"When does s* = s, where s is a compound proposition"
The dual of a compound proposition that contains only the logical operators AND, OR, and NOT is the compound proposition obtained by replacing each AND by OR, each OR by AND, each T by F, and each F by T. The dual of s is denoted by s*
Let s = P(a1, a2, a3,…, an) where ai is a single proposition, and P is the relationship among ai.
So s* = (¬P)(a1, a2, a3,…, an) where (¬P) is the inverse relationship of P
s* = ¬(¬(¬P)(a1, a2, a3,…, an) ) double negating
s* = ¬( P(¬a1,¬ a2,¬ a3,…,¬ an) )
If s*= s
Then ¬P(a1, a2, a3,…, an) = P(¬a1,¬ a2,¬ a3,…,¬ an)
It means when we inverse every single proposition in s, if the truth value of s is also inversed, proposition s is self-dual (or s*=s)