Set Difference of Relations Compatible with Group Operation is Compatible

Theorem

Let $\struct {G, \circ}$ be a group.

Let $\RR, \QQ$ be relations on $G$ which are compatible with $\circ$.

Then the difference $\RR \setminus \QQ$ is compatible with $\circ$.

Proof

By Complement of Relation Compatible with Group is Compatible, $\relcomp {G \times G} \QQ$ is compatible with $\circ$.

Thus by Intersection of Relations Compatible with Operation is Compatible, $\RR \cap \relcomp {G \times G} \QQ$ is compatible with $\circ$.

But:

$\RR \cap \relcomp {G \times G} \QQ = \RR \setminus \QQ$

so the theorem holds.

$\blacksquare$