My book says that the negation of "Everyone likes coffee" is "Not everyone likes coffee". But if i apply that to quantors (c = likes coffee), then...
$\neg(\forall x(c(x))) \Longleftrightarrow \exists x (\neg c(x))$
Which means that "Not everyone doesn't like coffee".
Can someone explain? Is this the same as the sentence above?
On
The negation of all moroccan like coffee is, there is at least one moroccan who does not like coffee. If there is one, two, more moroccan who do not like coffee, We cannot say that all moroccan like coffee.
If the proposition "All bresilian like coffee" is false or if its negation is true, that means that you can find At least one bresilian who dislike coffee. So the negation of $$\forall x \;\;\; c(x)$$ is
$$\exists x \;\; : \;\; not \; c(x)$$
As an exercice, what is the negation of
There is one and only one american who is crazy.
What you have written here:
$\exists x (\neg c(x))$
does not mean "Not everyone doesn't like coffee".
It means "There exists someone who is not a coffee-liker."
That is: "at least someone doesn't like coffee"
which is the same thing as "Not everyone likes coffee".