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.