Symmetric Closure of Symmetric Relation

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Theorem

Let $\mathcal R$ be a relation on a set $S$.

Let $\mathcal R^\leftrightarrow$ be the symmetric closure of $\mathcal R$.


Then $\mathcal R = \mathcal R^\leftrightarrow$.


Proof

\(\displaystyle \mathcal R^\leftrightarrow\) \(=\) \(\displaystyle \mathcal R \cup \mathcal R^{-1}\) Definition of Symmetric Closure
\(\displaystyle \) \(=\) \(\displaystyle \mathcal R \cup \mathcal R\) Inverse of Symmetric Relation is Symmetric
\(\displaystyle \) \(=\) \(\displaystyle \mathcal R\) Union is Idempotent

$\blacksquare$