What is the meaning of $\forall x: \exists y:(x+y=0)$?

Question

Domain sets:

x = [1 2 3] y = [-1 -4 -5]

For all x there exists at least one y that satisfies x + y = 0.

Evaluation:

A:

1 + -1 = 0 ok
1 + -4 = 0
1 + -5 = 0

B:

2 + -1 = 0
2 + -4 = 0
2 + -5 = 0

C:

3 + -1 = 0
3 + -4 = 0 
3 + -5 = 0

Should one from A or B or C be true, or one from A and B and C?

Answer 1

As written $\forall x\exists y : x+y=0$ requires at least one in each of A, B and C to be correct.

If we only require one to be correct in A, B or C the expression becomes $\exists x\exists y : x+y=0$.

Thus, for your domain sets, the first (your) statement is false and the second is true.