To prove that there exists a unique set $\displaystyle N \in \mathcal{P}(X) $ such that $\displaystyle A \Delta N = A $ for all $\displaystyle A \in \mathcal{P}(X) $ is it as simple as choosing $\displaystyle N = \emptyset $ and so:

$\displaystyle (A \cup \emptyset) - (A \cap \emptyset) = A - \emptyset = A $?