Inverse of Reflexive Relation is Reflexive

Theorem

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

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

$\ds x$

$\in$

$\ds S$

$\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.

$\blacksquare$

1971: Robert H. Kasriel: Undergraduate Topology ... (previous) ... (next): $\S 1.19$: Some Important Properties of Relations: Exercise $9$