Inverse of Asymmetric Relation is Asymmetric

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

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

Proof
Let $\RR$ be asymmetric.

Let $\tuple {x, y} \in \RR^{-1}$.

Then:

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