Inverse of Reflexive Relation is Reflexive
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Theorem
Let $\RR$ be a relation on a set $S$.
If $\RR$ is reflexive, then so is $\RR^{-1}$.
Proof
| \(\ds x\) | \(\in\) | \(\ds S\) | ||||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \tuple {x, x}\) | \(\in\) | \(\ds \RR\) | Definition of Reflexive Relation | ||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \tuple {x, x}\) | \(\in\) | \(\ds \RR^{-1}\) | Definition of Inverse Relation |
Hence the result by definition of reflexive relation.
$\blacksquare$
Sources
- 1971: Robert H. Kasriel: Undergraduate Topology ... (previous) ... (next): $\S 1.19$: Some Important Properties of Relations: Exercise $9$