Discrete Structures: Trying Correcting my Predicate Logic with the appropriate quantifiers

Question

I am trying to correctly use predicate symbols and using the appropriate quantifiers were I have to write each English language statement in predicate logic and the domain is the whole word.

$P(x)$ is "$x$ is a person."

$T(x)$ is "$x$ is a time."

$F(x_1,x_2)$ is "$x_1$ is fooled at $x_2$."

$1.$ You can fool some of the people all of the time.

My Answer at best: $\exists x(P(x) \rightarrow \forall y(T(y) \rightarrow F(x_1,x_2))$

$2.$ You can fool all of the people some of the time.

My answer at best: $\forall x(P(x) \rightarrow \exists y(T(y) \rightarrow F(x_1,x_2))$

$3.$ You can’t fool all of the people all of the time.

My answer at best: $\neg (\forall x(P(x) \rightarrow \forall y(T(y) \rightarrow F(x_1,x_2)))$

Answer 1

1. $$\exists x \exists y (P(x) \wedge P(y) \wedge \forall z T(z) \wedge F(x,y)) $$

2. $$ \exists x (P(x) \wedge \exists z T(z) \wedge (\forall y P(y) \rightarrow F(x,y)) ) $$

3. $$ \neg ( \exists x (P(x) \wedge ((\forall z T(z) \wedge \forall y P(y) ) \rightarrow F(x,y)) )) $$

In you answer, $x_1$ and $x_2$ are free variables. Make sure all variables are bounded.