(1) $ \forall x ( P(x) \lor Q(x)) \iff \forall x (P(x)) \lor \forall x (Q(x) ) $, is this the same as writing $ \forall x ( P(x) \lor Q(x)) \equiv \forall x (P(x)) \lor \forall x (Q(x) $ ?
As the truth table for iff requires both sides to evaluate to true or both sides to evaluate to false, in order for the statement to be true?
(2) I cannot convince myself that the above iff statement is false, I tried this counter-example: Let the domain of discourse be {2}, P(x) = x is even, Q(x) = x is odd. Then wouldn't both sides of the iff statement evaluate to true, since P(x) is true, Q(x) is false, then $P(x) \lor Q(x)$ is true?. I am not exactly sure how the scope of the quantifier affects the truth values of the predicates?
Would appreciate any help to clear this doubt, thanks!