Difference in placement of negation symbol?

Question

Let's say we had a dictionary Axy = x agrees with y; b = Bob. Consider the following:

1. $\neg \forall x Axb$
2. $\forall x \neg Axb$

They both seem to translate to "There is no one that agrees with Bob". Would there any difference between the placement of the negation?

Answer 1

The second statement $\forall x\neg Axb$ means "for every person, it is not true that this person agrees with Bob", in other words,

everybody disagrees with Bob.

$\forall xAxb$ means "everybody agrees with Bob", so the first statement $\neg\forall xAxb$ means "it is not true that everybody agrees with Bob", that is,

somebody disagrees with Bob.

These statements are not the same.

Answer 2

They're not the same, ther first one says: "Is not the case which all individues agree with Bob." The second one is: "For all individues, is not the case whic they agree with Bob."

Answer 3

$\neg \forall x~Axb$ : not everyone does agree with Bob.   There may be some who do agree.

$\forall x~\neg Axb$ : everyone does not agree with Bob.   There is certainly no one who does agree.

The sentences are distinct.