Let S be a non-empty set and P(S) be the power set of set S. Find the identity element for the union (∪) as a binary operation on P(S).
Given a none-empty set X,let∗:P(X)×P(X)→P(X) be defined as A×B=(A−B)∪(B−A),∀A,B∈P(X). Show that the empty set ϕ is the identity for the operation ∗ and all the elements A of P(X) are invertible with A−1=A.