Digital Teaching Aid (DED Philippinen, 86 p.) 
Karnaugh Mapping II  Lesson 5 

Titel: Karnaugh Mapping II
Objectives:
 Understand the product of sums method
 Able to use “Don't care conditions” for more effective simplifications
Time 
Method 
Topic 
Way 
Remark  


* Review Lesson 4 
  


* Introduction 
  


* Product of sums method    

 
 Fundamental sums 
 

 
 Product off sums simplifications   


* Don't care conditions    

 
Truth table 
 

 
 Karnaugh map 
 


* Review exercise 

Worksheet No. 5  

S: Speech  
B: Boardscript  
Karnaugh Mapping II
Instead of using the sum of products method we can also take the product of sums method which is based on the fundamental sums:
Fig. 51: Truth table with fundamental sums
A 
B 
C 
X  
Fundamental sums 
0 
0 
0 
0 
® 
_{} 
0 
0 
1 
1   
0 
1 
0 
1   
0 
1 
1 
0 
® 
_{} 
1 
0 
0 
1   
1 
0 
1 
1   
1 
1 
0 
0 
® 
_{} 
1 
1 
1 
1   
Product of sums simplification
A equation gained through the sum of product method can be realized into a NANDNAND logic circuit.
Fig. 52: NANDNAND logic
circuit
If we simplify the Karnaugh map in Fig. 52 we get the following equation:
_{}
Based on this equation we can draw the logic circuit like in Fig. 52. This solution is made with the help of the sum of product method. If we want to try the product of sums method we have to go through the following steps:
1. Complement the Karnaugh map and draw the complementary NANDNAND circuit:
Fig. 53: Complementary NANDNAND
circuit
The simplified equation is:
_{}
2. Convert the complementary NANDNAND circuit to a NORNOR circuit:
Fig. 54: NORNOR circuit
we have changed all NAND to NOR and complemented all signals.
3. Now we got a product of sums solution for X. The last step is to compare both circuits (NANDNAND, NORNOR) to find out which circuit is the simpler one, meaning the cheaper one. Because we need less gates. In our case the sum of product solution (NORNOR circuit) is the better solution.
Summary of process
1. Convert the truth table into a Karnaugh map
 Write the sum of products equation
 Draw the NANDNAND circuit
2. Complement the Karnaugh map
 Draw the complementary NANDNAND circuit _{}
3. Convert the complementary NANDNAND circuit to a NORNOR circuit.
 Change all NAND to NOR
 Complement all signals
4. Compare NANDNAND circuit with NORNOR circuit.
In some digital systems, certain input conditions never occur during normal operations; therefore, the corresponding output never appears.
Þ It is indicated in the truth table by an X.
Fig. 55: Don't care conditions
A 
B 
C 
D 
Z  
0 
0 
0 
0 
0  
0 
0 
0 
1 
0  
0 
0 
1 
0 
0  
0 
0 
1 
1 
0  
0 
1 
0 
0 
0  
0 
1 
0 
1 
0  
0 
1 
1 
0 
0  
0 
1 
1 
1 
0  
1 
0 
0 
0 
0  
1 
0 
0 
1 
1  
1 
0 
1 
0 
X  
1 
0 
1 
1 
X  
1 
1 
0 
0 
X 
Don't care conditions 
1 
1 
0 
1 
X  
1 
1 
1 
0 
X  
1 
1 
1 
1 
X  
Don't care conditions are like wild cards, you can let them stand for what ever you like:
Fig. 56: Karnaugh map and
simplified equation
HO: Simplify the following Boolean function:
_{}
That has the don't care conditions:
_{}
Solution:
First we have to translate the function into a truth table:
W 
X 
Y 
Z 
F 
0 
0 
0 
0 
X 
0 
0 
0 
1 
1 
0 
0 
1 
0 
X 
0 
0 
1 
1 
1 
0 
1 
0 
0 
0 
0 
1 
0 
1 
X 
0 
1 
1 
0 
0 
0 
1 
1 
1 
1 
1 
0 
0 
0 
0 
1 
0 
0 
1 
0 
1 
0 
1 
0 
0 
1 
0 
1 
1 
1 
1 
1 
0 
0 
0 
1 
1 
0 
1 
0 
1 
1 
1 
0 
0 
1 
1 
1 
1 
1 
Now we have to convert it into a Karnaugh map and simplify it:
Figure
No. 1
A 
B 
C 
D 
Y 
0 
0 
0 
0 
0 
0 
0 
0 
1 
1 
0 
0 
1 
0 
0 
0 
0 
1 
1 
0 
0 
1 
0 
0 
0 
0 
1 
0 
1 
1 
0 
1 
1 
0 
0 
0 
1 
1 
1 
0 
1 
0 
0 
0 
0 
1 
0 
0 
1 
0 
1 
0 
1 
0 
1 
1 
0 
1 
1 
1 
1 
1 
0 
0 
1 
1 
1 
0 
1 
1 
1 
1 
1 
0 
1 
1 
1 
1 
1 
1 
a) Draw the Karnaugh map.
b) Encircle all octets, quads and pairs you can find.
c) What is the simplified Boolean equation for the Karnaugh map?
d) Draw the logic circuit.
e) Suppose the last six entries of the truth table are changed to don't cares. Using the Karnaugh map, show the simplified circuit.
f) What is the simplified NORNOR circuit?