# How to simplify sum of products boolean expression?

I started with this sum of products:

abc’d’ + abc’d + ab’cd’ + a’b’cd’ + a’bc’d + a’bcd + ab’c’d + a’b’c’d


I have been able to simplify to this:

b'c'd + abc' + a'bc + b'cd'


I can't seem to find a way to simplify it anymore. I have tried reading and watching a ton of tutorials but nothing is clicking.

Some things I have tried is finding the common values among the or expressions such c' or b'.

combining c'

c'(b'd + ab) + b'cd' + a'bc


combining b'

b'(c'd + cd') + abc' + a'bc


I get stuck with the simplification of combining b' or c'.

Can anyone provide any guidance as to what I am doing wrong or missing?

• Sometimes things just cannot be simplified any further. – Yuval Filmus Feb 21 '17 at 2:54
• Your simplified formula is not equivalent to the original. Let A=C=0 and B=D=1. The original would evaluate to true, but the simplified formula would not. (I used an online logic minimizer) – Dmitri Chubarov Feb 22 '17 at 5:27

From a Karnaugh map you can get a one group of 4s and three group of 2s. So you can reduce sum of 4 term in which three will have 3 literals and one with 2 literals. By doing it use K-map you will use few 1s for more than one terms so if you don't want to use a K-map just use idempotent rule($x = x + x$) and then simplify.