5. (9 marks) Let A() and B(x) be predicates over the same nonempty domain. For each of the following assertions, state whether it is true or false. If an assertion is true then give a carefully reasoned argument in English to show that it is true (you do not have to give a formal proof). If an assertion is false, give a counterexample, i.e., an interpretation that satisfies one formula and falsifies the other in a way that violates the assertion. (a) 3x, (~ A(2)) logically implies 3r, (A(x) B(x)) (b) Vz, (A(z) (Vz. A(z))^ (Vr, B(z)) (e) 3rVy, (A(z) B(y)) is logically equivalent to (3x, ~ A(r)) V(w, B(y)) B()) logically implies 5. (9 marks) Let A() and B(x) be predicates over the same nonempty domain. For each of the following assertions, state whether it is true or false. If an assertion is true then give a carefully reasoned argument in English to show that it is true (you do not have to give a formal proof). If an assertion is false, give a counterexample, i.e., an interpretation that satisfies one formula and falsifies the other in a way that violates the assertion. (a) 3x, (~ A(2)) logically implies 3r, (A(x) B(x)) (b) Vz, (A(z) (Vz. A(z))^ (Vr, B(z)) (e) 3rVy, (A(z) B(y)) is logically equivalent to (3x, ~ A(r)) V(w, B(y)) B()) logically implies