hcju/proofwikips huggingface dataset:
Definition:Corollary
A '''corollary''' is a proof which is a direct result, or a direct application, of another proof. It can be considered as being a '''proof for free''' on the back of a proof which has been paid for with blood, sweat and tears.
Definition:Corresponding Conditional
Let $p_1, p_2, p_3, \ldots, p_n \vdash q$ be a sequent of propositional logic. It can be expressed as a theorem as follows: : $\vdash p_1 \implies \left({p_2 \implies \left({p_3 \implies \left({\ldots \implies \left({p_n \implies q}\right) \ldots }\right)}\right)}\right)$ This is known as the sequent's '''corresponding conditional'''. This is proved in the Extended Rule of Implication. {{wtd|This pertains probably to the classical interpretation of propositional calculus, and needs to be reformulated and put in the correct category in due time}}  fpshpqhmzdo2bsytg3qk4e0pvlsyucd
Definition:By Hypothesis
When demonstrating a proof, it is frequently necessary to refer to the specification of the result which is to be proved. This result often contains wording along the lines '''Suppose that ...''' or '''Let ...''' It is convenient to refer back to these specifications during the course of the proof. To do that, the term '''by hypothesis''' is often used.
Definition:Proper Name
A '''proper name''' (or just '''name''') is a symbol or collection of symbols used to identify a ''particular'' object uniquely. In contrast with natural language, a proper name has a wider range than being the particular identifying label attached to a particular single entity (be it a person, or a place, or whatever else). For example: : '''Sloth''' is a '''proper name''' for ''the concept'' of ''being lazy''. : '''Rain''' is a '''proper name''' for ''the meteorological phenomenon of water falling from the sky''.
Definition:Property
A '''property''' is a concept that specifies an aspect of an object. In the phrase ''living thing'', ''living'' is a property that ''thing'' has. It creates a distinction between ''things'' that are ''living'' and ''things'' that are ''not living''.
Definition:Abelian Group
=== Definition 1 === {{:Definition:Abelian Group/Definition 1}} === Definition 2 === {{:Definition:Abelian Group/Definition 2}}
Definition:Commutative
=== Commuting Elements === {{:Definition:Commutative/Elements}} === Commutative Operation === {{:Definition:Commutative/Operation}} === Commuting Set of Elements === {{:Definition:Commutative/Set}} === Commutative Algebraic Structure === {{:Definition:Commutative/Algebraic Structure}}
Definition:Anticommutative
=== Structure with One Operation === {{:Definition:Anticommutative/Structure with One Operation}} === Structure with Two Operations === {{:Definition:Anticommutative/Structure with Two Operations}}
Definition:Normal Subgroup
Let $G$ be a group. Let $N$ be a subgroup of $G$. $N$ is a '''normal subgroup of $G$''' : === Definition 1 === {{:Definition:Normal Subgroup/Definition 1}} === Definition 2 === {{:Definition:Normal Subgroup/Definition 2}} === Definition 3 === {{:Definition:Normal Subgroup/Definition 3}} === Definition 4 === {{:Definition:Normal Subgroup/Definition 4}} === Definition 5 === {{:Definition:Normal Subgroup/Definition 5}} === Definition 6 === {{:Definition:Normal Subgroup/Definition 6}} === Definition 7 === {{:Definition:Normal Subgroup/Definition 7}}
Definition:Idempotence
'''Idempotence''' is a property of an algebraic system, or an element of an algebraic system such that: :$E \circledcirc E = E$ for an object $E$ and an operation $\circledcirc$.
Definition:Subgroup
Let $\struct {G, \circ}$ be an algebraic structure. $\struct {H, \circ}$ is a '''subgroup''' of $\struct {G, \circ}$ : :$(1): \quad \struct {H, \circ}$ is a group :$(2): \quad H$ is a subset of $G$.
Definition:Group
A '''group''' is a semigroup with an identity (that is, a monoid) in which every element has an inverse.
Definition:Center (Abstract Algebra)/Group
The '''center of a group''' $G$, denoted $\map Z G$, is the subset of elements in $G$ that commute with every element in $G$. Symbolically: :$\map Z G = \map {C_G} G = \set {g \in G: g x = x g, \forall x \in G}$ That is, the '''center''' of $G$ is the centralizer of $G$ in $G$ itself.
Definition:Distributive Operation
Let $S$ be a set on which is defined ''two'' binary operations, defined on all the elements of $S \times S$, which we will denote as $\circ$ and $*$. The operation $\circ$ '''is distributive over''' $*$, or '''distributes over''' $*$, : :$\circ$ is right distributive over $*$ and: :$\circ$ is left distributive over $*$.
Definition:Self Distributive
Let $\circ$ be a binary operation on the set $S$. Then $\circ$ is '''self-distributive''' iff: * $\forall a, b, c \in S: \left({a \circ b}\right) \circ c = \left ({a \circ c}\right) \circ \left({b \circ c}\right)$ * $\forall a, b, c \in S: a \circ \left({b \circ c}\right) = \left({a \circ b}\right) \circ \left({a \circ c}\right)$
Definition:Ordered Pair
The definition of a set does not take any account of the order in which the elements are listed. That is, $\set {a, b} = \set {b, a}$, and the elements $a$ and $b$ have the same status - neither is distinguished above the other as being more "important". === Informal Definition === {{:Definition:Ordered Pair/Informal Definition}} === Kuratowski Formalization === {{:Definition:Ordered Pair/Kuratowski Formalization}} === Empty Set Formalization === {{:Definition:Ordered Pair/Empty Set Formalization}} === Wiener Formalization === {{:Definition:Ordered Pair/Wiener Formalization}}
Definition:Cartesian Product
Let $S$ and $T$ be sets or classes. The '''cartesian product''' $S \times T$ of $S$ and $T$ is the set (or class) of ordered pairs $\tuple {x, y}$ with $x \in S$ and $y \in T$: :$S \times T = \set {\tuple {x, y}: x \in S \land y \in T}$
Definition:Relation
Let $S \times T$ be the cartesian product of two sets $S$ and $T$. A '''relation''' on $S \times T$ is an ordered triple: :$\RR = \tuple {S, T, R}$ where $R \subseteq S \times T$ is a subset of the Cartesian product of $S$ and $T$.
Definition:Mapping
Let $S$ and $T$ be sets. Let $S\times T$ be their cartesian product. === Definition 1 === {{:Definition:Mapping/Definition 1}} === Definition 2 === {{:Definition:Mapping/Definition 2}} === Definition 3 === {{:Definition:Mapping/Definition 3}} === Definition 4 === {{:Definition:Mapping/Definition 4}}
Definition:Composition of Relations
Let $\RR_1 \subseteq S_1 \times T_1$ and $\RR_2 \subseteq S_2 \times T_2$ be relations. Then the '''composite of $\RR_1$ and $\RR_2$''' is defined and denoted as: :$\RR_2 \circ \RR_1 := \set {\tuple {x, z} \in S_1 \times T_2: \exists y \in S_2 \cap T_1: \tuple {x, y} \in \RR_1 \land \tuple {y, z} \in \RR_2}$
Definition:Algebraic Structure
An '''algebraic structure''' is an ordered tuple: :$\struct {S, \circ_1, \circ_2, \ldots, \circ_n}$ where $S$ is a set which has one or more binary operations $\circ_1, \circ_2, \ldots, \circ_n$ defined on all the elements of $S \times S$. An '''algebraic structure''' with one (binary) operation is thus an ordered pair which can be denoted $\struct {S, \circ}$ or $\struct {T, *}$, and so on.
Definition:Restriction
=== Restriction of a Relation === {{:Definition:Restriction/Relation}} === Restriction of a Mapping === {{:Definition:Restriction/Mapping}} === Restriction of an Operation === In the same way that a restriction is defined on a relation, it can be defined on a binary operation. {{:Definition:Restriction/Operation}}
Definition:Closure (Abstract Algebra)
=== Algebraic Structures === {{:Definition:Closure (Abstract Algebra)/Algebraic Structure}} === Scalar Product === {{:Definition:Closure (Abstract Algebra)/Scalar Product}}
Definition:Magma
A '''magma''' is an algebraic structure $\struct {S, \circ}$ such that $S$ is closed under $\circ$.
Definition:Submagma
Let $\struct {S, \circ}$ be a magma. Let $T \subseteq S$ such that $\struct {T, \circ}$ is a magma. Then $\struct {T, \circ}$ is a '''submagma''' of $\struct {S, \circ}$. This relation can be denoted: : $\struct {T, \circ} \subseteq \struct {S, \circ}$ === Induced Operation === {{:Definition:Operation Induced by Restriction}}
Definition:Cancellable Element
Let $\left ({S, \circ}\right)$ be an algebraic structure.
Definition:Semigroup
: A '''semigroup''' is an algebraic structure which is closed and whose operation is associative.
Definition:Subsemigroup
Let $\struct {S, \circ}$ be an algebraic structure. Let $T \subseteq S$ such that $\struct {T, \circ {\restriction_T} }$, where $\circ {\restriction_T}$ is the restriction of $\circ$ to $T$, is a semigroup. Then $\struct {T, \circ {\restriction_T} }$ is a '''subsemigroup''' of $S$.
Definition:Commutative Semigroup
Let $\left({S, \circ}\right)$ be a semigroup such that the operation $\circ$ is a commutative operation. Then $\left({S, \circ}\right)$ is a '''commutative semigroup'''.
Definition:Regular Representations
Let $\struct {S, \circ}$ be a magma. === Left Regular Representation === {{:Definition:Regular Representations/Left Regular Representation}} === Right Regular Representation === {{:Definition:Regular Representations/Right Regular Representation}}
Definition:Subset Product
Let $\struct {S, \circ}$ be an algebraic structure. We can define an operation on the power set $\powerset S$ as follows: :$\forall A, B \in \powerset S: A \circ_\PP B = \set {a \circ b: a \in A, b \in B}$ This is called the '''operation induced on $\powerset S$ by $\circ$''', and $A \circ_\PP B$ is called the '''subset product''' of $A$ and $B$.
Definition:Cyclic Group
=== Definition 1 === {{:Definition:Cyclic Group/Definition 1}} === Definition 2 === {{:Definition:Cyclic Group/Definition 2}}
Definition:Inverse Relation
Let $\RR \subseteq S \times T$ be a relation. The '''inverse relation to''' (or '''of''') $\RR$ is defined as: :$\RR^{-1} := \set {\tuple {t, s}: \tuple {s, t} \in \RR}$
Definition:Reflexivity
Let $\mathcal R \subseteq S \times S$ be a relation in $S$. === Reflexive === {{:Definition:Reflexive Relation}} === Coreflexive === {{:Definition:Coreflexive Relation}} === Antireflexive === {{:Definition:Antireflexive Relation}} === Non-reflexive === {{Definition:Non-reflexive Relation}}
Definition:Transitivity (Relation Theory)
Let $\mathcal R \subseteq S \times S$ be a relation in $S$.
Definition:Symmetry (Relation)
Let $\mathcal R \subseteq S \times S$ be a relation in $S$. === Symmetric === {{:Definition:Symmetric Relation}} === Asymmetric === {{:Definition:Asymmetric Relation}} === Antisymmetric === {{:Definition:Antisymmetric Relation}} === Non-symmetric === {{:Definition:Non-symmetric Relation}}
Definition:Equivalence Relation
Let $\RR$ be a relation on a set $S$.
Definition:Equivalence Class
Let $S$ be a set. Let $\RR \subseteq S \times S$ be an equivalence relation on $S$. Let $x \in S$. Then the '''equivalence class of $x$ under $\RR$''' is the set: :$\eqclass x \RR = \set {y \in S: \tuple {x, y} \in \RR}$
Definition:Power Set
The '''power set''' of a set $S$ is the set defined and denoted as: :$\powerset S := \set {T: T \subseteq S}$ That is, the set whose elements are all of the subsets of $S$.
Definition:Universal Quantifier
The symbol $\forall$ is called the '''universal quantifier'''. It expresses the fact that, in a particular universe of discourse, all objects have a particular property. That is: :$\forall x:$ means: :'''For all objects $x$, it is true that ...''' In the language of set theory, this can be formally defined: :$\forall x \in S: \map P x := \set {x \in S: \map P x} = S$ where $S$ is some set and $\map P x$ is a propositional function on $S$.
Definition:Existential Quantifier
The symbol $\exists$ is called the '''existential quantifier'''. It expresses the fact that, in a particular universe of discourse, there exists (at least one) object having a particular property. That is: :$\exists x:$ means: :'''There exists at least one object $x$ such that ...''' In the language of set theory, this can be formally defined: :$\exists x \in S: \map P x := \set {x \in S: \map P x} \ne \O$ where $S$ is some set and $\map P x$ is a propositional function on $S$.
Definition:Propositional Expansion
Suppose our universe of discourse consists of the objects $\mathbf X_1, \mathbf X_2, \mathbf X_3, \ldots$ and so on. (There may be an infinite number of objects in this universe.) === Universal Quantifier === {{:Definition:Propositional Expansion/Universal Quantifier}} === Existential Quantifier === {{:Definition:Propositional Expansion/Existential Quantifier}}
Definition:Disjoint Sets
Two sets $S$ and $T$ are '''disjoint''' : :$S \cap T = \O$ That is, '''disjoint sets''' are such that their intersection is the empty set -- they have no elements in common.
Definition:Pairwise Disjoint
=== Set of Sets === {{:Definition:Pairwise Disjoint/Set of Sets}} === Family of Sets === {{:Definition:Pairwise Disjoint/Family}}
Definition:Set Partition
Let $S$ be a set. === Definition 1 === {{:Definition:Set Partition/Definition 1}} === Definition 2 === {{:Definition:Set Partition/Definition 2}}
Definition:Relative Complement
Let $S$ be a set, and let $T \subseteq S$, that is: let $T$ be a subset of $S$. Then the set difference $S \setminus T$ can be written $\relcomp S T$, and is called the '''relative complement of $T$ in $S$''', or the '''complement of $T$ relative to $S$'''. Thus: :$\relcomp S T = \set {x \in S : x \notin T}$
Definition:Identity (Abstract Algebra)
Let $\left({S, \circ}\right)$ be an algebraic structure. === Left Identity === {{:Definition:Identity (Abstract Algebra)/Left Identity}} === Right Identity === {{:Definition:Identity (Abstract Algebra)/Right Identity}} === Two-Sided Identity === {{:Definition:Identity (Abstract Algebra)/Two-Sided Identity}}
Definition:Symmetric Group
Let $S$ be a set. Let $\map \Gamma S$ denote the set of permutations on $S$. Let $\struct {\map \Gamma S, \circ}$ be the algebraic structure such that $\circ$ denotes the composition of mappings. Then $\struct {\map \Gamma S, \circ}$ is called the '''symmetric group on $S$'''. If $S$ has $n$ elements, then $\struct {\map \Gamma S, \circ}$ is often denoted $S_n$.
Definition:Inverse (Abstract Algebra)
Let $\struct {S, \circ}$ be a monoid whose identity is $e_S$. === Left Inverse === {{:Definition:Inverse (Abstract Algebra)/Left Inverse}} === Right Inverse === {{:Definition:Inverse (Abstract Algebra)/Right Inverse}} === Inverse === {{:Definition:Inverse (Abstract Algebra)/Inverse}}
Definition:Symmetric Difference
The '''symmetric difference''' between two sets $S$ and $T$ is written $S * T$ and is defined as: === Definition 1 === {{:Definition:Symmetric Difference/Definition 1}} === Definition 2 === {{:Definition:Symmetric Difference/Definition 2}} === Definition 3 === {{:Definition:Symmetric Difference/Definition 3}} === Definition 4 === {{:Definition:Symmetric Difference/Definition 4}} === Definition 5 === {{:Definition:Symmetric Difference/Definition 5}}
Definition:Set Complement
The '''set complement''' (or, when the context is established, just '''complement''') of a set $S$ in a universe $\mathbb U$ is defined as: :$\map \complement S = \relcomp {\mathbb U} S = \mathbb U \setminus S$ See the definition of Relative Complement for the definition of $\relcomp {\mathbb U} S$.
Definition:Left Hand Side
In an equation: :$\text {Expression $1$} = \text {Expression $2$}$ the term $\text {Expression $1$}$ is the '''left hand side'''.
Definition:Right Hand Side
In an equation: :$\text {Expression $1$} = \text {Expression $2$}$ the term $\text {Expression $2$}$ is the '''right hand side'''.
Definition:Domain (Set Theory)
=== Relation === {{:Definition:Domain (Set Theory)/Relation}} === Mapping === The term '''domain''' is usually seen when the relation in question is actually a mapping. {{:Definition:Domain (Set Theory)/Mapping}} === Binary Operation === {{:Definition:Domain (Set Theory)/Binary Operation}}
Definition:Range of Relation
Let $\RR \subseteq S \times T$ be a relation, or (usually) a mapping (which is, of course, itself a relation). The '''range''' of $\RR$, denoted  is defined as one of two things, depending on the source. On  it is denoted $\Rng \RR$, but this may non-standard. === Range as Codomain === The '''range''' of a relation $\RR \subseteq S \times T$ can be defined as the set $T$. As such, it is the same thing as the term '''codomain''' of $\RR$. === Range as Image === The '''range''' of a relation $\RR \subseteq S \times T$ can also be defined as: :$\Rng \RR = \set {t \in T: \exists s \in S: \tuple {s, t} \in \RR}$ Defined like this, it is the same as what is defined as the '''image set''' of $\RR$.
Definition:Image (Set Theory)
== Relation == {{:Definition:Image (Set Theory)/Relation}} == Mapping == {{:Definition:Image (Set Theory)/Mapping}}
Definition:Trivial Relation
The '''trivial relation''' is the relation $\RR \subseteq S \times T$ in $S$ to $T$ such that ''every'' element of $S$ relates to ''every'' element in $T$: :$\RR: S \times T: \forall \tuple {s, t} \in S \times T: \tuple {s, t} \in \RR$
Definition:Null Relation
The '''null relation ''' is a relation $\RR$ in $S$ to $T$ such that $\RR$ is the empty set: :$\RR \subseteq S \times T: \RR = \O$ That is, ''no'' element of $S$ relates to ''any'' element in $T$: :$\RR: S \times T: \forall \tuple {s, t} \in S \times T: \neg s \mathrel \RR t$
Definition:Diagonal Relation
Let $S$ be a set. The '''diagonal relation on $S$''' is a relation $\Delta_S$ on $S$ such that: :$\Delta_S = \set {\tuple {x, x}: x \in S} \subseteq S \times S$ Alternatively: :$\Delta_S = \set {\tuple {x, y}: x, y \in S: x = y}$
Definition:Proper Subset
If a set $S$ is a subset of another set $T$, that is, $S \subseteq T$, and also: :$S \ne T$ :$S \ne \O$ then $S$ is referred to as a '''proper subset''' of $T$. The set $T$ '''properly contains''', or '''strictly contains''', the set $S$. If $S \subseteq T$ and $S \ne T$, then the notation $S \subsetneqq T$ is used. If we wish to refer to a set which we specifically require not to be empty, we can denote it like this: :$\O \subsetneqq S$ and one which we want to specify as possibly being empty, we write: :$\O \subseteq S$ Thus for $S$ to be a '''proper subset''' of $T$, we can write it as $\O \subsetneqq S \subsetneqq T$. === Proper Superset === {{:Definition:Proper Subset/Proper Superset}}
Definition:Complement of Relation
Let $\RR \subseteq S \times T$ be a relation. The '''complement of $\RR$''' is the relative complement of $\RR$ with respect to $S \times T$: :$\relcomp {S \times T} \RR := \set {\tuple {s, t} \in S \times T: \tuple {s, t} \notin \RR}$ If the sets $S$ and $T$ are implicit, then $\map \complement \RR$ can be used.
Definition:Extension of Relation
Let: : $\mathcal R_1 \subseteq X \times Y$ be a relation on $X \times Y$ : $\mathcal R_2 \subseteq S \times T$ be a relation on $S \times T$ : $X \subseteq S$ : $Y \subseteq T$ : $\mathcal R_2 \restriction_{X \times Y}$ be the restriction of $\mathcal R_2$ to $X \times Y$. Let $\mathcal R_2 \restriction_{X \times Y} = \mathcal R_1$. Then $\mathcal R_2$ '''extends''' or '''is an extension of''' $\mathcal R_1$.
Definition:Agreement
=== Relations === {{:Definition:Agreement/Relations}} === Mappings === The concept is usually seen in the context of mappings: {{:Definition:Agreement/Mappings}}
Definition:Combinable
=== Relations === {{:Definition:Combinable/Relations}} === Mappings === The concept is usually seen in the context of mappings: {{:Definition:Combinable/Mappings}}
Definition:Union Relation
Let: :$(1): \quad \mathcal R_1 \subseteq S_1 \times T_1$ be a relation on $S_1 \times T_1$ :$(2): \quad \mathcal R_2 \subseteq S_2 \times T_2$ be a relation on $S_2 \times T_2$ Let $\mathcal R_1$ and $\mathcal R_2$ be combinable, that is, that they agree on $S_1 \cap S_2$. Then the '''union relation''' (or '''combined relation''') $\mathcal R$ of $\mathcal R_1$ and $\mathcal R_2$ is: :$\mathcal R \subseteq \left({S_1 \cup S_2}\right) \times \left({T_1 \cup T_2}\right): \mathcal R \left({s}\right) = \begin{cases}  \mathcal R_1 \left({s}\right) : & s \in S_1 \\  \mathcal R_2 \left({s}\right) : & s \in S_2 \end{cases}$
Definition:Relational Structure
A '''relational structure''' is an ordered pair $\struct {S, \RR}$, where: :$S$ is a set :$\RR$ is an endorelation on $S$.
Definition:Many-to-One Relation
A relation $\RR \subseteq S \times T$ is '''many-to-one''' : :$\forall x \in \Dom \RR: \forall y_1, y_2 \in \Cdm \RR: \tuple {x, y_1} \in \RR \land \tuple {x, y_2} \in \RR \implies y_1 = y_2$ That is, every element of the domain of $\RR$ relates to no more than one element of its codomain.
Definition:One-to-Many Relation
A relation $\RR \subseteq S \times T$ is '''one-to-many''' : :$\RR \subseteq S \times T: \forall y \in \Img \RR: \tuple {x_1, y} \in \RR \land \tuple {x_2, y} \in \RR \implies x_1 = x_2$ That is, every element of the image of $\RR$ is related to by exactly one element of its domain. Note that the condition on $t$ concerns the elements in the ''image'', not the codomain. Thus a '''one-to-many relation''' may leave some element(s) of the codomain unrelated.
Definition:One-to-One Relation
A relation $\RR \subseteq S \times T$ is '''one-to-one''' if it is both many-to-one and one-to-many. That is, every element of the domain of $\RR$ relates to no more than one element of its codomain, and every element of the image is related to by exactly one element of its domain.
Definition:Many-to-Many Relation
A relation $\mathcal R \subseteq S \times T$ is '''many-to-many''' if it is neither many-to-one nor one-to-many. That is, there is no restriction to the number of elements relating to or being related to by any individual element.  9pp6l117o3zdptpdhqwbjgxldvc5l89
Definition:Graph (Graph Theory)
A '''graph''' is intuitively defined as a pair consisting of a set of vertices and a set of edges. 400px
Definition:Quotient Set
Let $\RR$ be an equivalence relation on a set $S$. For any $x \in S$, let $\eqclass x \RR$ be the $\RR$-equivalence class of $x$. The '''quotient set of $S$ induced by $\RR$''' is the set $S / \RR$ of $\RR$-classes of $\RR$: :$S / \RR := \set {\eqclass x \RR: x \in S}$
Definition:Proper Coloring
=== Proper Vertex Coloring === A '''proper (vertex) $k$-coloring''' of a simple graph $G = \left({V, E}\right)$ is defined as a vertex coloring from a set of $k$ colors such that no two adjacent vertices share a common color. That is, a $k$-coloring of the graph $G = \left({V, E}\right)$ is a mapping $c: V \to \left\{{1, 2, \ldots k}\right\}$ such that: :$\forall e = \left\{{u, v}\right\} \in E: c \left({u}\right) \ne \left({v}\right)$ === Proper Edge Coloring === A '''proper (edge) $k$-coloring''' of a simple graph $G = \left({V, E}\right)$ is defined as an edge coloring from a set of $k$ colors such that no two adjacent edges share a common color. That is, a $k$-coloring of the graph $G = \left({V, E}\right)$ is a mapping $c: E \to \left\{{1, 2, \ldots k}\right\}$ such that: :$\forall v \in V: \forall e = \left\{{u_k, v}\right\} \in E: c \left\{{u_i, v}\right\} \ne c \left\{{u_j, v}\right\}$  gso9y1cj4k3eofv3qpouxubl0xume6b
Definition:Chromatic Number
The '''chromatic number''' $\map \chi G$ of a graph $G$ is the smallest positive integer $k$ such that there exists a proper vertex $k$-coloring of $G$.
Definition:Graph of Mapping
Let $S$ and $T$ be sets. Let $f: S \to T$ be a mapping. The '''graph''' of $f$ is the relation  $\RR \subseteq S \times T$ defined as $\RR = \set {\tuple {x, \map f x}: x \in S}$ Alternatively, this can be expressed: :$G_f = \set {\tuple {s, t} \in S \times T: \map f s = t}$ where $G_f$ is the '''graph of $f$'''.
Definition:Composition of Mappings
Let $S_1$, $S_2$ and $S_3$ be sets. Let $f_1: S_1 \to S_2$ and $f_2: S_2 \to S_3$ be mappings such that the domain of $f_2$ is the same set as the codomain of $f_1$. === Definition 1 === {{:Definition:Composition of Mappings/Definition 1}} === Definition 2 === {{:Definition:Composition of Mappings/Definition 2}} === Definition 3 === {{:Definition:Composition of Mappings/Definition 3}}
Definition:Inverse of Mapping
Let $S$ and $T$ be sets. Let $f: S \to T$ be a mapping The '''inverse''' of $f$ is its inverse relation, defined as: :$f^{-1} := \set {\tuple {t, s}: \map f s = t}$ That is: :$f^{-1} := \set {\tuple {t, s}: \tuple {s, t} \in f}$ That is, $f^{-1} \subseteq T \times S$ is the relation which satisfies: :$\forall s \in S: \forall t \in T: \tuple {t, s} \in f^{-1} \iff \tuple {s, t} \in f$