Inverses of Composite Relations

Why

How do inverse and converse relations interact.

Results

Let $R$ be a relation between $X$ and $Y$ and let $S$ be a relation between $Y$ and $Z$.

$(RS)^{-1} = S^{-1}R^{-1}$

Identity relations

Recall that $I$ is the identity relation on $X$ if $x\,I\,y$ if and only if $x = y$.

Let $R$ be a relation on $X$. Let $I$ be the identity relation on $X$. Then $RI = IR = R$.

One would like $RR^{-1} \supset I$, $R^{-1}R \supset I$. The father of the son is the father and the son of the father is the son. But the empty relation violates these claims.

Relation properties

$R$ is symmetric if and only if $R \subset R^{-1}$

$R$ is reflextive if and only if $I \subset R$

$R$ is transitive if and only if $RR \subset R$.