Relation

Relation

Definition 1. A relation $R$ from $A$ to $B$ is a subset of $A \times B$. It is customary to write $aRb$ for $(a, b) \in R$. The symbol $aRb$ is read $a$ is $R$-related to $b$.

많은 경우 $A = B$이며, 이때 관계 $R$은 relation in $A$ 라고 말한다. 

Inverse Relation

Definition 2. Let $A, B$ be sets, not necessarily distinct, and let $R$ be a relation from $A$ to $B$. Then inverse $R^{-1}$ of $R$ is the relation from $B$ to $A$ such that $bR^{-1}a \iff aRb$. That is, $R^{-1} = \{(b,a) | (a,b) \in R\}$.

Domain and Image

Definition 3. Let $R$ be a relation from $A$ to $B$.
(1) The domain of $R$, denoted by Dom($R$), is the set of all those $a \in A$ such that $aRb$ for some $b \in B$;
(2) The image of $R$, denoted by Im($R$), is the set of all those $b \in B$ such that $aRb$ for some $a \in A$. 

당연하게도 Dom($R$) = Im($R^{-1}$), Im($R$) = Dom($R^{-1}$)이 성립한다. Dom($R \subset A \times A$) = $A$ 일 때 $R$은 relation on $A$라고 부른다.