# Boolean Algebra: Sum of Products

Discussion in 'Homework Help' started by Zachlr_ms, Sep 16, 2013.

1. ### Zachlr_ms Thread Starter New Member

Sep 16, 2013
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0
Hey all,
I'm trying to wrap my head around boolean algebra, specifically sum of products and product of sums. I know it has something to do with the distributive law, and possibly DeMorgan's laws, but I cannot figure it out.

I'm stuck on a problem on my homework. The instructions are to multiply out and simplify to obtain a product of sums. The problem goes as follows:
(A' + B + C')(A' + C' + D)(B' + D')

I grouped the A' + C' together using the law (X + Y)(X + Z) = X + YZ and got
A' + C' + BD(B'D')
Then I thought I could distribute the BD to make it BDB' + BDD'. Using the X + X' = 0 law I came up with A' + C' +B +D. The solution in the book is A'B' + A'D' + C'B' + C'D'. I'm way off and I have no idea what I'm doing. I'm sure if I could just grasp the concept of POS and SOP It would become easy, but so far I have not been able to understand.

Thanks for the help,
Zach

2. ### WBahn Moderator

Mar 31, 2012
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4,917
Take your best shot at describing what YOUR current understanding of each is. Then we can look at what you are thinking and try to fix any misconceptions or holes you might have.

Are you SURE the problem told you to obtain a product of sums? The answer you say the book provides is a sum of products and the problem is given as a product of sums.

3. ### djsfantasi AAC Fanatic!

Apr 11, 2010
2,911
880
Whoa there! Take baby steps.

This "law" simply doesn't make sense to me.

You could also approach this using simple algebra (Boolean algebra) and multiply the terms out and simplify later. Step by step... Don't make any leaps of faith!

Like WBahn said, "Take your best shot at describing what YOUR current understanding of each is. Then we can look at what you are thinking and try to fix any misconceptions or holes you might have." He is a great teacher.

4. ### Zachlr_ms Thread Starter New Member

Sep 16, 2013
2
0
My current understanding is that product of sums means to reduce it into different products added together. As in ab + cd + ef etc. Sum of products is the opposite. I wanted added terms multiplied together, such as (a+b)(c+d) etc.

The problem in the book (Fundamentals of Logic Design by Roth and Kinney) states the following: Multiply out and simplify to obtain a sum of products:
(a) (A+B)(C+B)(D'+B)(ACD'+E)
(b) another problem
(c) so on and so forth.

The answer in the back is ACD' + BE.

djsfantasi, I was mistaken. Looking back, the Laws of complementarity say that X + X' = 1 and X * X' = 0. I must have mixed them up.

Thanks for the replies. I've been very busy lately, but I'll try and stay on top of this thread.

Nov 9, 2007
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6. ### WBahn Moderator

Mar 31, 2012
18,088
4,917
This understanding is fine. Now consider your original question in light of this understanding:

The problem is already in product of sums as given!

The solution given is in sum of products.

This is why I asked you to verify that you were stating the problem and what you were trying to do correctly.

So have you done this? If not, do it here and show your work, in detail, for as far as you can get and we will help you get the rest of the way.

You shouldn't have to "look back" for these. They should be very obvious to you in short order -- you need to get to that point. X+X' requires that the two terms be opposite, meaning that one of them WILL be True. Since OR is True if ANY term is True, the result must always be True. Similarly, X*X' requires that the two factors be opposite, meaning that one of them WILL be False. Since AND is False if ANY factor is False, the result must always be False.

7. ### djsfantasi AAC Fanatic!

Apr 11, 2010
2,911
880
Ok, that makes much more sense to me.

WBahn beat me to the explanation!