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:

\(\ds \tuple {x, y}\)

\(\in\)

\(\ds \RR^{-1}\)

\(\ds \leadsto \ \ \)

\(\ds \tuple {y, x}\)

\(\in\)

\(\ds \RR\)

Inverse of Inverse Relation

\(\ds \leadsto \ \ \)

\(\ds \tuple {x, y}\)

\(\notin\)

\(\ds \RR\)

Definition of Asymmetric Relation

\(\ds \leadsto \ \ \)

\(\ds \tuple {y, x}\)

\(\notin\)

\(\ds \RR^{-1}\)

Definition of Inverse Relation

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

$\blacksquare$