Predicate Logic and Negation Assistance

Question

I just want to make sure I'm on the right path with these:

Using the predicate symbols shown and appropriate quantifiers, write each English language statement in predicate logic. (The domain is the whole world.)

P(x) is ”x is a person.”

T(x) is ”x is a time.”

F(x,y) is ”x is fooled at y.”

1. You can fool some of the people all of the time.
2. You can fool all of the people some of the time.
3. You can’t fool all of the people all of the time.

My answers:

1. $\exists x F(P(x), T(x))$
2. $\forall x F(P(x), T(x))$
3. $\neg \forall x F(P(x), T(x))$

Which of the following is the correct negation for “Nobody is perfect.”

1. Everyone is imperfect.
2. Everyone is perfect.
3. Someone is perfect.

My answer:

Symbolic form of the above is $\neg \forall x P(x)$

Symbolic form negating the above is $\neg (\neg \forall x P(x))$

In words: "Everyone is perfect."

Am I on the right track with these?