For a set S, let P(S) be the collection of all subsets of S. Let binary operations + and · on P(S) be defined by 

A+ B = (A U B) - (A ∩ B) = {x | x ∈ A or x ∈ B but x ∉ (A ∩ B)} and A . B=A ∩ B for A, B ∈ P(S). 

a. Give the tables for + and • for P(S), where S = {a, b}.

b. Show that for any set S, (P(S), +, •) is a Boolean ring (see Exercise 55). 

Data from in Exercise 55

A ring R is a Boolean ring if a² = a for all a ∈ R, so that every element is idempotent. Show that every Boolean ring is commutative.