# Symmetric Closure of Symmetric Relation

## 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

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

$\blacksquare$