Intersection of Reflexive Relations is Reflexive

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Theorem

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


Proof

Let $\mathcal R_1$ and $\mathcal R_2$ be reflexive relations on a set $S$.

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

$\Delta_S \subseteq \mathcal R_1$
$\Delta_S \subseteq \mathcal R_2$

Hence from Intersection is Largest Subset:

$\Delta_S \subseteq \mathcal R_1 \cap \mathcal R_2$

Hence the result, from Relation Contains Diagonal Relation iff Reflexive.

$\blacksquare$