Y6

Problem 3. Let X be a nonempty finite set of alternatives, A the collect tion of all nonempty subsets S of X, and C a choice correspondence on A. Let us say that C is idempotent if C(C(S)) = C(S) for all S E A. Prove or falsifying statements. (1) If C satisfies a-axiom, then C is idempotent. (2) If C satisfies B-axiom, then C is idempotent. (3) If C satisfies a-axiom and X| = 3, then C is rationalizable. (4) If C satisfies B-axiom and |X| = 3, then C is rationalizable