# Inverse of Antisymmetric Relation is Antisymmetric

## Theorem

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

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

## Proof

Let $\RR$ be antisymmetric.

Then:

$\tuple {x, y} \land \tuple {y, x} \in \RR \implies x = y$

It follows that:

$\tuple {y, x} \land \tuple {x, y} \in \RR^{-1} \implies x = y$

Thus it follows that $\RR^{-1}$ is also antisymmetric.

$\blacksquare$