Diagonal Complement Relation Compatible with Group Operation

Theorem

Let $\left({G, \circ}\right)$ be a group.

Let $\Delta_G$ be the diagonal relation on $G$.

Then $\Delta_G^c = \complement_{G \times G} \Delta_G$ is a relation compatible with $\circ$.

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In other words, $\ne$ is a relation compatible with $\circ$.

Proof

By Diagonal Relation is Universally Compatible, $\Delta_G$ is compatible with $\circ$.

By Complement of Relation Compatible with Group is Compatible , $\Delta_G^c$ is also compatible with $\circ$.

$\blacksquare$