Simplifying a boolean expression without using sum-of-products

Question

Yes this is for a school class, and no I'm not asking anyone to do my homework. This is from an example.

I have a boolean algebra expression that I have to simplify. I also have the answer simplified expression. However I can't seem to get from the problem to the simplified answer using only standard boolean rules (can't use sum-of-product reduction).

Original:

1. P = (ABC' + D)' + (AB' + BC')'

Using Demorgan + some clean up I arrive at

2. P = D'(A' + B' + C) + (A' + B)(B' + C) or 3. P = A'D' + B'D' + CD' + A'B' + A'C + BC

This is where I get stuck, as the simplified result is 3. 'A'B + BC + A'D' + CD'

Is it possible to get from #2 to #3 without using a sum-of-products reduction? If so, what are the algebraic steps?

Answer 1

Consensus will get you the rest of the way.

$$ X Y + \overline X Z + Y Z = X Y + \overline X Z $$

$$ [BC + \overline B \ \overline D + C \overline D] + [BC + \overline B\ \overline A + \overline AC] + \overline A\ \overline D$$

Although I do not get your answer.

$$ BC + \overline B \ \overline D + \overline B\ \overline A + \overline A\ \overline D$$