Inverse of Transitive Relation is Transitive

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

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

Proof
Let $\mathcal R$ be transitive.

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

Thus:
 * $\left({y, x}\right), \left({z, y}\right) \in \mathcal R^{-1} \implies \left({z, x}\right) \in \mathcal R^{-1}$

and so $\mathcal R^{-1}$ is transitive.