Relation

set-theory

Definition

Relation

A relation $R$ is a set of ordered pairs that associates elements from set $A$, the domain, with elements of set $B$, the codomain, denoted as $R\subseteq A\times B$ (using the Cartesian Product).

An element $a\in A$ is related to an element $b\in B$ by the relation $R$ if $(a,b)\in R$, which can be denoted:

$$aRb:=(a,b)\subseteq R$$

Domain

Definition

Domain (Relation )

Let $R$ b a relation. Then the domain of $R$ is defined as:

$$\mathrm{dom}(R)=\{x:\exists y(x,y)\in R\}$$

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Range

Definition

Range (Relation)

Let $R$ be a relation. The range of $R$ is defined as:

$$\mathrm{ran}(R)=\{y:\exists x(x,y)\in R\}$$

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