Intersection of Symmetric Relations is Symmetric

Theorem

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


Proof

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

Let $\RR_3 = \RR_1 \cap \RR_2$.

Then:

\(\ds \tuple {x, y}\) \(\in\) \(\ds \RR_3\)
\(\ds \leadsto \ \ \) \(\ds \tuple {x, y}\) \(\in\) \(\ds \RR_1\) Definition of Set Intersection
\(\, \ds \land \, \) \(\ds \tuple {x, y}\) \(\in\) \(\ds \RR_2\)
\(\ds \leadsto \ \ \) \(\ds \tuple {y, x}\) \(\in\) \(\ds \RR_1\) Definition of Symmetric Relation
\(\, \ds \land \, \) \(\ds \tuple {y, x}\) \(\in\) \(\ds \RR_2\)
\(\ds \leadsto \ \ \) \(\ds \tuple {y, x}\) \(\in\) \(\ds \RR_3\) Definition of Set Intersection

$\blacksquare$

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