# simplifying using Boolean Algebra.

I was doing the following question. Using the following rules of boolean algebra:

         _
law 1: X+X=1

law 2: X.1=X

law 3:X.Y+X.Z = X.(Y+Z)


simplify:

    __  _   __
ABC+ABC+ABC+ABC


I have tried to simplify starting off with law 1, but I get stuck because the complements somewhat confuse me.

The bar meaning complement or 'not'

Could someone explain?

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This is extremely confusing since it looks like you've switched from "primes" for complements to "overbars", and you're using juxtaposition for "and" instead of &. Can you please switch it to be consistent? –  rschwieb May 8 '12 at 13:16
@rschwieb I edited, sorry for confusion. –  Xabi May 8 '12 at 13:23
The notation in the laws and the notation in the question are still completely different. If you think my solution below is answering the question you asked, I can attempt to make your post clear. –  rschwieb May 8 '12 at 13:31
Are the second and third terms $\overline{A}\overline{B}C$ or are they $\overline{AB}C$? –  robjohn Jan 17 '13 at 21:20

If I understand correctly, the problem is to reduce $$(\overline{A \& B} \& C) + (\overline{A} \& B \& C)+ (\overline{A \& B} \& C) +(A\& B \& C)$$

You can deduce from the laws that $X+X=X$, so this is clearly already

$$=(\overline{A \& B} \& C) + (\overline{A} \& B \& C)+(A\& B \& C)$$ By law 3, then law 1, then law 2: $(\overline{A} \& B \& C)+(A\& B \& C)=(\overline{A}+A)\&B\& C=1\&B\& C=B\&C$, so the original expression is now:

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

By De Morgan's laws $\overline{A \& B}=(\overline{A}+\overline{B})$, and you can deduce the rest from your laws:

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

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

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This is perfect Thank you! –  Xabi May 8 '12 at 13:50

=C(AB+A'B')+ A'C(B+B') =C(1) + A'C(1) =C + A'C =C(1+A') =C according to (1+x=1)

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There are four variables P,B,M,O

The expression is

(P&B&M)+(B&O&-M)+(M&O&-P&-B)

What would be the simplified form of the above expression using boolean laws which are : _ law 1: X+X=1

law 2: X.1=X

law 3:X.Y+X.Z = X.(Y+Z) and so on ...!