Digital Teaching Aid (DED Philippinen, 86 p.) 
Karnaugh Mapping  Lesson 4 

Titel: Karnaugh Mapping
Objectives:
 Able to derive Karnaugh maps from a given truth table
 Know how to simplify Karnaugh maps
Time 
Method 
Topic 
Way 
Remark  


* Review Lesson 3 
  
 
* Introduction 
  
 

 Karnaugh map 
 
 
* Karnaugh mapping 
  
 

 Truth table to Karnaugh map   
 

 Two variable map 
 
 

 Three variable map 
 
 

 Four variable map 
 
 

 Pairs, Quads, and Octets   
 

 Overlapping groups 
 
 

 Rolling the map 
 
 
* Review exercise 
WS 
Worksheet No. 4  

S: Speech  
B: Boardscript  
Karnaugh Mapping
Simplify a Boolean equation
Ex: 
_{}  

_{} 
_{}is common in each term 

_{} 
Again factor to get 

_{} 
Simplify 

_{} 
Simplify 

_{}  
As you see, Boolean algebra can be used to create simpler circuits. But if you not quite familiar with Boolean theorems it is difficult to find the best simplification. Therefore Karnaugh mapping is the better alternative for simplification.
The starting point for digital circuit design is usually the truth table which gives us the following information:
Under which input condition occurs a certain output condition.
Two variable map
Ex:
Fig. 41: Two variable Karnaugh
map
The truth table outputs are translated into the Karnaugh map. Each position in the map represents a certain fundamental product.
Three variable map
Ex:
Fig. 42: Three variable Karnaugh
map
Four variable map
Many digital systems process 4 bit numbers. For this reason, logic circuits are often designed to handle 4 input variables.
Ex:
(see Fig. 43)
Fig. 43: Four variable Karnaugh
map
Pairs
Fig. 44: Four variable
simplification
As you see in Fig. 44, only one variable goes from uncomplement to complement. Whenever this happens, you can eliminate the variable that changes form.
Proof: 
_{} 

_{} 

X = A B C 
Ex:
Fig. 45: Pairs
Whenever you see a pair first encircle it and then simplify to get the simplified Boolean expression:
_{}
Quad
Fig. 46: Quad
Quad: A group of 4 one's that are horizontally or vertically adjacent. End to end or in form of a square.
A quad eliminates two variables and their complements.
Proof: 
_{} 
(two pairs) 

X = A B (C + C) 


X = A B  
Encircle the quad and step through the different one's in the quad and determine which two variables go from complement to uncomplement (or vs), these are the variables that drop out.
Ex:
Fig. 47: Quad
The variables B and D can be eliminated. So we get the following equation:
X = A C
Octet
Fig. 48: Octet
An octet eliminates three variables and their complements.
Proof: 
_{} 
(two quads) 

X = A (C + C) 


X = A  
Karnaugh Simplifications
Process:
1. Draw the Karnaugh map
2. Look for octets and encircle them.
3. Look for quads and encircle them.
4. Look for pairs and encircle them.
5. Simplify and write down the equation.
Ex:
Fig. 49: Karnaugh map
_{}
Overlapping groups
Ex:
Fig. 410: Karnaugh map
Groups can overlap to get a simpler equation:
_{}
Rolling the map
Ex:
Fig. 411: Karnaugh map
Instead of encircling two pairs:
_{}
We can roll the map and encircle a quad:
_{}
HO: Simplify the following
map.
Solution:
_{}
HO: Simplify the following
map.
Solution:
_{}
No. 1 Simplify the following Karnaugh maps and write down the Boolean equation for every map.

_{} 
_{} 
_{} 
0 
0 
_{} 
1 
0 
_{} 
1 
1 
_{} 
0 
0 

_{} 
_{} 
_{} 
1 
1 
_{} 
0 
0 
_{} 
1 
0 
_{} 
1 
1 

_{} 
_{} 
_{} 
1 
0 
_{} 
0 
0 
_{} 
0 
0 
_{} 
1 
0 

_{} 
_{} 
_{} 
_{} 
_{} 
0 
1 
0 
0 
_{} 
1 
0 
1 
1 
_{} 
1 
0 
0 
1 
_{} 
0 
1 
1 
0 

_{} 
_{} 
_{} 
_{} 
_{} 
1 
1 
0 
1 
_{} 
0 
0 
1 
1 
_{} 
0 
0 
0 
1 
_{} 
1 
1 
0 
1 
No. 2 Translate each output (v, w, x, y, z) into a Karnaugh map, do the simplification and write down the Boolean equation.
A 
B 
C 
D 
v 
w 
x 
y 
z 
0 
0 
0 
0 
1 
1 
0 
1 
0 
0 
0 
0 
1 
0 
1 
0 
1 
1 
0 
0 
1 
0 
0 
1 
0 
0 
0 
0 
0 
1 
1 
0 
0 
0 
0 
1 
0 
1 
0 
0 
0 
0 
1 
0 
1 
0 
1 
0 
1 
1 
0 
1 
1 
0 
0 
1 
1 
0 
1 
1 
1 
1 
0 
0 
1 
1 
1 
1 
0 
1 
1 
0 
1 
0 
0 
0 
0 
1 
0 
0 
1 
1 
0 
0 
1 
1 
1 
1 
1 
0 
1 
0 
1 
0 
0 
1 
0 
1 
0 
1 
0 
1 
1 
1 
1 
1 
1 
1 
1 
1 
0 
0 
0 
0 
0 
0 
1 
1 
1 
0 
1 
1 
0 
0 
0 
1 
1 
1 
1 
0 
0 
0 
0 
1 
1 
1 
1 
1 
1 
0 
1 
0 
0 
0 