Reflexive Closure is Reflexive

Theorem

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

Then $\RR^=$, the reflexive closure of $\RR$, is reflexive.

Proof

Recall the definition of reflexive closure:

$\RR^= := \RR \cup \Delta_S$

From Set is Subset of Union:

$\Delta_S \subseteq \RR^=$

The result follows directly from Relation Contains Diagonal Relation iff Reflexive.

$\blacksquare$

Sources

• 1971: Robert H. Kasriel: Undergraduate Topology ... (previous) ... (next): $\S 1.19$: Some Important Properties of Relations: Exercise $2$