# Boolean algebra simplification

#### retlig

Joined Sep 8, 2021
7
A´BC´ + A´D + BC´D´

Could anyone point me in the right direction in simplifying this expression. I've been trying to solve it for awhile but I'm stuck.

Cheers!

#### dl324

Joined Mar 30, 2015
16,696
Could anyone point me in the right direction in simplifying this expression.
I'd use a Kmap unless you were required to use boolean algebra.

#### RBR1317

Joined Nov 13, 2010
712
Logisim is public-domain software that does logic minimization & K-maps. (I use the version 2.7.1 jar file)
Of course, this won't help you to pass an exam on minimization; but then you could ask why be tested for a skill that is better done with software.

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#### retlig

Joined Sep 8, 2021
7
Thanks for additional replies. This was a part of an recommended assignment in which we should simplify the expressions with Boolean algebra. But this was the only one out of approximately 20 that I couldn't solve. For the actual exam I don't think it will matter, but I'm gonna discuss it with the examiner beforehand.

Cheers and have a wonderful weekend!

#### MrAl

Joined Jun 17, 2014
11,272
A´BC´ + A´D + BC´D´

Could anyone point me in the right direction in simplifying this expression. I've been trying to solve it for awhile but I'm stuck.

Cheers!
Just a little note...

Sometimes it helps to just negate the entire expression, simplify that, then negate that result and that brings you to an expression that is logically equivalent to the first but is often simpler. This may not work for every problem but it works for a lot of them and negating an expression is not hard to do.

Also, since you are working with a small number of variables (4 here) you could even print out or just write down the truth table for this expression and then see if that gives you any insight. With four variables you only have 16 different possible combinations for the variables so that will lead to a table with only 16 entries in it. When you get an expression with a very large number of variables this gets more difficult though because each new variable brings in twice the number of rows in the table. So with 5 variables there will be 32 rows, with 6 vars 64 rows, ..., and now jumping ahead with 32 variables there will be over 4 billion rows!.

You can also use that method to check your results. If the truth table of the original matches exactly to your solution truth table, then you did it right (although it may not be in the simplest form yet).

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#### JBMuhandis

Joined Sep 11, 2021
2
Logisim is public-domain software that does logic minimization & K-maps. (I use the version 2.7.1 jar file)
Of course, this won't help you to pass an exam on minimization; but then you could ask why be tested for a skill that is better done with software.
Please how to I access the logisim tool ?

#### JBMuhandis

Joined Sep 11, 2021
2
Logisim is public-domain software that does logic minimization & K-maps. (I use the version 2.7.1 jar file)
Of course, this won't help you to pass an exam on minimization; but then you could ask why be tested for a skill that is better done with software.
how do I access the software please ?

#### retlig

Joined Sep 8, 2021
7
how do I access the software please ?

#### MrAl

Joined Jun 17, 2014
11,272
how do I access the software please ?
Hello again,

Are you sure you want to use software? You probably wont learn as much that way.

I did verify that this expression can be reduced.
I also verified that negating, simplifying, negating then simplifying again did not seem to work for this expression as i got the original expression back again which of course does not help. There may be alternate ways of simplifying the resulting expressions but i did not look for them.
The way i ended up simplifying it was i simply printed out the truth table and then by inspection found a reduced expression. It was not hard to do because there are only 4 inputs.

#### RBR1317

Joined Nov 13, 2010
712
The way i ended up simplifying it was i simply printed out the truth table and then by inspection found a reduced expression.
That is essentially what a Karnaugh Map does. But the K-map is easier to apply. And K-map software is easier yet.

#### click_here

Joined Sep 22, 2020
548
First step when doing any logic problems is to do a k-map - This will tell you the answer.

Or if you're forced to do the boolean algebra, the trick for this one is to break up A'BC' into A'BC'D+A'BC'D'.

Once you've done that it's easy

#### click_here

Joined Sep 22, 2020
548
I should also add how I knew that splitting A'BC' into A'BC'D+A'BC'D' would work - I looked at the k-map that I made on a whiteboard and saw that half of A'BC' was in the circle of A'D and the other part was in the circle of BC'D'. Therefore if I split A'BC' into two, it would disappear with the other two terms.

Let me know if you need any more help

#### click_here

Joined Sep 22, 2020
548
I'm guessing that you have finished your assignment, so I'll add my working out for you if you're still curious/stuck...

#### MrAl

Joined Jun 17, 2014
11,272
This simple problem interested me at first because it was in the homework section, but then i found that the "invert, simplify, invert, simplify" method did not work. Although i knew that method does not always work, i decided to try to find out exactly why it did not work for this problem. What i ran into was an entirely different way to solve logic problems that do not involve maps or manually scanning truth tables (although i did solve it with a simple truth table too). I think this new method would be very amenable to quantum computing so it is a little more interesting to me now.

#### click_here

Joined Sep 22, 2020
548
This simple problem interested me at first because it was in the homework section, but then i found that the "invert, simplify, invert, simplify" method did not work. Although i knew that method does not always work, i decided to try to find out exactly why it did not work for this problem. What i ran into was an entirely different way to solve logic problems that do not involve maps or manually scanning truth tables (although i did solve it with a simple truth table too). I think this new method would be very amenable to quantum computing so it is a little more interesting to me now.
What specifically did you do?

#### MrAl

Joined Jun 17, 2014
11,272
What specifically did you do?
What you mean with the truth table solution?
I am not sure i want to reveal the other method just yet i dont see it mentioned anywhere else in some 50 years so far. I even studied automated reasoning and yet never ran into anything like this before.
Come to think of it, i dont think i ever saw that other method of inverting and simplifying either, but then again that's not always applicable.

#### click_here

Joined Sep 22, 2020
548
What you mean with the truth table solution?
I am not sure i want to reveal the other method just yet i dont see it mentioned anywhere else in some 50 years so far. I even studied automated reasoning and yet never ran into anything like this before.
Come to think of it, i dont think i ever saw that other method of inverting and simplifying either, but then again that's not always applicable.
Sounds like you have come up with something great - I once came up with a way using venn diagrams.

I found my math professor at university very excited and went through it with him.

After that he said that it was great that I had discovered that, but that was actually where boolean algebra was developed from!

I'll show you how I did it later, but I'd love to hear how you did yours

#### click_here

Joined Sep 22, 2020
548

You can use that to solve simple bool equations by colouring in sections and describing the final diagram...

e.g.
A´BC´ + A´D + BC´D´

From there you could describe the 2 shaded areas as BC'D' + A'D

You can also see other ways of describing it quite quickly by breaking it into different sections

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