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
Recall the definition of reflexive closure:


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

Now from Subset of Union: Binary Union:


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

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