# 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

 $\displaystyle \mathcal R^\leftrightarrow$ $=$ $\displaystyle \mathcal R \cup \mathcal R^{-1}$ $\quad$ 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 $\quad$

$\blacksquare$