Definition:Symmetric Closure

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:


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

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.