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I'm having problems with the following expression:

(A'+B)'+B(A'+AC)+ABC'

And here is what I tried to simplify:

AB' + B(A'+AC) + ABC' (De Morgan's) AB' + B(A'+C) + ABC' (Identity) AB' + A'B + BC + ABC' (Distribuitive) 

Here I convert to Standard SOP Form (first term is missing C or C'; second term is missing C or C'; third term is missing A or A')

I run this on Karnaugh maps, and final result is: A+B

For the Identity rule, I read: Boolean Simplification of A'B'C'+AB'C'+ABC'

(A'+AB = A'+B)

What exactly am I doing wrong here?

  • 0
    $AB' + B = A(B'+B)$? Which law are you using here?2012-09-06

3 Answers 3

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We'll start with the most obvious simplifications.

$(A' + B')' + B(A' + AC) + ABC'$

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

$AB' + BA' + BC + ABC'$

$A(B' + BC') + BA' + BC$

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

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

At this point the expression cannot be obviously reduced further. The secret is to use the consensus theorem to introduce redundancy. The consensus theorem states that $XY + X'Z + YZ = XY + X'Z$ If we set $XY = A'B$ and $X'Z = AC'$ then our expession becomes

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

From here we can do the standard reductions.

$AB' + AC' + A'B + B$

$AB' + AC' + B$

$A + B + AC'$

$A + B$

  • 0
    Cheers for your help, seems the online tool is giving out the wrong result then...2012-09-06
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$(A'+B)'+B(A'+AC)+ABC'=AB'+BA'+BAC+ABC'=AB'+BA'+AB(C+C')=AB'+BA'+AB=AB'+B(A+A')=AB'+B=A+B$

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(A'+B)'+B(A'+AC)+ABC' =  (A'+B)'+B(A'+C)+ABC'         by absorption [ A'+AC  =  A'+C ] =  (A)(B')+B(A'+C)+ABC'         by DeMorgan [ (A'+B)'  =  (A)(B') ] =  (A)(B')+(BA'+BC)+ABC'        by distributivity [ B(A'+C)  =  (BA'+BC) ] =  AB'+BA'+BC+ABC'      by associativity =  AB'+A'B+BC+ABC'      by commutativity =  AB'+AC'+A'B+BC       by absorption [ AB'+ABC'  =  AB'+AC' ] =  A'B+AB'+BC'+BC       by consensus [ AB'+AC'+A'B  =  A'B+AB'+BC' ] =  A'B+AB'+B        by absorption [ BC'+BC  =  B ] =  B+AB'        by absorption [ A'B+B  =  B ] =  A+B      by absorption [ B+AB'  =  A+B ] 

Answer: A+B

Using tool at www.logicminimizer.com