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 |
$\blacksquare$