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.
Thanks for your help.
You are correct about all your conclusions, and your reason for the first being false is spot on. But the reasons you give for why the second is false and the last is true are a bit off target.
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.