Is this formula logically valid

Question

Is: $\exists x (P(x) \land Q(x)) \rightarrow \exists x P(x) \land \exists x Q(x) $ logically valid?.

I cant found an intepretation in wich the formula is false.

Answer 1

Yes, indeed it is valid. No counterexample to be found.

If there exists an $x$ for which both ($P(x)$ and $Q(x)$) hold, then there certainly exists an $x$ for which $P(x)$ holds, and there exists an $x$ for which $Q(x)$ holds.

The converse implication is not valid, however. If there exists an $x$ that's a pumpkin and there exists an $x$ that is green, it does not follow that there exists an $x$ that is a green pumpkin.

Answer 2

Yes, it's valid. If there is something which is both $P$ and $Q$, then there is something which is $P$ AND there is something which is $Q$ (viz. the same thing which was both $P$ and $Q$).