discrete mathematics nested quantifiers

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So I have a questions about nested quantifiers that I am really not catching on.

I have E(x) = x is even and P(x) = x is prime.

The questions is asking if the following holds for :

1) natural numbers

2) even numbers

3) integers

4) rational number

∀x∃y, x^2 = y

Why isn't the answer natural numbers, even numbers, and rational numbers??

I also need some more clarification for universal quantifiers and existential quantifiers.