Q: 2-27. Use decomposition to find minimum gate input count, multiple-level implementations for the functions using AND and OR gates and inverters.

a) F(A,B,C,D) = AB'C + A'BC + AB'D + A'BD
b) F(W,X,Y,Z) = WY + XY + W'XZ + WX'Z

I'm not sure where I am stuck . . . the very beginning? We had snow days and this material has not been covered in class yet, but the homework is due never the less. I'm not sure why, but that's not something I can change so it doesn't matter anyway.

The book (Logic and Computer Design Fundamentals 4th (Mano & Kime)) discusses the topic on page 96, but in it's illustration of decomposition, it picks up after a function has already undergone a different multilevel optimization transformation by algebraic factoring.

If I could see one example worked, I could do this, I know it.

Let me try a little here and explain my thoughts as I go.
AB'C + A'BC + AB'D + A'BD can be factored somehow maybe.
I think I will pick C and D because they ... because of intuition? So...
= C(AB' + A'B) + D(AB'+A'B)
Okay. It looks kinda cool, like there is a pattern here maybe. Patterns are good. . . So... then it says that I can redefine groups of variables as other single term variables and then they can be manipulated in some other sort of way that will undoubtedly show itself to be somehow beneficial to my goal here.

Okay, lets let X1 = AB'+A'B and X2 be . . . uh, maybe I don't need an X2 like the example says. And I don't know if what I am doing is valid, but lets go with it.

=C(X1) + D(X1)
with X1 = AB' + A'B

so... now what? They mention gate count, but I'm not really clear on it yet.

Anyway, any help would be appreciated.