Definition 1.12

Given sets A and B, any subset $R$ of $A\times <!--MULTIPLICATION\; SIGN-->B$ is a relation between A and B. If $\left(\mathrm{a,b}\right)\in <!--ELEMENT\; OF-->R$ 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 $\{a:a\in <!--ELEMENT\; OF-->A\mathrm{ and\; there\; exists\; }b\in <!--ELEMENT\; OF-->B\mathrm{ so\; that\; }aRb\}$

The range of a relation $R$ between $A$ and $B$ is the set $\{b:b\in <!--ELEMENT\; OF-->B\mathrm{ and\; there\; exists\; }a\in <!--ELEMENT\; OF-->A\mathrm{ so\; that\; }aRb\}$

Definition 1.14

Let $R$ be a relation between $A$ and $B$. The inverse of the relation $R$ dentoed by $R^{\mathrm{ -1}}$ is a relation between $B$ and $A$, defined by
$R^{\mathrm{ -1}}=\left\{\right(b,a):(a,b)\in <!--ELEMENT\; OF-->R\}$

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 $(a,b)\in <!--ELEMENT\; OF-->S○R\mathrm{ \; if\; there\; exists\; }b\in <!--ELEMENT\; OF-->B\mathrm{ such\; that }(a,b)\in <!--ELEMENT\; OF-->R\mathrm{ \; and }(b,c)\in <!--ELEMENT\; OF-->S$

Definition 1.16

A relation $R$ on $A$ is reflexive if $aRa$ for all $a\in <!--ELEMENT\; OF-->A$.

A relation $R$ on $A$ is symmetric if $aRb\to bRa$ for all $a,b\in <!--ELEMENT\; OF-->A$.

A relation $R$ on $A$ is antisymmetric if $aRb\mathrm{ and }bRa\mathrm{implies}a=b$.

A relation is transitive if whenever $aRb\mathrm{ and }bRc$, then $aRc$