So basically there's a mathvideo in which there are some examples about compound quantifiers, but of which the answer is not provided. So I have no clue that I'm right or wrong, could someone please check my answers? Thank you in advance.
ex.1: ∀y∃x(x<y) "For all y there exist at least one x such that x < y" True because: x = y - 1
ex.2 ∃x∀y(x<y) "There exists an x such that for all y (x<y). False because: x=x
ex.3 ∀x∀y(x<y) "For all x, for all y (x<y)" False because: x<x
ex.4 ∃x∃y(x<y) "There exists at least one x and one y such that (x<y) True because: x < y + 1
It is important to keep in mind the domain of $x,y$. While all your answers are correct for, say real numbers, they may not be correct for $x,y\in\Bbb Z_{\ge1}$. The statement $(1)$ will then be false because there is no element in $\Bbb Z_{\ge1}$ smaller than $y=1$.