Problem #WSP-000320

Problem

Let $X$ be a finite set, and let $\mathcal{P}X$ be the power set of $X$ - that is, the set of subsets of $X$. For subsets $A$ and $B$ of $X$, define $A\ast B$ as the symmetric difference of $A$ and $B$ - that is, those elements that are in either $A$ or $B$, but not both. In formal set theory notation, this is $A\ast B=(A\cup B)\mathrm{\setminus }(A\cap B)$.

Prove that $(\mathcal{P}X,\ast )$ forms a group.