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.
$\blacksquare$