predicate and logical equivalency question

Question

i've been given a question

"Are the statements ¬(∀xA(x) → ∀x∀yB(x, y)) and ∀xA(x) ∧ ∃x∃y¬B(x, y) logically equivalent? If they are equivalent, prove that they are. If not, give an interpretation under which they have different truth values."

I put in an interpretation and the answer was that they ARENT logically equivalent but i'd just like to double check as i'm new to predicates. also, i don't understand "If not, give an interpretation under which they have different truth values." If i worked out the answer through interpretation for A(x) and B(x,y), is that what the question wants?

thankyou

Answer 1

Yes, they are.

The formulae: $\lnot (p \to q)$ and $p \land \lnot q$ are tautologically equivalent.

Thus, applying this transformation to $¬(∀xA(x) → ∀x∀yB(x,y))$ we get the equivalent:

$(∀xA(x) \land \lnot ∀x∀yB(x,y))$.

The final step is to "move inside" the negation sign with the equivalence between $\lnot \forall$ and $\exists \lnot$.