How to simplify sum of products boolean expression?

Question

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?

Answer 1

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.