Discrete Maths:Predicate Logic Negation

Question

Let's say we have ∀xR(x).
Is ¬(∀xR(x)) the same as ¬∀xR(x)? Does that mean that the negation goes only to the quantitative indicator?

Answer 1

Yes, that's correct. The negation is the negation f the quantifificational formula ... it does not also (i.e. In addition to that) negate what is inside the quantifier.

So, yes, you have that:

$$\neg (\forall x \ P(x)) \Leftrightarrow \neg \forall x \ P(x)$$

In fact, this is not just an equivalence, but these two statements are really one and the same: the parentheses would merely there for your reading convenience.

Finally, it should be noted that the negation can be brought inside the quantifier, and when you do that, the negation will negate the formula on the inside, but the quantifier changes its sign:

$$\neg \forall x \ P(x) \Leftrightarrow \exists x \ \neg P(x)$$

Answer 2

Negation goes to quantifiers and changes them at the same time the truth value of statement changes w.r.t that quantifiers.

but an idea or say the meaning of a statement is still the same.