Definition:Symmetric Closure

Definition

Let $\RR$ be a relation on a set $S$.

Definition 1

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

$\RR^\leftrightarrow = \RR \cup \RR^{-1}$

Definition 2

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

Examples

Arbitrary Example $1$

Let $S = \set {1, 2, 3, 4, 5}$ be a set.

Let $\RR$ be the relation on $S$ defined as:

$\RR = \set {\tuple {1, 2}, \tuple {2, 3}, \tuple {3, 4}, \tuple {5, 4} }$

The symmetric closure $\RR^\leftrightarrow$ of $\RR$ is given by:

$\RR^* = \set {\tuple {1, 2}, \tuple {2, 3}, \tuple {3, 4}, \tuple {5, 4}, \tuple {2, 1}, \tuple {3, 2}, \tuple {4, 3}, \tuple {4, 5} }$

Also see

• Equivalence of Definitions of Symmetric Closure

• Definition:Reflexive Reduction
• Definition:Transitive Reduction

• Results about symmetric closures can be found here.