# Definition:Symmetric Closure/Definition 1

Let $\mathcal R$ be a relation on a set $S$.
The symmetric closure of $\mathcal R$ is denoted $\mathcal R^\leftrightarrow$, and is defined as the union of $\mathcal R$ with its inverse:
$\mathcal R^\leftrightarrow = \mathcal R \cup \mathcal R^{-1}$