I hate this stuff. Just to note. + means OR * means AND ! means NOT.
(A+B) * (A+C) * (!B + !C)
(A | B) & (A | C) & (!B | !C) // more conventnal
The answer is A(!B + !C)
I'm trying to get there.
So I start off with using Distributive rule which gets me here (A + B) * C * (!B + !C)
and that's where I'm stuck. I know I some how have to get rid of B and C but I see no way using any of the rules. I've got Identity, Null, Itempotent, Inverse, Commutative, Associative, Distributive, De Morgan's, and Cancellation to work with.
Am I starting off wrong? I really just used the only rule that I could see possible to even use. I was horrible with doing Proofs in Geometry and this stuff just makes me feel like that all over again.
Your first step is wrong.
(A+B) * (A+C) is (A+(B*C)).
Next, (!B + !C) is !(B*C).
So we get A*(!(B*C)) + (B*C)*(!(B*C)), which gives the desired result.