Let X be a nonempty set, A a collection of nonempty subsets of X, and C a choice correspondence on A. Recall the following axioms. (a) (Axiom ) If S,TA and TS, then C(S)TC(T). (b) (Axiom ) If S,TA,TS, and C(S)T=, then C(T)C(S). (c) (WARP) If S,TA and C(T)S=, then C(S)TC(T). When we proved that Axioms and together are equivalent to WARP in class, we assumed that A is the collection of all nonempty subsets of X. But assuming observability of a decision maker's choice from every choice set is sometimes too strong. In this problem, by weakening this assumption, let us instead assume that A is just a nonempty collection of nonempty subsets of X. Prove or falsify each of the following claims. (To prove, give a proof. To falsify, give a counterexample.) (1) If C satisfies Axioms and , then C satisfies WARP. (2) If C satisfies WARP, then C satisfies Axioms and . (3) If C is rationalizable, then C satisfies WARP. (4) If C satisfies WARP, then C is rationalizable. Let X be a nonempty set, A a collection of nonempty subsets of X, and C a choice correspondence on A. Recall the following axioms. (a) (Axiom ) If S,TA and TS, then C(S)TC(T). (b) (Axiom ) If S,TA,TS, and C(S)T=, then C(T)C(S). (c) (WARP) If S,TA and C(T)S=, then C(S)TC(T). When we proved that Axioms and together are equivalent to WARP in class, we assumed that A is the collection of all nonempty subsets of X. But assuming observability of a decision maker's choice from every choice set is sometimes too strong. In this problem, by weakening this assumption, let us instead assume that A is just a nonempty collection of nonempty subsets of X. Prove or falsify each of the following claims. (To prove, give a proof. To falsify, give a counterexample.) (1) If C satisfies Axioms and , then C satisfies WARP. (2) If C satisfies WARP, then C satisfies Axioms and . (3) If C is rationalizable, then C satisfies WARP. (4) If C satisfies WARP, then C is rationalizable