Intersection of Relation with Inverse is Symmetric Relation

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Let $\RR$ be a relation on a set $S$.

Then $\RR \cap \RR^{-1}$, the intersection of $\RR$ with its inverse, is symmetric.


Let $\tuple {x, y} \in \RR \cap \RR^{-1}$

By definition of intersection:

$\tuple {x, y} \in \RR$
$\tuple {x, y} \in \RR^{-1}$

By definition of inverse relation:

$\tuple {x, y} \in \RR \implies \tuple {y, x} \in \RR^{-1}$
$\tuple {x, y} \in \RR^{-1} \implies \tuple {y, x} \in \paren {\RR^{-1} }^{-1}$

By Inverse of Inverse Relation the second statement may be rewritten:

$\tuple {x, y} \in \RR \implies \tuple {y, x} \in \RR^{-1}$
$\tuple {x, y} \in \RR^{-1} \implies \tuple {y, x} \in \RR$

Then by definition of intersection:

$\tuple {y, x} \in \RR \cap \RR^{-1}$

Hence $\RR \cap \RR^{-1}$ is symmetric.