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Karnaugh MapsKarnaugh Maps (K maps)(K maps)

2

What are KarnaughWhat are Karnaugh11 maps? maps?

Karnaugh maps provide an alternative way of simplifying logic circuits.

Instead of using Boolean algebra simplification techniques, you can transfer logic values from a Boolean statement or a truth table into a Karnaugh map.

The arrangement of 0's and 1's within the map helps you to visualise the logic relationships between the variables and leads directly to a simplified Boolean statement.

1Named for the American electrical engineer Maurice Karnaugh. 3

Karnaugh mapsKarnaugh maps

Karnaugh maps, or K-maps, are often used to simplify logic problems with 2,

3 or 4 variables. BA

For the case of 2 variables, we form a map consisting of 22=4 cellsas shown in Figure

AB

0 1

0

1

Cell = 2n ,where n is a number of variables

00 10

01 11

AB

0 1

0

1

AB

0 1

0

1

BA

BA AB

BA+ BA +

BA+ BA +

Maxterm Minterm

0 2

1 3

4

Karnaugh mapsKarnaugh maps

3 variables Karnaugh map

CBA

ABABCC 00 01 11 10

0

1

CBA CAB CBA

CBA BCA ABC CBA

0 2 6 4

531 7

Cell = 23=8

5

Karnaugh mapsKarnaugh maps

4 variables Karnaugh map

ABABCDCD 00 01 11 10

00

01

11

10

5

3

1

7

62

0 4

9

15

13

11

1014

12 8

6

Karnaugh mapsKarnaugh maps

The Karnaugh map is completed by entering a '1‘(or ‘0’) in each of the appropriate cells.

Within the map, adjacent cells containing 1's (or 0’s) are grouped together in twos, fours, or eights.

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ExampleExample

A B Y

0 0 0

0 1 1

1 0 1

1 1 1

2-variable Karnaugh maps are trivial but can be used to introduce the methods you need to learn. The map for a 2-input OR gate looks like this:

AB

0 1

0

1

1

11

BB

AA

A+BA+B

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ExampleExample

AA BB CC YY

00 00 00 11

00 00 11 11

00 11 00 00

00 11 11 00

11 00 00 11

11 00 11 11

11 11 00 11

11 11 11 00

CA

CAB +B

ABABCC 00 01 11 10

0

1

1

1

1 1

1

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Truth Table to K-Map MappingTruth Table to K-Map Mapping

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Four Variable K-MapW X Y Z FWXYZ

0 0 0 0 1

0 0 0 1 0

0 0 1 0 0

0 0 1 1 0

0 1 0 0 1

0 1 0 1 0

0 1 1 0 0

0 1 1 1 1

1 0 0 0 1

1 0 0 1 0

1 0 1 0 0

1 0 1 1 0

1 1 0 0 1

1 1 0 1 0

1 1 1 0 0

1 1 1 1 1

V

0 1 3 2

4 5 7 6

12 13 15 14

8 9 11 10

X W

X W

X W

X W

Z Y Z Y ZY ZY

1 00 1

1 00 1

1 00 0

1 00 0

Only onevariable changes for every row

change

Only one variable changes for

every column change

Fwxyz= Y Z + X Y Z

Individual WorkIndividual Work

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Question(1Question(1((

A Karnaugh map is nothing more than a special form of truth table, useful for reducing logic functions into minimal Boolean expressions.

Here is a truth table for a specific three-input logic circuit:

Out= A D + A C + B C

A

A

D

C

C

B

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Question(2Question(2((

A Karnaugh map is nothing more than a special form of truth table, useful for reducing logic functions into minimal Boolean expressions.

Here is a truth table for a four-input logic circuit:

Out= B C

C

13

Question(3Question(3((

A Karnaugh map is nothing more than a special form of truth table, useful for reducing logic functions into minimal Boolean expressions.

Here is a truth table for a four-input logic circuit:

Out = A B

B

14

Question(4Question(4((

A Karnaugh map is nothing more than a special form of truth table, useful for reducing logic functions into minimal Boolean expressions.

Here is a truth table for a four-input logic circuit:

Out = B C D + B C D

B C

D15

steps work experiment:

1.Solution truth table by Karnaugh MapsKarnaugh Maps  (K maps).(K maps).

2.2.Gate work by Karnaugh MapsGate work by Karnaugh Maps  (K maps).(K maps).

3.3.Applied to the gate on test Applied to the gate on test board and make sure by truth board and make sure by truth table.table.

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Experiment :GROUP (AExperiment :GROUP (A((AA BB CC DD YY

00 00 00 00 11

00 00 00 11 00

00 00 11 00 00

00 00 11 11 00

00 11 00 00 11

00 11 00 11 00

00 11 11 00 11

00 11 11 11 00

11 00 00 00 00

11 00 00 11 11

11 00 11 00 00

11 00 11 11 11

11 11 00 00 00

11 11 00 11 00

11 11 11 00 11

11 11 11 11 00

11 00 00 00

11 00 00 11

00 00 00 11

00 11 11 00

C D

A B00

01

10

11

0100 1011

Y= A C D  + A B D + B C D

A C D A B D  B C D  17

Experiment :GROUP (BExperiment :GROUP (B((AA BB CC DD YY

00 00 00 00 00

00 00 00 11 00

00 00 11 00 11

00 00 11 11 00

00 11 00 00 00

00 11 00 11 11

00 11 11 00 11

00 11 11 11 00

11 00 00 00 00

11 00 00 11 00

11 00 11 00 00

11 00 11 11 00

11 11 00 00 00

11 11 00 11 11

11 11 11 00 00

11 11 11 11 00

00 00 00 11

00 11 00 11

00 11 00 00

00 00 00 00

AB

DC

00

1000 1101

01

11

10

Y= B D C + A D C

C

A D

C 18

Experiment :GROUP (CExperiment :GROUP (C((AA BB CC DD YY

00 00 00 00 00

00 00 00 11 00

00 00 11 00 00

00 00 11 11 00

00 11 00 00 11

00 11 00 11 11

00 11 11 00 00

00 11 11 11 00

11 00 00 00 00

11 00 00 11 00

11 00 11 00 11

11 00 11 11 11

11 11 00 00 11

11 11 00 11 11

11 11 11 00 00

11 11 11 11 00

00 00 00 00

11 11 00 00

11 11 00 00

00 00 11 11

AB

DC

00

00

01

01 11

11

10

10

Y=B D + A B D

A B

D

D

B

19

22

23