suppose that the domain of variable x is the set of people, and f(x) = "x is friendly" , t(x)= "x is tall" and a(x) = "x is angry". Write the statement using these predicates and any needed quantifiers.
1) some people are not angry
2) all tall people are friendly
3) No friendly people are angry
My solutions:
1) $∃x\sim A(x)$
2) $∀xF(x)$
3) $\sim ∀x A(x)$
I'd like to know if my answers are right or wrong.
On
As noted, your first translation is correct.
In your second translation, you address only "friendliness": "Everyone is friendly". What you want to say is "All tall people are friendly." To do this, we need to write $$\forall x\,\Big(T(x) \rightarrow F(x)\Big)$$
In your third translation, again, it only addresses one predicate, and actually says "Not everyone is angry." That would mean that the statement is true if some (possibly unfriendly) people are not angry and all friendly people are angry. What you want to say is "No friendly people are angry." Put differently: "There does not exist a person who is both friendly and angry."
$$\lnot \exists x\,\Big(F(x) \land A(x)\Big)$$
Note that this is equivalent to $$\begin{align} \forall x \;\lnot\Big(F(x) \land A(x)\Big) & \equiv \forall x\, \Big(\lnot F(x) \lor \lnot A(x)\Big) \\ \\ &\equiv \forall x\,\Big(F(x) \rightarrow \lnot A(x)\Big)\end{align}$$
The last translates directly as "Everyone who is friendly is not angry", which is the logical equivalent of the initial statement: "No friendly person is angry."
Your first one is correct.
Your second statement is saying, "all people are friendly" and your third one doesn't make sense since $\neg\forall A(x)$ doesn't mean anything.
Your second one should be $\forall x (t(x)\implies f(x))$ and your third one should be $\forall x (f(x)\implies \neg a(x)$ or $\neg \exists x(f(x)\land a(x))$.