Steps to Simplify Boolean Expression

Question

Could someone please example how

$(\bar A . \bar B . \bar C) + (\bar A . \bar B . C) + (A . B . \bar C) + (A . B . C)$

can be simplified to

$(\bar A . \bar B) + (A . B)$ ?

Here's what I've tried, but I don't trust all the steps:

$(\bar A . \bar B . \bar C) + (\bar A . \bar B . C) + (A . B . \bar C) + (A . B . C)$

$(\bar A . \bar B . \bar C) + (A . B . \bar C) + (\bar A . \bar B . C) + (A . B . C)$

$\bar C . ((\bar A . \bar B ) + (A . B)) + C . ((\bar A . \bar B) + (A . B))$

$(\bar C + C) . ((\bar A . \bar B ) + (A . B) + (\bar A . \bar B) + (A . B))$

$(\bar A . \bar B ) + (A . B) + (\bar A . \bar B) + (A . B)$

$2 . (\bar A . \bar B ) + 2 . (A . B)$

Now I'm stuck. Am I right so far? If so, why can I cancel those $2$s?

score 1 · Accepted Answer · answered Jan 14 '20 at 09:17

1

@Michael Rozenberg has already explained how to reduce the expression. Your method is also correct until this point:

$(\bar A . \bar B ) + (A . B) + (\bar A . \bar B) + (A . B)$

At this point, remember the laws of Boolean Algebra, which state that A + A = A. That allows you to simplify the expression further to $(\bar A . \bar B ) + (A . B) $, which is the desired result

answered Jan 14 '20 at 09:17

shree.pat18

141

Ah, so you never have coefficients of a term, like 2(AB), then? – Robin Andrews Jan 14 '20 at 09:36
1

Yeah, because we are dealing with binary values. There is nothing like more true or more false, only true or false, so A, 2A, 3A, and so on are all the same. – shree.pat18 Jan 14 '20 at 14:03

score 0 · Answer 2 · answered Jan 14 '20 at 09:01

0

Because $$\bar A \bar B \bar C + \bar A \bar BC + AB\bar C + ABC=$$ $$=\bar{A}\bar{B}(\bar{C}+C)+AB(\bar{C}+C)=\bar{A}\bar{B}+AB.$$

answered Jan 14 '20 at 09:01

Michael Rozenberg

194,933

Thanks for that. Would my steps also work, even though I took a less efficient route? – Robin Andrews Jan 14 '20 at 09:12
@Robin By your way we obtain: $(\bar{C}+C)(\bar{A}\bar{B}+AB)=\bar{A}\bar{B}+AB.$ – Michael Rozenberg Jan 14 '20 at 11:34
Can down-voter explain us why did you do it? – Michael Rozenberg Jan 14 '20 at 21:27

Steps to Simplify Boolean Expression

2 Answers2