Symmetric Closure of Relation Compatible with Operation is Compatible

Theorem

Let $\struct {S, \circ}$ be a magma.

Let $\RR$ be a relation compatible with $\circ$.

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

Then $\RR^\leftrightarrow$ is compatible with $\circ$.

Proof

By the definition of symmetric closure:

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

Here $\RR^{-1}$ is the inverse of $\RR$.

By Inverse of Relation Compatible with Operation is Compatible, $\RR^{-1}$ is compatible with $\circ$.

Thus by Union of Relations Compatible with Operation is Compatible:

$\RR^\leftrightarrow = \RR \cup \RR^{-1}$ is compatible with $\circ$.

$\blacksquare$