Prove equivalence in predicate logic

Question

I have to prove that these formulas are equivalent: $$\begin{align} \exists x \forall y P(x,y) \equiv \forall y \exists x P(x,y) \\ \end{align}$$ Can I say that $$\begin{align} \forall y \exists x P(x,y) \equiv \neg \exists x \forall y P(x,y) \\ \end{align}$$ And the result is $\neq$ ?

Answer 1

You can't prove that they are equivalent because they aren't.

For example, suppose $P(x,y)$ is $x=y$. Then $\exists x\forall y(x=y)$ means that there exists something that equals everything (which is very false in a universe with more than one individual), whereas $\forall y\exists x(x=y)$ merely says that everything has something it is equal to (which is trivially true -- everything is equal to itself).

$\forall y\exists x P(x,y)$ is not equivalent to $\neg \exists x\forall y P(x,y)$ either. Here, if $P(x,y)$ means $x=x$ (that is, always true), then $\forall y\exists x P(x,y)$ is true in all worlds with at least one inhabitant, whereas $\neg\exists x\forall y P(x,y)$ is false in all those worlds.