Symmetric Closure of Inverse Relation

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Definition

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

Let $\mathcal R^{-1}$ be the inverse of $\mathcal R$.

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


Then:

$\mathcal R^\leftrightarrow = \left({\mathcal R^{-1}}\right)^\leftrightarrow$


Proof

\(\displaystyle \mathcal R^\leftrightarrow\) \(=\) \(\displaystyle \mathcal R \cup \mathcal R^{-1}\) $\quad$ Definition of Symmetric Closure $\quad$
\(\displaystyle \) \(=\) \(\displaystyle \left({\mathcal R^{-1} }\right)^{-1} \cup \mathcal R^{-1}\) $\quad$ Inverse of Inverse Relation $\quad$
\(\displaystyle \) \(=\) \(\displaystyle \left({\mathcal R^{-1} }\right)^\leftrightarrow\) $\quad$ Definition of Symmetric Closure $\quad$

$\blacksquare$