Relation is Symmetric iff Inverse is Symmetric

Theorem

A relation $\RR$ is symmetric if and only if its inverse $\RR^{-1}$ is also symmetric.

Proof

Let $\RR$ be symmetric.

Then from Relation equals Inverse iff Symmetric:

$\RR = \RR^{-1}$

The result follows.

$\blacksquare$

Sources

• 1971: Robert H. Kasriel: Undergraduate Topology ... (previous) ... (next): $\S 1.19$: Some Important Properties of Relations: Exercise $8$