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