Inverse of Non-Reflexive Relation is Non-Reflexive

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Theorem

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


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


Proof

Let $\RR$ be non-reflexive.


Then:

\(\ds \exists x \in S: \, \) \(\ds \tuple {x, x}\) \(\in\) \(\ds \RR\) as $\RR$ is not antireflexive
\(\ds \leadsto \ \ \) \(\ds \exists x \in S: \, \) \(\ds \tuple {x, x}\) \(\in\) \(\ds \RR^{-1}\) Definition of Inverse Relation

Thus $\RR^{-1}$ is not antireflexive.


Also:

\(\ds \exists x \in S: \, \) \(\ds \tuple {x, x}\) \(\notin\) \(\ds \RR\) as $\RR$ is not reflexive
\(\ds \leadsto \ \ \) \(\ds \exists x \in S: \, \) \(\ds \tuple {x, x}\) \(\notin\) \(\ds \RR^{-1}\) Definition of Inverse Relation

Thus $\RR^{-1}$ is not reflexive.


Hence the result, by definition of non-reflexive relation.

$\blacksquare$