Help with quantifiers: $\forall y\in\mathbb{Z},\exists x\in\mathbb{Z},(x^2+y\geq1)$

Question

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.

Answer 1

The first interpretation is correct: $\forall y, \exists x$ means that for any single $y$ there is some $x$ which works. If asserting that there is a single $x$ that works simultaneously for all $y$, it would be $\exists x,\forall y$.

Answer 2

It means that if you fix an integer $y$ you can find an integer $x$ such that $x^2+y \geq 1$. There is no uniqueness of $x$ understood.

Answer 3

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 $y$.