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:


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

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.