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I need to simplify the following expression:

$$ P = AC + A\bar{B} + \bar{A}BC + \bar{A}\bar{B}\bar{C} $$

Using a K-Map I get the correct answer of:

$$ P =AC + A\bar{B} + BC + \bar{B}\bar{C} $$

My problem is that I don't understand how

$$ \bar{A}BC + \bar{A}\bar{B}\bar{C} = BC + \bar{B}\bar{C} $$

I've tried using logical adjacency, expanding variables, DeMorgan's law, and the basic properties like distributivity to manipulate the variables, but I can never end up with the answer I get for the K-Map. Am I missing some theorem or property?

Movers
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  • The equation $\bar{A}BC + \bar{A}\bar{B}\bar{C} = BC + \bar{B}\bar{C}$ does not hold – Ben Grossmann Sep 21 '16 at 00:44
  • The statement $\bar{A}BC+\bar{A}\bar{B}\bar{C}=BC+\bar{B}\bar{C}$ is not always true. However, the statement $AC+A\bar{B}+\bar{A}BC+\bar{A}\bar{B}\bar{C}=AC+A\bar{B}+BC+\bar{B}\bar{C}$ is always true. – JRN Sep 21 '16 at 00:44
  • How do you prove that algebraically? I tried expanding AC and A~B, but then I get 3 terms and not the four that I want. – Movers Sep 21 '16 at 00:48

1 Answers1

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Here's what's really going on: $$ AC + A \bar B + \bar A BC + \bar A \bar B \bar C = \\ AC + A(B + \bar B)C + A \bar B + A \bar B(C + \bar C) + \bar A BC + \bar A \bar B \bar C = \\ AC + A \bar B + [\bar ABC + A BC] + [A \bar B \bar C + \bar A \bar B \bar C] =\\ AC + A \bar B + [\bar A + A]BC + [A + \bar A]\bar B \bar C =\\ AC + A \bar B + BC + \bar B \bar C $$ what we're implicitly doing with the K-map is introducing redundant terms in the sum in order to allow for a tidy factorization.

Ben Grossmann
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  • I deleted my answer because yours is better (and came first). – JRN Sep 21 '16 at 01:00
  • @Omnomnomnom, do you think it's bad that I couldn't figure this out? It never even occurred to me that I could add the term A(B +~B)C. I learned about logical absorption, but I haven't thought of doing the reverse. Maybe I'm just not cut out for this. – Movers Sep 21 '16 at 01:11
  • @Movers the fact that you understand it now and can acknowledge the previous gap in your knowledge is, to me, a sign that you are cut out for this after all. This is also the kind of trick that you'll forget in a week, until you think about it and remember what it was after all. After that, it will stick. I think you're doing just fine. – Ben Grossmann Sep 21 '16 at 01:20
  • @Movers also, if this kind of manipulation were supposed to be easy or intuitive, they wouldn't be teaching you about K-maps. – Ben Grossmann Sep 21 '16 at 01:26
  • @Movers a handy tool for understanding these this is to remember that every expression can be broken down into a "sum of products". – Ben Grossmann Sep 21 '16 at 01:28