Union of Reflexive Relations is Reflexive

Theorem

The union of two reflexive relations is also a reflexive relation.

Proof

Let $\RR_1$ and $\RR_2$ be reflexive relations on a set $S$.

From Relation Contains Diagonal Relation iff Reflexive, we have that:

$\Delta_S \subseteq \RR_1$

$\Delta_S \subseteq \RR_2$

Hence from Subset Relation is Transitive:

$\Delta_S \subseteq \RR_1 \cup \RR_2$

Hence the result, by definition of reflexive relation.

$\blacksquare$