Definition:Symmetric Closure

Definition
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:


 * $\mathcal R^\leftrightarrow := \left\{{\left({y, x}\right): \left({x, y}\right) \in \mathcal R}\right\} \cup \mathcal R$

It follows immediately that $\mathcal R^\leftrightarrow$ is the union of $\mathcal R$ with its inverse.

That is:
 * $\mathcal R^\leftrightarrow = \mathcal R \cup \mathcal R^{-1}$

From Union Smallest, it follows that $\mathcal R^\leftrightarrow$ is the smallest symmetric relation on $S$ which contains $\mathcal R$.

Thus if $\mathcal R$ is symmetric, then $\mathcal R = \mathcal R^\leftrightarrow$.

It also follows from Relation equals Inverse iff Symmetric that $\left({\mathcal R^{-1}}\right)^\leftrightarrow = \mathcal R^\leftrightarrow$

Note
In contrast to reflexive and transitive relations, there is no concept of symmetric reduction. A moment's thought will establish why.