How would you use predicate logic to check if a statement is true or false?

Question

Let $P(x,y,z)$ be the predicate $x+y<z$. Over which set is the statement $∀z∃x∃y\ P(x,y,z)$ true?

$\Bbb Z^+=\{1,2,3,\dots\}$ or $\Bbb Z$?

I had thought that it would be neither, but that is not an option. Can someone please explain to me which one is correct and why? I am quite new to discrete mathematics so would greatly appreciate a walkthrough. Thanks!

Answer 1

Over $\Bbb Z^+$, because the smallest number is 1, for any $x$ and $y$ we have $x+y>1$, so the predicate is always false if $z=1$. Therefore the statement is false (we have shown its negation to be true).

Over $\Bbb Z$, the statement is true because for any $z$ we can take (there exists) $x=z$ and $y=-1$ such that $x+y=z-1<z$.