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