Karnaugh map simplification

I'm working through an example that looks like a fairly simple Karnaugh map and simplification, but I feel stupid that I can't seem to understand the correct answer.

This is the map:

My groupings:

This is the equation:

But I don't know how it simplifies to the correct answer here:

I understand why the NOT A goes outside of the bracket, but I don't see how the C OR NOT C just becomes a C inside the bracket?

• Welcome to Computer Science! Note that you can use LaTeX here to typeset mathematics in a more readable way. See here for a short introduction. – Raphael Nov 20 '14 at 10:26
• Create a full table for both formulae. Are they equivalent? – Raphael Nov 20 '14 at 10:27

2 Answers

Check for solution. Grouping should be done using 2 quads

• Note that you can use Markdown and LaTeX here to typeset mathematics, rather than using an image. See here for a short introduction to Latex. – D.W. Dec 30 '15 at 1:02

First, when you are trying to group, you must use the biggest possible group. A group's size has to be a power of 2 (1, 2, 4, 8...).

In this case you need to make 2 groups:

cells 4, 5, 12, 13: $A'D$
cells 8, 9, 12, 13: $A'C$

Represented as SOP: $A'D+A'C$

Using the Distributive Law:

$A'D+A'C=A'(C+D)$

About the way that you tried:

You grouped:

cells 4, 5: $A'C'D$
cells 8, 9, 12, 13: $A'C$

Assuming that you didn't notice the bigger grouping option, the expression that you have can still be simplified using Boolean Laws:

$A'C'D+A'C=A'(C'D+C)$ Distributive Law
According to one of the laws: $A+A'B=A+B$
So: $A'(C'D+C)=A'(C+D)$