Inverse of Non-Symmetric Relation is Non-Symmetric

Theorem

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

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

Proof

Let $\RR$ be non-symmetric.

Then:

$\exists \tuple {x_1, y_1} \in \RR \implies \tuple {y_1, x_1} \in \RR$

and also:

$\exists \tuple {x_2, y_2} \in \RR \implies \tuple {y_2, x_2} \notin \RR$

Thus:

$\exists \tuple {y_1, x_1} \in \RR^{-1} \implies \tuple {x_1, y_1} \in \RR^{-1}$

and also:

$\exists \tuple {y_2, x_2} \in \RR^{-1} \implies \tuple {x_2, y_2} \notin \RR^{-1}$

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

$\blacksquare$