Negating quantifiers with brackets My book and my teacher both said that these two statements are different:
1) $∀x∈A,∃y∈B, F(x, y)⇒G(x)$
2) $∀x∈A,(∃y∈B, F(x, y))⇒G(x)$
How can just adding an extra parenthesis make the statement different?
I am not quite sure I understand. 
If they are different, how are there negation different?
Would the negation be as follows:
1) $∀x∈A,∃y∈B, F(x, y) ∧ ¬ G(x)$
2) $∀x∈A,(∀y∈B, ¬ F(x, y)) ∧ ¬ G(x)$

SIDE NOTE:
I tried taking contrapositives, but ended up with the same expression for both. So how and why are they diffrent?
$¬ G(x) ⇒ ∃x∈A,∀y∈B, ¬ F(x, y) $

Thanks
 A: 
How can just adding an extra parenthesis make the statement different?

Because adding parentheses changes the scope of quantifiers.
Thus, the two formulas are not equivalent, provided that we read 1) as: $∀x \ [∃y \ (F(x,y) \to G(x))]$.
See Prenex normal form: 

$(\exists x\phi )\rightarrow \psi$  is equivalent to $\forall x(\phi \rightarrow \psi )$.

Formula 2) is $∀x \ [(∃y \ F(x,y)) \to G(x)]$, where the scope of the existential quantifier is restricted to sub-formula $F(x,y)$.
Thus, according to the equivalence above, 2) is equivalent to: $∀x \ [∀y \ (F(x,y) \to G(x))]$.

An easy check: formula 2) is equivalent to: $∀x \ [\lnot (∃y \ F(x,y)) \lor G(x)]$, i.e. to: $∀x \ [(∀y \lnot F(x,y)) \lor G(x)]$.
In general, the universal quantifier does not "distribute" over $\lor$ but, due to the fact that $y$ does not occur in $G(x)$, we can move it to get: $∀x \ [∀y \ ( \lnot F(x,y) \lor G(x))]$.


Maybe your perplexities are due to the (not so good) practice of mixing commas and parentheses to identify the scope of quantifiers.
Formula 1): $∀x∈A,∃y∈B,F(x,y)⇒G(x)$ is $∀x∈A \ [∃y∈B \ (F(x,y)⇒G(x))]$ where the scope of the existential quantifier is the sub-formula $F(x,y)⇒G(x)$.
Formula 2): $∀x∈A,(∃y∈B,F(x,y))⇒G(x)$, instead, is $∀x∈A \ [∃y∈B \ (F(x,y)) ⇒G(x)]$ where the scope of the existential quantifier is only the sub-formula $F(x,y)$.
