1. Consider the set S = {0, 1, 2, 3, 4,5, 6, 7}, where for a, be S we define a + b = max(a, b) and a x b = min(a, b). Define the complement as a = 7 - a. (a) Is 3 + (4 x 7) = (3+4) x (3+7)? (b) Find the identities for each of the operations, + and X. (c) Suppose that (S, +, x) was a Boolean algebra system. Given a E S, what should a + a and a x a be? What is 2 + 2? What is 3 x 3? (d) Is (S, +, x) a Boolean algebra? Justify your answer. 2. Let B be a Boolean algebra and let a, be B. Prove the following: (a) at Idx = Idx (b) (a x b) ta =a