# Inverse of Reflexive Relation is Reflexive

## Theorem

Let $\RR$ be a relation on a set $S$.

If $\RR$ is reflexive, then so is $\RR^{-1}$.

## Proof

 $\ds x$ $\in$ $\ds S$ $\ds \leadsto \ \$ $\ds \tuple {x, x}$ $\in$ $\ds \RR$ Definition of Reflexive Relation $\ds \leadsto \ \$ $\ds \tuple {x, x}$ $\in$ $\ds \RR^{-1}$ Definition of Inverse Relation

Hence the result by definition of reflexive relation.

$\blacksquare$