Premise: Let $F(x,y)$ be the statement $x$ can fool $y$, where the domain for discourse for both $x$ and $y$ is all people.
I converted the following corresponding statements:
1) Nobody can fool me
2) Anybody can Fool Fred
3) Everyone can fool someone
1) $\neg \exists xF(x,me)$
2) $\forall xF(x,Fred)$
3) $\forall x\exists yF(x,y)$
Now I am trying to negate these and I can't just throw in a not in front of the statements and I don't know how to go about this.
I looked at the negation rules and came up with this but I doubt it's correct:
1) $\neg\exists xF(x,me) \iff \forall x\neg F(x,me)$
2) $\forall xF(x,Fred) \iff \forall x\neg\neg F(x,Fred) \iff \neg[\exists x\neg F(x,Fred)]$
3) $\forall x\exists yF(x,y) \iff \neg\forall x\exists yF(x,y)$