Intersection of Reflexive Relations is Reflexive
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Theorem
The intersection 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 Intersection is Largest Subset:
- $\Delta_S \subseteq \RR_1 \cap \RR_2$
Hence the result, from Relation Contains Diagonal Relation iff Reflexive.
$\blacksquare$