hcju/proofwikips huggingface dataset:
Definition:Matrix/Column
Let $\mathbf A$ be an $m \times n$ matrix. For each $j \in \closedint 1 n$, the '''columns''' of $\mathbf A$ are the ordered $m$-tuples: : $c_j = \tuple {a_{1 j}, a_{2 j}, \ldots, a_{m j} }$ where $c_j$ is called the '''$j$th column of $\mathbf A$'''. A '''column''' of an $m \times n$ matrix can also be treated as a $m \times 1$ column matrix in its own right: :$c_j = \begin {bmatrix} a_{1 j} \\ a_{2 j} \\ \vdots \\ a_{m j} \end {bmatrix}$ for $j = 1, 2, \ldots, n$.
Definition:Matrix/Element
Let $\mathbf A$ be an $m \times n$ matrix over a set $S$. The individual $m \times n$ elements of $S$ that go to form $\mathbf A = \sqbrk a_{m n}$ are known as the '''elements of the matrix'''. The '''element''' at row $i$ and column $j$ is called '''element $\tuple {i, j}$ of $\mathbf A$''', and can be written $a_{i j}$, or $a_{i, j}$ if $i$ and $j$ are of more than one character. If the indices are still more complicated coefficients and further clarity is required, then the form $a \tuple {i, j}$ can be used. Note that the first subscript determines the row, and the second the column, of the matrix where the '''element''' is positioned.
Definition:Matrix/Square Matrix
An $n \times n$ matrix is called a '''square matrix'''. That is, a '''square matrix''' is a matrix which has the same number of rows as it has columns. A '''square matrix''' is usually denoted $\sqbrk a_n$ in preference to $\sqbrk a_{n n}$. In contrast, a non-'''square matrix''' can be referred to as a '''rectangular matrix'''.
Definition:Matrix/Diagonal Elements
Let $\mathbf A = \sqbrk a_{m n}$ be a matrix. The elements $a_{j j}: j \in \closedint 1 {\min \set {m, n} }$ constitute the '''main diagonal''' of the matrix. The elements themselves are called the '''diagonal elements'''.
Definition:Matrix/Underlying Structure
Let $\mathbf A$ be a matrix over a set $S$. The set $S$ can be referred to as the '''underlying set of $\mathbf A$'''. In the context of matrices, however, it is usual for $S$ itself to be the underlying set of an algebraic structure in its own right. If this is the case, then the structure $\struct {S, \circ_1, \circ_2, \ldots, \circ_n}$ (which may also be an ordered structure) can be referred to as the '''underlying structure of $\mathbf A$'''. When the '''underlying structure''' is not specified, it is taken for granted that it is one of the standard number systems, usually the real numbers $\R$.
Definition:Matrix/Zero Row or Column
Let $\mathbf A = \sqbrk a_{m n}$ be an $m \times n$ matrix whose underlying structure is a ring or field (usually numbers). If a row or column of $\mathbf A$ contains only zeroes, then it is a '''zero row''' or a '''zero column'''.
Definition:Matrix/Indices
Let $\mathbf A$ be an $m \times n$ matrix. Let $a_{i j}$ be the element in row $i$ and column $j$ of $\mathbf A$. Then the subscripts $i$ and $j$ are referred to as the '''indices''' (singular: '''index''') of $a_{i j}$.
Definition:Module Automorphism
Let $\left({G, +_G, \circ}\right)_R$ be an $R$-module. Let $\phi: G \to G$ be a module isomorphism to itself. Then $\phi$ is a '''module automorphism'''.
Definition:Vector Space Automorphism
Let $V$ be a $K$-vector space. Let $\phi: V \to V$ be a vector space isomorphism to itself. Then $\phi$ is a '''vector space automorphism'''.
Definition:Pointwise Maximum of Mappings/Extended Real-Valued Functions
Let $X$ be a set, and let $f, g: X \to \overline{\R}$ be extended real-valued functions. Let $\max$ be the max operation on $\overline{\R}$ (Ordering on Extended Real Numbers is Total Ordering ensures it is in fact defined). Then the '''pointwise maximum of $f$ and $g$''', denoted $\max \left({f, g}\right)$, is defined by: :$\max \left({f, g}\right): X \to \overline{\R}: \max \left({f, g}\right) \, \left({x}\right) := \max \left({f \left({x}\right), g \left({x}\right)}\right)$ '''Pointwise maximum''' thence is an instance of a pointwise operation on extended real-valued functions. Since the ordering on $\overline{\R}$ coincides on $\R$ with the standard ordering, this definition incorporates the definition for real-valued functions.
Definition:Pointwise Maximum of Mappings/Real-Valued Functions
Let $S$ be a set. Let $f, g: S \to \R$ be real-valued functions. Let $\max$ be the max operation on $\R$ (Ordering on Real Numbers is Total Ordering ensures it is in fact defined). Then the '''pointwise maximum of $f$ and $g$''', denoted $\map \max {f, g}$, is defined by: :$\map \max {f, g}: S \to \R: \map {\map \max {f, g} } x := \map \max {\map f x, \map g x}$ '''Pointwise maximum''' thence is an instance of a pointwise operation on real-valued functions.
Definition:Non-Invertible Matrix
{{:Definition:Non-Invertible Matrix/Definition 1}}
Definition:Pointwise Maximum of Mappings
Let $X$ be a set. Let $\struct {S, \preceq}$ be a toset. Let $f, g: X \to S$ be mappings. Let $\max$ be the max operation on $\struct {S, \preceq}$. Then the '''pointwise maximum of $f$ and $g$''', denoted $\map \max {f, g}$, is defined by: :$\map \max {f, g}: X \to S: \map {\map \max {f, g} } x := \map \max {\map f x, \map g x}$ Hence '''pointwise maximum''' is an instance of a pointwise operation on mappings.
Definition:Pointwise Minimum of Mappings
Let $X$ be a set. Let $\struct {S, \preceq}$ be a toset. Let $f, g: X \to S$ be mappings. Let $\min$ be the min operation on $\struct {S, \preceq}$. Then the '''pointwise minimum of $f$ and $g$''', denoted $\map \min {f, g}$, is defined by: :$\map \min {f, g}: X \to S: \map {\map \min {f, g} } x := \map \min {\map f x, \map g x}$ Hence '''pointwise minimum''' is an instance of a pointwise operation on mappings.
Definition:Pointwise Minimum of Mappings/Real-Valued Functions
Let $X$ be a set, and let $f, g: X \to \R$ be real-valued functions. Let $\min$ be the min operation on $\R$ (Ordering on Real Numbers is Total Ordering ensures it is in fact defined). Then the '''pointwise minimum of $f$ and $g$''', denoted $\min \left({f, g}\right)$, is defined by: :$\min \left({f, g}\right): X \to \R: \min \left({f, g}\right) \, \left({x}\right) := \min \left({f \left({x}\right), g \left({x}\right)}\right)$ '''Pointwise minimum''' thence is an instance of a pointwise operation on real-valued functions.
Definition:Pointwise Minimum of Mappings/Extended Real-Valued Functions
Let $X$ be a set, and let $f, g: X \to \overline{\R}$ be extended real-valued functions. Let $\min$ be the min operation on $\overline{\R}$ (Ordering on Extended Real Numbers is Total Ordering ensures it is in fact defined). Then the '''pointwise minimum of $f$ and $g$''', denoted $\min \left({f, g}\right)$, is defined by: :$\min \left({f, g}\right): X \to \overline{\R}: \min \left({f, g}\right) \, \left({x}\right) := \min \left({f \left({x}\right), g \left({x}\right)}\right)$ '''Pointwise minimum''' thence is an instance of a pointwise operation on extended real-valued functions. Since the ordering on $\overline{\R}$ coincides on $\R$ with the standard ordering, this definition incorporates the definition for real-valued functions.
Definition:Simultaneous Equations/Linear Equations
A '''system of simultaneous linear equations''' is a set of equations: :$\displaystyle \forall i \in \set {1, 2, \ldots, m} : \sum_{j \mathop = 1}^n \alpha_{i j} x_j = \beta_i$ That is: {{begin-eqn}} {{eqn | l = \beta_1       | r = \alpha_{1 1} x_1 + \alpha_{1 2} x_2 + \cdots + \alpha_{1 n} x_n }} {{eqn | l = \beta_2       | r = \alpha_{2 1} x_1 + \alpha_{2 2} x_2 + \cdots + \alpha_{2 n} x_n }} {{eqn | o = \cdots}} {{eqn | l = \beta_m       | r = \alpha_{m 1} x_1 + \alpha_{m 2} x_2 + \cdots + \alpha_{m n} x_n }} {{end-eqn}}
Definition:Set of Residue Classes
The quotient set of congruence modulo $m$ denoted $\Z_m$ is: :$\Z_m = \dfrac \Z {\RR_m}$
Definition:Pointwise Scalar Multiplication of Extended Real-Valued Functions
Let $S$ be a non-empty set, and let ${\overline \R}^S$ be the set of all mappings $f: S \to \overline \R$. Here $\overline \R$ denotes the extended set of real numbers. Then '''pointwise ($\overline \R$-)scalar multiplication''' on ${\overline \R}^S$ is the binary operation $\cdot: \overline \R \times {\overline \R}^S \to {\overline \R}^S$ defined by: :$\forall \lambda \in \overline \R: \forall f \in {\overline \R}^S: \forall s \in S: \map {\paren {\lambda \cdot f} s := \lambda \cdot \map f s$ where the $\cdot$ on the right is extended real multiplication. '''Pointwise scalar multiplication''' thence is an instance of a pointwise operation on extended real-valued functions.
Definition:Addition/Real Numbers
The addition operation in the domain of real numbers $\R$ is written $+$. From the definition, the real numbers are the set of all equivalence classes $\eqclass {\sequence {x_n} } {}$ of Cauchy sequences of rational numbers. Let $x = \eqclass {\sequence {x_n} } {}, y = \eqclass {\sequence {y_n} } {}$, where $\eqclass {\sequence {x_n} } {}$ and $\eqclass {\sequence {y_n} } {}$ are such equivalence classes. Then $x + y$ is defined as: :$\eqclass {\sequence {x_n} } {} + \eqclass {\sequence {y_n} } {} = \eqclass {\sequence {x_n + y_n} } {}$
Definition:Multiplication/Real Numbers
The multiplication operation in the domain of real numbers $\R$ is written $\times$. From the definition, the real numbers are the set of all equivalence classes $\eqclass {\sequence {x_n} } {}$ of Cauchy sequences of rational numbers. Let $x = \eqclass {\sequence {x_n} } {}, y = \eqclass {\sequence {y_n} } {}$, where $\eqclass {\sequence {x_n} } {}$ and $\eqclass {\sequence {y_n} } {}$ are such equivalence classes. Then $x \times y$ is defined as: : $\eqclass {\sequence {x_n} } {} \times \eqclass {\sequence {y_n} } {} = \eqclass {\sequence {x_n \times y_n} } {}$
Definition:Ring (Abstract Algebra)/Addition
The distributand $*$ of a ring $\struct {R, *, \circ}$ is referred to as '''ring addition''', or just '''addition'''. The conventional symbol for this operation is $+$, and thus a general ring is usually denoted $\struct {R, +, \circ}$.
Definition:Rig
A '''rig''' is an additive semiring $\struct {S, *, \circ}$ in which $\struct {S, *}$ is a monoid. Alternatively, this is a semiring in which $\struct {S, *}$ is a commutative monoid. That is, $\struct {S, *, \circ}$ has the following properties: {{begin-axiom}} {{axiom | n = \text A 0         | q = \forall a, b \in S         | m = a * b \in S         | lc= Closure under $*$ }} {{axiom | n = \text A 1         | q = \forall a, b, c \in S         | m = \paren {a * b} * c = a * \paren {b * c}         | lc= Associativity of $*$ }} {{axiom | n = \text A 2         | q = \forall a, b \in S         | m = a * b = b * a         | lc= Commutativity of $*$ }} {{axiom | n = \text A 3         | q = \exists 0_S \in S: \forall a \in F         | m = a * 0_S = a = 0_S * a         | lc= Identity element for $*$: the zero }} {{axiom | n = \text M 0         | q = \forall a, b \in S         | m = a \circ b \in S         | lc= Closure under $\circ$ }} {{axiom | n = \text M 1         | q = \forall a, b, c \in S         | m = \paren {a \circ b} \circ c = a \circ \paren {b \circ c}         | lc= Associativity of $\circ$ }} {{axiom | n = \text M 2         | q = \forall a \in S         | m = a \circ 0_S = 0_S = 0_S \circ a         | lc= The zero is a zero element for $\circ$ }} {{axiom | n = \text D         | q = \forall a, b, c \in S         | m = a \circ \paren {b * c} = \paren {a \circ b} * \paren {a \circ c}, \paren {a * b} \circ c = \paren {a \circ c} * \paren {a \circ c}         | lc= $\circ$ is distributive over $*$ }} {{end-axiom}} Note that the zero element needs to be specified here as an axiom: $\text M 2$. By Ring Product with Zero, in a ring, the property $\text M 2$ of the zero element follows as a consequence of the ring axioms.
Definition:Congruence (Number Theory)/Modulo Operation
Let $\bmod$ be defined as the modulo operation: :$x \bmod y := \begin{cases} x - y \left \lfloor {\dfrac x y}\right \rfloor & : y \ne 0 \\ x & : y = 0 \end{cases}$ Then '''congruence modulo $z$''' is the relation on $\R$ defined as: :$\forall x, y \in \R: x \equiv y \pmod z \iff x \bmod z = y \bmod z$
Definition:Congruence (Number Theory)/Integer Multiple
Let $x, y \in \R$. Then '''$x$ is congruent to $y$ modulo $z$'''  their difference is an integer multiple of $z$: :$x \equiv y \pmod z \iff \exists k \in \Z: x - y = k z$
Definition:Congruence (Number Theory)/Integers
Let $m \in \Z_{> 0}$. === Definition by Remainder after Division === {{:Definition:Congruence (Number Theory)/Integers/Remainder after Division}} === Definition by Modulo Operation === {{:Definition:Congruence (Number Theory)/Integers/Modulo Operation}} === Definition by Integer Multiple === We also see that $a$ is congruent to $b$ modulo $m$ if their difference is a multiple of $m$: {{:Definition:Congruence (Number Theory)/Integers/Integer Multiple}}
Definition:Congruence (Number Theory)/Residue
Let $a, b \in \Z$. Let $a \equiv b \pmod m$. Then $b$ is a '''residue of $a$ modulo $m$'''. '''Residue''' is another word meaning remainder, and is ''any'' integer congruent to $a$ modulo $m$.
Definition:Congruence (Number Theory)/Integers/Modulo Operation
Let $\bmod$ be defined as the modulo operation: :$x \bmod m := \begin{cases} x - m \left \lfloor {\dfrac x m}\right \rfloor & : m \ne 0 \\ x & : m = 0 \end{cases}$ Then '''congruence modulo $m$''' is the relation on $\Z$ defined as: :$\forall x, y \in \Z: x \equiv y \pmod m \iff x \bmod m = y \bmod m$
Definition:Congruence (Number Theory)/Integers/Integer Multiple
Let $x, y \in \Z$. '''$x$ is congruent to $y$ modulo $m$'''  their difference is an integer multiple of $m$: :$x \equiv y \pmod m \iff \exists k \in \Z: x - y = k m$
Definition:Integral Multiple/Rings and Fields
Let $\struct {F, +, \times}$ be a ring or a field. Let $a \in F$. Let $n \in \Z$ be an integer. Then $n \cdot a$ is an integral multiple of $a$ where $n \cdot a$ is defined as in Powers of Ring Elements: :$n \cdot a := \begin {cases} 0_F & : n = 0 \\ \paren {\paren {n - 1} \cdot a} + a & : n > 1 \\ \size n \cdot \paren {-a} & : n < 0 \\ \end {cases}$ where $\size n$ is the absolute value of $n$.
Definition:Integral Multiple/Real Numbers
Let $x, y \in \R$ be real numbers. Then $x$ is an '''integral multiple''' of $y$  $x$ is congruent to $0$ modulo $y$: :$x \equiv 0 \pmod y$ That is: :$\exists k \in \Z: x = 0 + k y$
Definition:Trivial Annihilator
From Annihilator of Ring Always Contains Zero, we have that $0 \in \map {\mathrm {Ann} } R$ whatever the ring $R$ is. $R$ is said to have a '''trivial annihilator'''  its annihilator $\map {\mathrm {Ann} } R$ consists '''only''' of the integer $0$.
Definition:Pendant Vertex
Let $G$ be a graph. A vertex $v$ of $G$ is said to be a '''pendant vertex'''  it has degree $1$.
Definition:Pendant Edge
Let $G$ be a graph. An edge $e$ of $G$ is said to be a '''pendant edge'''  it is incident on a pendant vertex.
Definition:Parallel Edges
Let $G$ be a graph or a digraph. A pair of edges $e$ and $e'$ of $G$ are said to be a '''parallel'''  they are incident on precisely the same vertices.
Definition:Even Vertex (Graph Theory)
If the degree of $v$ is even, then $v$ is called an '''even vertex'''.
Definition:Odd Vertex (Graph Theory)
If the degree of $v$ is odd, then $v$ is an '''odd vertex'''.
Definition:Out-Degree
The '''out-degree of $v$ in $G$''' is the number of arcs which are incident from $v$. It is denoted $\map {\operatorname {outdeg}_G} v$, or just $\map {\operatorname {outdeg} } v$ if it is clear from the context which digraph is being referred to.
Definition:In-Degree
The '''in-degree of $v$ in $G$''' is the number of arcs which are incident to $v$.  It is denoted $\map {\operatorname {indeg}_G} v$, or just $\map {\operatorname {indeg} } v$ if it is clear from the context which digraph is being referred to.
Definition:Non-Empty Set
Let $S$ be a set. Then $S$ is said to be '''non-empty'''  $S$ has at least one element. By the Axiom of Extension, this may also be phrased as: :$S \ne \O$ where $\O$ denotes the empty set. Many mathematical theorems and definitions require sets to be '''non-empty''' in order to avoid erratic results and inconsistencies.
Definition:Left Cancellable Operation
The operation $\circ$ in $\struct {S, \circ}$ is '''left cancellable''' : :$\forall a, b, c \in S: a \circ b = a \circ c \implies b = c$ That is,  all elements of $\struct {S, \circ}$ are left cancellable.
Definition:Right Cancellable Operation
The operation $\circ$ in $\struct {S, \circ}$ is '''right cancellable''' : :$\forall a, b, c \in S: a \circ c = b \circ c \implies a = b$ That is,  all elements of $\struct {S, \circ}$ are right cancellable.
Definition:Cancellable Operation
Let $\left ({S, \circ}\right)$ be an algebraic structure. === Left Cancellable Operation === {{:Definition:Left Cancellable Operation}} === Right Cancellable Operation === {{:Definition:Right Cancellable Operation}} === Cancellable Operation === The operation $\circ$ in $\struct {S, \circ}$ is '''cancellable''' : * $\forall a, b, c \in S: a \circ b = a \circ c \implies b = c$ * $\forall a, b, c \in S: a \circ c = b \circ c \implies a = b$ That is,  it is both a left cancellable operation and a right cancellable operation.
Definition:Algebraic System
An '''algebraic system''' is a mathematical system $\SS = \struct {E, O}$ where: :$E$ is a non-empty set of elements :$O$ is a set of finitary operations on $E$.
Definition:Successor Mapping on Natural Numbers
Let $\N$ be the set of natural numbers. Let $s: \N \to \N$ be the mapping defined as: :$s = \set {\tuple {x, y}: x \in \N, y = x + 1}$ Considering $\N$ defined as a Peano structure, this is seen to be an instance of a successor mapping.
Definition:Formal Grammar/Top-Down
A '''top-down grammar''' for $\mathcal L$ is a formal grammar which allows well-formed formulas to be built from a single metasymbol. Such a grammar can be made explicit by declaring that: * A metasymbol may be replaced by a letter of $\mathcal A$. * A metasymbol may be replaced by certain collations labeled with metasymbols and signs of $\mathcal A$. From the words thus generated, those not containing any metasymbols are the well-formed formulas.
Definition:Formal Grammar/Bottom-Up
Let $\mathcal L$ be a formal language whose alphabet is $\mathcal A$. <section notitle="true" name="tc"> A '''bottom-up grammar''' for $\mathcal L$ is a formal grammar whose rules of formation allow the user to build well-formed formulas from primitive symbols, in the following way: * Letters are well-formed formulas. * A collection of specified collations of well-formed formulas, possibly labeled with additional signs, are also well-formed formulas. In certain use cases, the first clause is adjusted to allow for more complex situations, for example in the bottom-up specification of predicate logic {{transclude:Definition:Formal Grammar/Bottom-Up/Extremal Clause   |section = tc   |title = Extremal Clause   |header = 3   |link = true }} </section>
Definition:Finite Extended Real Number
An extended real number is defined as '''finite'''  it is a real number.
Definition:Infinite
=== Infinite Cardinal === {{:Definition:Infinite Cardinal}} === Infinite Set === {{:Definition:Infinite Set}} === Infinity === {{:Definition:Infinity}}
Definition:Cross-Relation
Let $\struct {S, \circ}$ be a commutative semigroup. Let $\struct {S_1, \circ_{\restriction_1} }, \struct {S_2, \circ_{\restriction_2} }$ be subsemigroups of $S$, where $\circ_{\restriction_1}$ and $\circ_{\restriction_2}$ are the restrictions of $\circ$ to $S_1$ and $S_2$ respectively. Let $\struct {S_1 \times S_2, \oplus}$ be the (external) direct product of $\struct {S_1, \circ_{\restriction_1} }$ and $\struct {S_2, \circ_{\restriction_2} }$, where $\oplus$ is the operation on $S_1 \times S_2$ induced by $\circ_{\restriction_1}$ on $S_1$ and $\circ_{\restriction_2}$ on $S_2$. Let $\boxtimes$ be the relation on $S_1 \times S_2$ defined as: :$\tuple {x_1, y_1} \boxtimes \tuple {x_2, y_2} \iff x_1 \circ y_2 = x_2 \circ y_1$ This relation $\boxtimes$ is referred to as the '''cross-relation on $\struct {S_1 \times S_2, \oplus}$'''.
Definition:Additive Semiring
An '''additive semiring''' is a semiring with a commutative distributand. That is, an '''additive semiring''' is a ringoid $\left({S, *, \circ}\right)$ in which: : $(1): \quad \left({S, *}\right)$ forms a commutative semigroup : $(2): \quad \left({S, \circ}\right)$ forms a semigroup. === Additive Semiring Axioms === {{:Definition:Additive Semiring/Axioms}}
Definition:A Fortiori
'''A fortiori''' knowledge arises from stronger facts already established. For example, in: :''I am too old for this game; therefore, my father is also too old.'' the knowledge: :''My father is too old for this game.'' arises '''a fortiori''' from the stronger knowledge that: :''I am too old for this game.'' An '''a fortiori''' argument is most commonly used by applying a general fact in a particular case.
Definition:Rule of Formation
Let $\mathcal F$ be a formal language whose alphabet is $\mathcal A$. The '''rules of formation''' of $\mathcal F$ are the rules which define how to construct collations in $\mathcal A$ which are well-formed. That is, the '''rules of formation''' tell you how to build collations featuring symbols from the alphabet $\mathcal A$ which are part of the formal language $\mathcal F$. The '''rules of formation''' of a formal language together constitute its formal grammar. There are no strict guidelines on how a '''rule of formation''' should look like, since they are employed to ''produce'' such strict guidelines. Thus, these '''rules of formation''' are often phrased in natural language, but their exact form is to some extent arbitrary.
Definition:Cross-Relation on Natural Numbers
Consider the commutative semigroup $\left({\N, +}\right)$ composed of the natural numbers $\N$ and addition $+$. Let $\left({\N \times \N, \oplus}\right)$ be the (external) direct product of $\left({\N, +}\right)$ with itself, where $\oplus$ is the operation on $\N \times \N$ induced by $+$ on $\N$. Let $\boxtimes$ be the relation on $\N \times \N$ defined as: :$\left({x_1, y_1}\right) \boxtimes \left({x_2, y_2}\right) \iff x_1 + y_2 = x_2 + y_1$ This relation $\boxtimes$ is referred to as the '''cross-relation on $\left({\N \times \N, \oplus}\right)$'''.
Definition:Definitional Abbreviation
When discussing a formal language, some particular WFFs may occur very often. If such WFFs are very unwieldy to write and obscure what the author tries to express, it is convenient to introduce a shorthand for them. Such a shorthand is called a '''definitional abbreviation'''. It does ''not'' in any way alter the meaning or formal structure of a sentence, but is purely a method to keep expressions readable to human eyes.
Definition:Function Symbol
Let $\mathcal L$ be a formal language (for example, the language of predicate logic $\mathcal L_1$). A '''function symbol''' is a letter of $\mathcal L$ used to describe a function. The name '''function symbol''' is a gesture to the reader to make clear what such a symbol should (intuitively) represent in the formal language $\mathcal L$.
Definition:Alphabetic Substitution
Consider the (abbreviated) WFF $Q x: \mathbf C$. Let $y$ be another variable such that $y$ does not occur in $\mathbf C$. Let $\mathbf C'$ be the WFF resulting from replacing all free occurrences of $x$ in $\mathbf C$ with $y$. Then to all intents and purposes, the WFFs: :$Q x: \mathbf C$ :$Q y: \mathbf C'$ will have the same interpretation. Thus we may change the free occurrences of any variable for another variable symbol. This change is called '''alphabetic substitution'''.