We have been given these exercises on predicate logic, and this is what I have for an answer, I would like to have some feedback to know if I am on the right track, or I have made some mistakes, this is what I've got.
MY ANSWERS:
1. ∀x∃y (A(x) -> H(x,y)
2. ∀x∃y∃z ((C(x,y) ^ H(x,z)) ^ (C(x,z) ^ H(x,z)))
3. A(e(n))
4. ∀x (A(x) -> H(x, e(x)))
5. ∀x (C(x,e(x))-> ¬H(x,e(x))
If I have something wrong, could you give me a brief explanation in how should I approach the problem? Thank you!
I assume both of those were typos. Other than that, the rest of it looks correct to me. Personally, I would write (2) as $$\forall x (\exists y(C(x,y)\land H(x,y))\land\exists z(C(x,z)\land \neg H(x,z)))$$
But this is equivalent to what you had (after the correction). I just prefer to keep quantifiers as close as possible to what they are quantifying.