How can I solve this $(∀x)(∃y)P(x,y) <-> (∃y)(∀x)P(x,y)$?

Question

Is the following example true? If yes give explanation why ? $$(∀x)(∃y)P(x,y) \leftrightarrow (∃y)(∀x)P(x,y)$$ I tried using values for $x$ and for $y$, and restricted them to the following: $x > 3$ and $y < 4$ but how can I be sure if this is true?
I have come across another question as well where for the function, $Q(x)$: $x$ is even $\to$ $x^2 - 4>0$ and the domain is $Z$. I tried to use the following: ${...-5,-3,-1,1,3,5,...}$ but I am not sure :D

Answer 1

1. For every integer $x$, there exists an integer $y$ such that $x<y$.
2. There exists an integer $y$ such that, for all integers $x$, we have $x<y$.

The first is clearly true, and the second is nonsense. Therefore, the original claim that you're asking about is false.

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Regarding your second question, it is not true that "$x$ is even" implies that $x^2-4>0$. Counterexamples are $x=-2, x=0$, and $x=2$. Whatever $Q(x)$ is supposed to be doing, it must have some way of excluding those values. I'm not sure why you'd be trying odd numbers, when they don't satisfy the condition in question.