[Math] simplifying using Boolean Algebra.

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?

Best Answer

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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