Chapter 1 - Section 1.4 - Predicates and Quantifiers - Exercises - Page 56: 47

a) If $(\forall x P(x)) \land A$ is true, then A is true and for all values y we have P(y) is true. Thus, $P(y) \land A$ is true for all values of y, which means that $\forall x (P(x) \land A)$ is true. If $(\forall x P(x)) \land A$ is false, then A is false or there is a value y such that P(y) is false. Then, $P(y) \land A$ is false, which means that $\forall x (P(x) \land A)$ is false. Thus the two expressions always have the same truth value and thus they are logically equivalent. b) If $(\exists x P(x)) \land A$ is true, then A is true and there exists a value y for which P(y) is true. Thus, $P(y) \land A$ is true, which means that $\exists x (P(x) \land A)$ is true. If $(\exists x P(x)) \land A$ is false, then A is false or for all values of y we have P(y) is false. Then, $P(y) \land A$ is false for all values y, which means that $\exists x (P(x) \land A)$ is false. Thus the two expressions always have the same truth value and thus they are logically equivalent.

See answer