Boolean Simplification Problem

Question

I have to simplify the figure

I came up with the equation $$(ac+\bar b)\cdot (a+b+\bar c) \cdot (b+ac).$$

Help me out

Attempted Answer:

\begin{align*} (ac+\bar b)\cdot (a+b+c)\cdot (b+ac) &=(ac+abc+ac+a\bar b+\bar b c)\cdot(b+ac)\\ &=abc+abc+abc+ac+abc+ac+a\bar b c+a\bar b c \\ &=abc+a\bar b c+ac\\ &=ac(b+\bar b)+ac\\ &=ac+ac=ac \end{align*}

What do you know about rules for manipulating these kinds of expressions? What laws are you allowed to use? — dfeuer, Oct 13 '13 at 04:55
Well, I'm hoping your teacher and/or textbook explained something about how to go about doing that. Karnaugh maps, absorption laws, distributive laws? Anything? — dfeuer, Oct 13 '13 at 05:00
its says but i usually prefer algebric method but this time i tried to do it and i ended up having the Answer: AC but not sure its right answer or not !!!! — k31453, Oct 13 '13 at 05:10
The second factor is supposed to be $a+b+\bar c$, but you switched it to $a+b+c$ in your solution attempt. — dfeuer, Oct 13 '13 at 05:47
Note: it would be better, in this case, to move your solution attempt to an answer section ("answer your own question"), because that way this question will show up as answered rather than staying in the unanswered queue forever and ever amen. — dfeuer, Oct 13 '13 at 05:49

score 0 · Answer 1 · answered Oct 13 '13 at 08:23

OK, here goes:

We need to simplify

$(ac + \bar b)(a + b + \bar c)(b + ac). \tag{1}$

I'll do this by a method which is a little different than the OP's, but will yield the same answer $ac$; this will corroborate k31453's work. First, I observe that

$(ac + \bar b)(a + b + \bar c)(b + ac) = (ac + \bar b)(ac + b)(a + b + \bar c), \tag{2}$

and we have

$(ac + \bar b)(ac + b) = (ac)(ac) + acb + ac \bar b + \bar b b = ac + acb + ac \bar b, \tag{3}$

since $(ac)(ac) = ac$ and $\bar b b = 0$. Next, I use

$acb + ac \bar b = ac(b + \bar b) = ac1 = ac \tag{4}$

to obtain

$(ac + \bar b)(ac + b) = ac + ac = ac. \tag{5}$

Substituting this back into (2) yields

$ac(a + b + \bar c) = ac + abc = ac(1 + b) = ac1 = ac, \tag{6}$

confirming k31453's calculation.

Hope this helps. Cheerio,

and as always,

Fiat Lux!!!

Boolean Simplification Problem

1 Answers1