Symmetric Difference on Power Set forms Abelian Group

Theorem
Let $$S$$ be a set such that $$\varnothing \subset S$$ (i.e. $$S$$ is not empty).

Let $$A * B$$ be defined as the symmetric difference between $$A$$ and $$B$$.

Let $$\mathcal{P} \left({S}\right)$$ be the power set of $$S$$.

Then the algebraic structure $$\left({\mathcal{P} \left({S}\right), *}\right)$$ is an abelian group.

Proof
First we show that, as an algebraic structure, $$\left({\mathcal{P} \left({S}\right), *}\right)$$ is closed.

Let $$A, B \subseteq S$$, i.e. $$A, B \in \mathcal{P} \left({S}\right)$$.

$$ $$ $$ $$

Thus we see that $$\left({\mathcal{P} \left({S}\right), *}\right)$$ is closed.

The result follows directly from Set System Closed with Symmetric Difference is Group.