Question about free variables in mathematical logic

Question

So the concept is from Herbert.B.enderton, A mathematical introduction to Logic.

Now consider the following expression:

1.) $\forall v_2 (Av_2 \rightarrow\ Bv_2)$

2.)$(\forall v_1Av_1 \rightarrow\ Bv_1)$

The books says 1.) is a sentence, meaning no variable occurs freely. But 2.) is not, as $v_1$ occurs freely.

But why is that ? I would really like an explanation on this. Thank you

Answer 1

In 2, the scope of the quantifier $\forall v_1$ is only the $Av_1$ part, hence the $v_1$ in $Bv_1$ is free, making 2 not a sentence.

In 1, the parentheses are used to make the scope of the quantifier $\forall v_2$ to be all of $Av_2 \rightarrow Bv_2$, so there are no free variables there.