Translation of statements to logical notation

Question

I have recently started learning logic and I came to know how to translate statement into logical notation using propositional functions and quantifiers. I came across the following problems which are a little bit heavy for me.

We have to translate them into logical notation.

1. Some horses are gentle and have been well trained.
2. Some horses are gentle only if they have been well trained.
3. Some horses are gentle if they have been well trained.
4. Any horse is gentle that has been well trained.
5. No horse is gentle unless it has been well trained.
6. Any horse is gentle if it has been well trained.
7. Any horse has been well trained if it is gentle.
8. Any horse is gentle if and only if it has been well trained.
9. Gentle horses have been well trained.

I have translated them as follows.

Here $H(x)$ ,$G(x)$, $T(x)$ means $x$ is a horse, $x$ is gentle and $x$ is well trained respectively.

1. ($\exists$ $x$) ($H(x)$$\cdot$$G(x)$ $\cdot$$T(x)$).

2. ($\exists$ $x$) ($H(x)$ $\cdot$ ($T(x)$ $\rightarrow$ $G(x)$ )).

3. ($\exists$ $x$) ($H(x)$ $\cdot$ ($G(x)$ $\rightarrow$ $T(x)$ )).

4. ($\forall$ $x$) ($ H(x)$ $\cdot$ $T(x)$ $ \rightarrow$ $ G(x)$ ).

5. ($\forall$ $x$) ($ H(x) $ $\cdot$ $G(x) $ $\rightarrow$ $ T(x)$ ).

6. Same as 4.

7. Same as 5.

8. Unable to translate.(help needed)

9. Same as 4.

   Here $\cdot$ means $and$.

   Are my translations right ? If I have done any mistake please let me know.

   It's a long question ! So sorry for that. But I am stuck at these problems. Hope to get help. Thanks.

Answer 1

You should swap the answers to 2 and 3:

'Gentle if well-trained' means 'if well-trained then gentle', while 'gentle only if well-trained' means 'if gentle then well-trained'

4 through 9 all need universals ($\forall $)

9 is like 5, not 4

8 is $\forall x (H(x) \rightarrow (G(x) \leftrightarrow T(x)))$