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close this folderKarnaugh Mapping - Lesson 4
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View the documentIntroduction
View the documentKarnaugh map
View the documentTruth table to Karnaugh map
View the documentPairs, Quads, and Octets
View the documentOverlapping and Rolling

Pairs, Quads, and Octets

Pairs


Fig. 4-4: Four variable simplification

As you see in Fig. 4-4, 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. 4-5: Pairs

Whenever you see a pair first encircle it and then simplify to get the simplified Boolean expression:


Quad


Fig. 4-6: 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. 4-7: Quad

The variables B and D can be eliminated. So we get the following equation:

X = A C

Octet


Fig. 4-8: 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. 4-9: Karnaugh map