Reflexive Closure is Reflexive

Theorem
Let $\mathcal R$ be a relation on a set $S$.

Then $\mathcal R^{=}$, the reflexive closure of $\mathcal R$, is reflexive.

Proof
From Reflexive Closure/Union with Diagonal:


 * $\mathcal R^= := \mathcal R \cup \Delta_S$

From Subset of Union/Binary Union:


 * $\Delta_S \subseteq \mathcal R^=$

The result follows directly from Reflexive Relation contains Diagonal Relation.