Inverse of Antisymmetric Relation is Antisymmetric
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Theorem
Let $\RR$ be a relation on a set $S$.
If $\RR$ is antisymmetric, then so is $\RR^{-1}$.
Proof
Let $\RR$ be antisymmetric.
Then:
- $\tuple {x, y} \land \tuple {y, x} \in \RR \implies x = y$
It follows that:
- $\tuple {y, x} \land \tuple {x, y} \in \RR^{-1} \implies x = y$
Thus it follows that $\RR^{-1}$ is also antisymmetric.
$\blacksquare$