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?