$\exists \ x \ \forall \ y\ (x\leq y^2)$
$\mathbb{N}:$ True
$\mathbb{Z}:$ True
$\mathbb{R}^+:$ False
I understand why it is true with natural numbers and integers, but why is it false for the set of positive real numbers? What is an example of $(x\leq y^2)$ for $\mathbb{R}^+$?
There is no such real positive number that it is smaller than the square of any positive real number you pick, since the squares get arbitrarily small.
For the natural numbers it would be 0 or 1, depends on whether you include 0. For integers it would be zero.