Inverse of Antitransitive Relation is Antitransitive

Theorem

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

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

Proof

Let $\RR$ be antitransitive.

Then:

$\tuple {x, y}, \tuple {y, z} \in \RR \implies \tuple {x, z} \notin \RR$

Thus:

$\tuple {y, x}, \tuple {z, y} \in \RR^{-1} \implies \tuple {z, x} \notin \RR^{-1}$

and so $\RR^{-1}$ is antitransitive.

$\blacksquare$