Boolean Algebra - Product of Sums

Question

I converted from a truth table to sum of products and simplified that easily. What I am having problems with is simplifying the product of sums for that same truth table. I have: NOTE: $A' = \text{not} A$

$$(A+B+C)(A+B+C')(A+B'+C)(A'+B+C)$$

which I have simplified to (so far):

$$A + AB + AC' + B + BC' + AC + BC + A'B' + B'C + A'C + C$$

which I know should simplify to:

$$AB + BC + AC$$

I have used the Boolean algebra rules that I know, I just need help learning the rules that I don't know.

Thanks!

Answer 1

$$\begin{align} (A+B+C)(A+B+\overline C)(A+\overline B + C)(\overline A+B+C) &=\\ (A+B+C\overline C)(A+\overline B + C)(\overline A+B+C) &=\\ (A+B)(C+(A+ \overline B)(\overline A + B)) &=\\ (A+B)(C+AB+\overline A \overline B) &=\\ AC + AB + 0 + BC+AB + 0 &=\\ AB + BC + AC&\end{align}$$