Symmetric Relation equals its Symmetric Closure

Theorem

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

Let $\RR^\leftrightarrow$ be the symmetric closure of $\RR$.

Then:

$\RR = \RR^\leftrightarrow$

Proof

\(\ds \RR^\leftrightarrow\)

\(=\)

\(\ds \RR \cup \RR^{-1}\)

Definition of Symmetric Closure

\(\ds \)

\(=\)

\(\ds \RR \cup \RR\)

Inverse of Symmetric Relation is Symmetric

\(\ds \)

\(=\)

\(\ds \RR\)

Set Union is Idempotent

$\blacksquare$