# Relations

## Definition 1.12

Given sets A and B, any subset $R$ of $A×B$ is a relation between A and B. If $\left(\mathrm{a,b}\right)\in R$ this is often denoted by $aRb$

If $A=B$, $R$ is said to be a relation on $A$

## Definition 1.13

The domain of a relation $R$ between $A$ and $B$ is the set

The range of a relation $R$ between $A$ and $B$ is the set

## Definition 1.14

Let $R$ be a relation between $A$ and $B$. The inverse of the relation $R$ dentoed by is a relation between $B$ and $A$, defined by

## Definition 1.15

Let $R$ be a relation between $A$ and $B$, and let $S$ be a relation between $B$ and $C$.

The composition of $R$ and $S$, denoted by $S○R$, is a relation between $A$ and $C$ defined by

## Definition 1.16

A relation $R$ on $A$ is reflexive if $aRa$ for all $a\in A$.

A relation $R$ on $A$ is symmetric if $aRb\to bRa$ for all $a,b\in A$.

A relation $R$ on $A$ is antisymmetric if .

A relation is transitive if whenever , then $aRc$