Help with quantifiers: $\forall y\in\mathbb{Z},\exists x\in\mathbb{Z},(x^2+y\geq1)$ As the title suggests, I am revisiting quantifiers and am having trouble deciphering the proper meaning. I know it is one of these two:
Does it mean that for each individual $y\in\mathbb{Z}$ we can find an $x\in\mathbb{Z}$ pertaining to that individual $y\in\mathbb{Z}$ such that the statement is true
or
Does it mean that for all $y\in\mathbb{Z}$ collectively, there must exist a single unique $x\in\mathbb{Z}$ that satisfies the above proposition for any $y\in\mathbb{Z}$. 
I'm leaning towards the latter statement. Thank you.

It means the first. Think of it as $\forall y \in \mathbb{Z}: (\exists x \in \mathbb{Z}: (x^2+y^2 \geq 1))$ -- that is, each quantifier "quantifies" a single expression and you can nest them to build more complex statements.
To obtain the latter meaning, you'd invert the order of the quantifiers: $\exists x \in \mathbb{Z}: (\forall y \in \mathbb{Z}: (x^2+y^2 \geq 1))$ means that at least one $x \in \mathbb{Z}$ works for each $y$.
