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.

answer; the department of energy's