# Simplifying ambiguous statements

I have been working on the following question from Velleman's How to prove book:

Let S stand for the statement “Steve is happy” and G for “George is happy.” What English sentences are represented by the following expressions? (a) (S ∨ G) ∧ (¬S ∨ ¬G). (b) [S ∨ (G ∧ ¬S)] ∨ ¬G. (c) S ∨ [G ∧ (¬S ∨ ¬G)].

Now, this is how I'm solving each of these:

(a)

(S ∨ G) = Steve is happy or George is happy
(¬S ∨ ¬G) = Steve is not happy or George is not happy

(S ∨ G) ∧ (¬S ∨ ¬G) = Steve is happy or George is
and Steve is not happy or George is not happy.

(b)

¬G = George is not happy
S ∨ (G ∧ ¬S) = Steve is happy or George is happy and steve ins't happy

[S ∨ (G ∧ ¬S)] ∨ ¬G = Either George isn't happy or Steve is or George
is happy and steve isn't.

(c)

[G ∧ (¬S ∨ ¬G)] = George is happy and steve isn't.
S ∨ [G ∧ (¬S ∨ ¬G)] = Either Steve is happy or George is and steve isn't.


But the problem which I feel that are present in the solutions are that are ambiguous and doesn't really present a concise information to the readers ? Is my solutions correct or more simplification needed for it ? How do logicians provide statements for that ?