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

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
_{}