Inverse of Asymmetric Relation is Asymmetric

Theorem
Let $\mathcal R$ be a relation on a set $S$.

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

Proof
Let $\mathcal R$ be asymmetric.

Then:
 * $\left({x, y}\right) \in \mathcal R \implies \left({y, x}\right) \notin \mathcal R$

Thus if $\left({x, y}\right) \in \mathcal R$ then:
 * $\left({y, x}\right) \in \mathcal R^{-1}$ and $\left({x, y}\right) \notin \mathcal R^{-1}$

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