Set Equivalence of Regular Representations

Theorem
If $$S$$ is a finite subset of a group $$G$$, then:

$$\left|{a \circ S}\right| = \left|{S}\right| = \left|{S \circ a}\right|$$

That is, $$a \circ S$$, $$S$$ and $$S \circ a$$ are equivalent: $$a \circ S \sim S \sim S \circ a$$.

Proof
Follows immediately from the fact that both the left and right regular representation are permutations, and therefore bijections.