# Simplify a logic expression

I need to simplify the following logic expression:

$$\begin{array}{|c|c|c|c|c|} \hline A & B & C & X & SOP \\ \hline 0 & 0 & 0 & 0 & \\ \hline 0 & 0 & 1 & 1 & \overline{A}\ \overline{B}C \\ \hline 0 & 1 & 0 & 1 & \overline AB\overline C\\ \hline 0 & 1 & 1 & 1 & \overline ABC \\ \hline 1 & 0 & 0 & 0 & \\ \hline 1 & 0 & 1 & 1 & A\overline BC\\ \hline 1 & 1 & 0 & 0 &\\ \hline 1 & 1 & 1 & 1 & ABC \\ \hline \end{array}$$

$$X=\overline{A}\ \overline{B}C+\overline AB\overline C+\overline ABC+A\overline BC+ABC$$

I get the following logic expression after simplification:

\begin{aligned}X &=\overline{A}\ \overline{B}C+\overline AB\overline C+\overline ABC+A\overline BC+ABC\\ &=ABC+\overline ABC+\overline{A}\ \overline{B}C+A\overline BC+\overline AB\overline C\\ &=BC(A+\overline A)+\overline BC(A+\overline A)+\overline AB\overline C\\ &=BC+\overline BC+\overline AB\overline C\\ &=C(B+\overline B)+\overline AB\overline C\\ &=C+\overline AB\overline C \end{aligned}

However, K-MAP and Logic Friday each give me a different answer.

Edit: I changed the SOP, there was a mistake.

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Can you post how you get that result? –  Vladimir Cravero Aug 16 '14 at 10:32
Your formula is wrong anyway, it's not just "not simplified enough" –  Vladimir Cravero Aug 16 '14 at 10:34
I posted how I did the simplification. I don't see where I went wrong. –  user51414 Aug 16 '14 at 10:43
$\overline{AB}\ne\overline{A}\cdot\overline{B}$ –  jippie Aug 16 '14 at 12:08
@Ruslan it appeared like ~(A&B). Reading the simplification steps in the original post says that I am correct. –  nidhin Aug 16 '14 at 17:47

The point where you arrived is almost right:

$$C+\overline{A}B\overline{C}=C(1+\overline{A}B)+\overline{A}B\overline{C}=\\= C+\overline{A}BC+\overline{A}B\overline{C}=C+\overline{A}B(C+\overline{C})=\\= C+\overline{A}B$$

That should be the same answer the K-map and whatever reduction software should give you.

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Thank you very much, it makes sense now! –  user51414 Aug 16 '14 at 11:01
@user2357111317192329 yeah, these things just need a TON of practice –  Vladimir Cravero Aug 16 '14 at 11:02
I am still in first year EECS. Do you think I will be doing a lot of those in later years? I find them very interesting for some reason! –  user51414 Aug 16 '14 at 11:25
I found them interesting too because of the somewhat different algebra. You won't do much of these, automated tools usually simplify expressions, it will be needed for the exam ofc. –  Vladimir Cravero Aug 16 '14 at 11:47

If you are getting two different answer don't worry

Just make the truth table for both the expressions and compare if both give the same answer then you are correct.

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