Boolean Logic Simplification Issue

Question

I hate this stuff. Just to note. + means OR * means AND ! means NOT.

(A+B) * (A+C) * (!B + !C)

(A | B) & (A | C) & (!B | !C) // more conventnal

The answer is A(!B + !C)

I'm trying to get there.

So I start off with using Distributive rule which gets me here (A + B) * C * (!B + !C)

and that's where I'm stuck. I know I some how have to get rid of B and C but I see no way using any of the rules. I've got Identity, Null, Itempotent, Inverse, Commutative, Associative, Distributive, De Morgan's, and Cancellation to work with.

Am I starting off wrong? I really just used the only rule that I could see possible to even use. I was horrible with doing Proofs in Geometry and this stuff just makes me feel like that all over again.

UncleO · Accepted Answer · 2010-09-24 01:55:07Z

1

Your first step is wrong.

(A+B) * (A+C) is (A+(B*C)).

Next, (!B + !C) is !(B*C).

So we get A*(!(B*C)) + (B*C)*(!(B*C)), which gives the desired result.

answered Sep 24, 2010 at 1:55

UncleO

8,45924 silver badges30 bronze badges

Sign up to request clarification or add additional context in comments.

3 Comments

Doug Over a year ago

How do you go from (A+(BC)) * !(BC) to A*(!(BC)) + (BC) * (!(B*C)) I missed something big there.

dash-tom-bang Over a year ago

@Doug - he's distributing (X + Y) * !Y to (X * !Y) + (Y * !Y).

UncleO Over a year ago

dash-tom-bang is right. Distribute the !(BC) over the +. The point of that is we recognize (BC)*(!(BC)) is "false", so it drops out of the or statement. Convert !(BC) back to (!B + !C) and you're done.

Bear Monkey · Accepted Answer · 2010-09-24 02:18:50Z

1

(A | B) & (A | C) & (!B | !C) = (A | (B & C)) & (!B | !C)
                              = (A | (B & C)) & !(B & C)

substitute D = (B & C)

                              = (A | D) & !D 
                              = A & !D
                              = A & !(B & C)
                              = A & (!B | !C)

answered Sep 24, 2010 at 2:18

Bear Monkey

5212 silver badges9 bronze badges

Collectives™ on Stack Overflow

Boolean Logic Simplification Issue

2 Answers 2

3 Comments

Comments

Your Answer

Hot Network Questions

Collectives™ on Stack Overflow

2 Answers 2

3 Comments

Comments

Your Answer

Sign up or log in

Post as a guest

Related