Predicate Calculus: Clarification on Switching Quantifiers

Question

Consider X={9, 10, 15}, Y={2,3}. Let Q(x,y): y divides x Then, is ∀x∃y Q(x,y) ≡ ∃y∀x Q(x,y)?

I answered yes, ∀x∃y Q(x,y) ≡ ∃y∀x Q(x,y).

Am I correct? If not, then why are these quantified statements not equal?

Answer 1

$\exists y\,\forall x\, Q(x,y)$ would mean there is an element of $Y$ that divides $9, 10, $ and $15$.

But $2$ does not divide $9$, and $3$ does not divide $10$, and there is no other element of $Y$.

Therefore, it is not true that $\exists y\,\forall x\, Q(x,y)$, even though it is true that $\forall x\, \exists y\,Q(x,y)$

(take $3$ for $9$ and $15$, and $2$ for $10$); these quantified statements are not equivalent.