Symmetry of Relations is Symmetric

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Theorem

Let $\RR$ be a relation on $S$ which is symmetric. Then:

$\tuple {x, y} \in \RR \iff \tuple {y, x} \in \RR$.


Proof

Let $\RR$ be symmetric.

\(\ds \tuple {x, y} \in \RR\) \(\implies\) \(\ds \tuple {y, x} \in \RR\) Definition of Symmetric Relation
\(\ds \tuple {y, x} \in \RR\) \(\implies\) \(\ds \tuple {x, y} \in \RR\) Definition of Symmetric Relation
\(\ds \leadsto \ \ \) \(\ds \leftparen {\tuple {x, y} \in \RR}\) \(\iff\) \(\ds \rightparen {\tuple {y, x} \in \RR}\) Definition of Biconditional

$\blacksquare$