Definition:Symmetric Closure

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Definition

Let $\mathcal R$ be a relation on a set $S$.


Definition 1

The symmetric closure of $\mathcal R$ is denoted $\mathcal R^\leftrightarrow$, and is defined as the union of $\mathcal R$ with its inverse:

$\mathcal R^\leftrightarrow = \mathcal R \cup \mathcal R^{-1}$


Definition 2

The symmetric closure of $\mathcal R$ is denoted $\mathcal R^\leftrightarrow$, and is defined as the smallest symmetric relation on $S$ which contains $\mathcal R$.


Equivalence of Definitions

See Equivalence of Definitions of Symmetric Closure for a proof that the definitions above are equivalent.


Note

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