# Simplifying a boolean expression to a minimum number of literals

#### mcc123pa

Joined Sep 12, 2010
40
Hi everyone:

I was assigned the following problem for homeowork:

A'B'D+A'C'D+BD (a ' mark after a letter means a bar)

The directions read simplify the expression to an expression containing a minimum number of literals.

Here is my attempt, could someone please tell me if it is right? If I'm wrong, could you please post the correct way to do this?

A'B'D+BD+A'C'D (original function re-arranged)
(B+B')(A'D+D)+A'C'D
(1)(A'D+D)+A'C'D
A'D+D+A'C'D
A'D+A'C'D+D
A'D(1+C')+D
A'D(1)+D
D(A'+1)
D(1)
D

Thanks!

#### Georacer

Joined Nov 25, 2009
5,182
Unfortunately, it's not quite correct. The error is in the second line where you rush to make the factorization. Take your time and make one factorization at a time. Try it once more. If you find it too hard, feel free to ask again.

#### mcc123pa

Joined Sep 12, 2010
40
Hi, thanks for the response!

Here is my latest attempt:

A'B'D+A'C'D+BD (original function)

D(A'B'+A'C'+B) (factored out a D)

D((B+B')(A'+1)+A'C') (factored out B+B')

D(A'+1+A'C') (after converting B+B' to 1)

Is this as far as I can go with this? Is D(A'+1+A'C') my final answer?

#### Georacer

Joined Nov 25, 2009
5,182
You can't just factor out B+B' out of two terms and hope to get it right. Factor out one letter at a time to ensure you don't make logic mistakes.
Here's how it goes:

$$A'B'D+BD+A'C'D =\\ D(A'B'+B) +A'C'D=\ \ \ \text{% Factoring out D}\\ D(A'+B)(B'+B)+A'C'D=\ \ \ \text{% Using the identity X+YZ=(X+Y)(X+Z)}\\ D(A'+B)+A'C'D=\ \ \ \text{% B+B'=1 wich is a neutral term}\\ D(A'+B+A'C')=\ \ \ \text{% Factoring out D}\\ D(A'(C'+1)+B)=\ \ \ \text{% Factoring out A'}\\ D(A'+B)=\ \ \ \text{% C'+1=1}\\ A'D+BD$$

This is the final result.

• jmhutchins

#### mcc123pa

Joined Sep 12, 2010
40
Thanks so much for your help!!

#### Georacer

Joined Nov 25, 2009
5,182
The "Thanks" button is down on the right!
(Yeah, we all have our weaknesses... )

#### jmhutchins

Joined Feb 14, 2018
1
I know this thread is super old, but if I found it and someone else might as well.

I did a k-map from the truth table and ended up with ~A*D+A*D Which can be simplified to D. I'd be interested if someone else would do the same to see which answer they got.