The coins are then uncovered. If they both show the same side, $A$ is deemed to have won, and he takes both coins. If they show different sides, $B$ is deemed to have won, and he takes both coins.