Consider the following first order logic statements:
I: $\forall_x\forall_y p(x,y)$
II: $\forall_x\exists_y p(x,y)$
III: $\exists_y\exists_x p(x,y)$
IV: $\exists_y\forall_x p(x,y)$
Which of the following is not true about above statements?
(A) if I is true then II, III, IV are true.
(B) if II is true then III, IV are true.
(C) is IV is true then II, III are true.
(D) None of these.
Ans given is (B).
Now i think that statement of (A) is also false (and can be answer too) because if domain of $x$ and $y$ is empty then statement I will trivially be true. But in this case III and IV are false. So options (A) will be false.
Same for (C) if domain of $x$ is empty then IV will be true for all $y$ but in this case III will be false. So, options C is also false and hence answer too.
Is my argument correct or not?