# Nested Quantifiers - Differentiating between $\forall x \forall y$, $\forall x \exists y$, and $\exists x \exists y$

I have a few questions regarding quantifiers which I'm still not clear about.

1) $\forall x \forall y (x^2 + y^2 = 9)$

I believe this is false as x and y could be 2 and results in 8.

2) $\forall x \exists y (x^2 + y^2 = 9)$

I believe this is false as well as x could be 25 and y could be 1 and still not result in 9.

3) $\exists x \exists y (x^2 + y^2 = 9)$

I believe this is true as x could be 3 and y could be 1 and results in 9.

A correct example for $(3)$ would be to let $x = 3$ and $y = 0$. Then, because there exists an x and there exists a y for which $x^2 + y^2 = 9$, the statement is true.
A better way to understand why $(2)$ is false is that it is claiming that for *every $x \in \mathbb R$, there is some $y$ such that $x^2 + y^2 = 9$. But then, what about, say, $x = 10$. Then we'd need some $y$ such that $$(10)^2 + y^2 = 9 \iff y^2 = 9 - 100 = -91$$ This is absurd, since $y^2 \geq 0 \;\;\forall y \in \mathbb R$. Indeed, for each $x \gt 3$, there is no $y$ that satisfies the equation. So since it is NOT true that for each $x$, there exists some $y$ whose square is negative, the claim is false.