# Boolean Simplification Help

Discussion in 'Homework Help' started by Pariah, Oct 5, 2008.

1. ### Pariah Thread Starter New Member

Oct 5, 2008
3
0
1. The problem statement, all variables and given/known data
We have been given a task to develop a circuit which displays the square of a binary number on a 3 x 7 seq displays.

I have already gone through and done up the Karnaugh Maps for the task and have identified the minterms. However, I believe that these can still be simplified even more.

2. Relevant equations
Karnaugh Maps Output = A'B'CD + A'BC'D + A'BCD' + ABCD + B'C'D' + AB'C'

3. The attempt at a solution
A'B'CD + A'BC'D + A'BCD' + ABCD + B'C'D' + AB'C'
Factor out A' and D from minterms 1 and 2
A'D(b' + b) (c + c')
= A'D + A'BC'D + ABCD + B'C'D' + AB'C'
Factor out B and D from minterms 2 and 3
bd(a' + a)(c' + c)
= A'D + BD + B'C'D' + AB'C'

This is where I am getting stuck. Is it possible to further simplify the equation or is this the final solution???

2. ### mik3 Senior Member

Feb 4, 2008
4,846
70
Well, i think you didnt get the Karnaugh map correct because you simplified it again using the boolean algebra. This is not correct because if you do solve correct a Karnaugh map you get the simplest solution. However, i can't see a simpler solution to your final answer.

3. ### Dave Retired Moderator

Nov 17, 2003
6,960
172
Common factor simplification will reduce the gate count:

A'D + BD = D(A' + B)

This reduces from 2x 2I/P AND-gates and 1x 2I/P OR-gate to 1x 2I/P AND-gate and 1x 2I/P OR-gate

B'C'D' + AB'C' = B'C'(A + D')

This reduces from 2x 3I/P AND-gates and 1x 2I/P OR-gate to 1x 3I/P AND-gate and 1x 2I/P OR-gate.

So the simplified expression would be: D(A' + B) + B'C'(A + D')

Dave

4. ### Pariah Thread Starter New Member

Oct 5, 2008
3
0
Thanks alot for the help.

Also, I understand that the Karnaugh Maps usually produce the simplest forms but I saw that the equation could be further simplified because of the common factors.

So Am I right in thinking that the equation that Dave gave is correct or should I not try to further minimize the equation???

I am confused

5. ### hgmjr Retired Moderator

Jan 28, 2005
9,029
219
Take a look at the material on boolean algebra in the AAC ebook.

hgmjr

6. ### Dave Retired Moderator

Nov 17, 2003
6,960
172
You may be able to DeMorganise the expression to less gates, however this may not translate to a more simplified expression in terms of transistors or terms used. You would need to play around with DeMorgans Theorems to see if you can actually simplify the expression further - it isn't obvious from the above expression if it would work.

Your current expression utilises only one of each term and its compliment (actually there is no C); I would commend that this expression is fully simplified (note there are probably several simplifications of the original expression which would be equally simplified).

Dave

7. ### Pariah Thread Starter New Member

Oct 5, 2008
3
0
Dave, so your saying that my original equation should not be further simplified???

Also, thanks alot for the help guys.

I think you are correct in saying that the original form is already simplified and I am only complicating everything by overthinking

Once again thanks alot

If I have some more questions I will be sure to ask

8. ### Dave Retired Moderator

Nov 17, 2003
6,960
172
I cannot see that it can be logically simplified further. Like I said previously, you may be able to DeMorganise the expression to remove a term and its compliment, however this is not obvious nor would it yield a more simplified expression depending on how you make the judgement of simplification.

I would commend that your expression is in its simplified form.

No problems, that is why this site is here. Be sure to check the e-book and worksheets to help you with your studies.

Dave

9. ### Ratch New Member

Mar 20, 2007
1,068
4
Pariah,

Sorry I did respond to this sooner, but I just got back from vacation.

I did the simplification using the Quine-McCluskey tabulation method http://www.cs.ualberta.ca/~amaral/courses/329/webslides/Topic5-QuineMcCluskey/sld001.htm , and verified that you cannot simplify the above expression any more. You should be able to verify that by using a K-map.

Your Boolean algebra is in error. There is no way that Minterms(3,5) are going to reduce to a two variable term (A'D).