Given a non-empty set \( X,\) let \(\ast :\; P(X)\; \times\; P(X) \to P(X) \) be defined as \(A \ast B = \; ( A-B)\; \cup \; (B-A),\; \forall A, B \in \; P(X).\). Show that the empty set \(\emptyset \) is the identity for the operation $\ast$ and all the elemnets \(A\) of \( P(X) \) are invertible with \( A^{-1} \;= A\).

Question

    \((Hint:\; (A- \emptyset )\; \cup \; (\emptyset - A)=A\; and ( A-A) \cup \; (A-A)= A \ast A = \emptyset) \)