Provided that Alice but not the Hatter is cross unless mad, the Hatter, but not Alice, is mad unless frowning." How do you symbolize this sentence logically?

Well, it is a little tricky, but here is what I got:

(P ∨ R) ∧ ∼ (S ∨ U) → (U ∨ T) ∧ ∼ (R ∨ Q)

Explanation:

First the Symbols:

P: Alice is mad.

Q: Alice frowns.

R: Alice is mad

S: The hatter is cross

T: The hatter frowns

U: The Hatter is mad

*It is arbitrary, though, you can choose other symbols.

I think the difficult part is that the "but not the Hatter" means that the whole statement that applies to Alice is negated for the Hatter.