TITLE: Quantifiers, predicates, logical equivalence
QUESTION [12 upvotes]: I am asked if $(\exists x) (P(x) \rightarrow Q(x))$ and $\forall x P(x) \rightarrow \exists xQ(x)$ are logically equivalent. 
I don't think they are but how will I prove it?
Am I supposed to use either of the direct proof, contrapositive or contradiction proofs? Or give an interpretation?

REPLY [3 votes]: Yet another way to do this is with the method of analytic tableaux. It's a systematic search for contradictions, so we assume the negation of $$(\exists x(Px\to Qx))\leftrightarrow (\forall xPx\to\exists xQx)\tag{1}$$ and show that no matter what we do, we end up with a contradiction; look:
.
Can you spot the contradictions?

 From left to right: $\neg Pa$ with $Pa$, $\neg Qa$ with $Qa$, $\neg Pb$ with $Pb$, and $\neg Qc$ with $Qc$.

Hence $(1)$ is indeed an equivalence.
This method is explained in detail in M. D'Agostino's Handbook of Tableau Methods.