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

Let $$P(x,y,z)$$ be the predicate $$x+y. Over which set is the statement $$∀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!

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.