# Boolean Algebra - Product of Sums

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!

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If you take $A=1, B=C=0$, what does the middle expression evaluate to? What about the other two? –  Karolis Juodelė Feb 10 '13 at 16:52
B = C = 0? I'm not sure that I understand. –  mkjo0617 Feb 10 '13 at 17:01
In the middle expression $A = 1$ would make the whole expression true. That is not the case for the other two, if $B$ and $C$ are false. –  Karolis Juodelė Feb 10 '13 at 17:05
It seems that I'm doing it totally wrong. What does $A+B$ and $AB$ means in this context? Does $A+B$ mean $A\vee B$ and $AB$ mean $A\wedge B$? –  zaarcis Feb 10 '13 at 17:53
Yes, it seems so. Sorry for disturbance, will get used to this notation ASAP. –  zaarcis Feb 10 '13 at 17:58

\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}

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I think I multiplied it out too much and that's why I got confused. I understand now. Thank you so much! –  mkjo0617 Feb 10 '13 at 17:25