Symmetric Closure of Inverse Relation

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

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

$\blacksquare$