Evaluate $ (\forall y)(\exists x)(x \times x = y) $

Question

$\times$ is as you would think, the multiplication operation.

$ (\forall y)(\exists x)(x \times x = y) $

For the domains $\mathbb{N}$ and $\mathbb{Z}$ I found this formula fairly simple to evaluate, false in both cases.

I'm not certain with the following domains:

Domain: $\mathbb{Q}$

With this domain I concluded it evaluated to false, as for y=2, the $\sqrt[]{2}$ is not a rational number.

Is that correct?

Domain: $\mathbb{R}$

Would this evaluate to true, based off every real number $y$ having a real square root $x$?

Answer 1

With the help of the users in the comment section:

"Not every real number has a real square root" - @Henry

I have worked out that indeed this formula would evaluate to false for domains $ \mathbb{Q} $ (take y=2) and $\mathbb{R} $ (take y=-1). In both domains, no such x exists.