Symmetric Closure of Inverse Relation

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Definition

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

Let $\RR^{-1}$ be the inverse of $\RR$.

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


Then:

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


Proof

\(\ds \RR^\leftrightarrow\) \(=\) \(\ds \RR \cup \RR^{-1}\) Definition of Symmetric Closure
\(\ds \) \(=\) \(\ds \paren {\RR^{-1} }^{-1} \cup \RR^{-1}\) Inverse of Inverse Relation
\(\ds \) \(=\) \(\ds \RR^{-1} \cup \paren {\RR^{-1} }^{-1}\) Union is Commutative
\(\ds \) \(=\) \(\ds \paren {\RR^{-1} }^\leftrightarrow\) Definition of Symmetric Closure

$\blacksquare$