Predicate Logic-Discrete Math

Hello all! I came across this problem on my homework and I would like for somebody to check and see if my answers are correct.
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.
-For this I put ∃xF(P(x),T(x))
2. You can fool all of the people some of the time.
-∀xF(P(x),T(x))
3. You can’t fool all of the people all of the time.
I put ¬∀xF(P(x),T(x)) for this.
Thanks in advance!!