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Karnaugh Map Method

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Karnaugh Map Technique

K-Maps, like truth tables, are a way to showthe relationship between logic inputs anddesired outputs.

K-Maps are a graphical technique used tosimplify a logic equation.

K-Maps are much cleanerthan Booleansimplification.

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K-Map Format

Each minterm in a truth table corresponds to acell in the K-Map.

Once a K-Map is filled (0s & 1s) the sum-of-products expression for the function can be

obtained by OR-ing together the cells thatcontain 1s.

Since the adjacent cells differ by only onevariable, they can be grouped to create simpler

terms in the sum-of-product expression.

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Y

Y

X X

0

1

2

3

Truth Table -TO- K-Map

Y

0

1

0

1

Z

1

0

1

1

X

0

0

1

1

minterm 0

minterm 1

minterm 2

minterm 3

1

1

0

1

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Y

Y

X X

0

0

1

0

X Y

Y

Y

X X

0

0

0

1 X Y

Y

Y

X X

1

0

0

0

X Y

Y

Y

X X

0

1

0

0 X Y

2 Variable K-Map : Groups of One

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Adjacent CellsX Y

Y

Y

X X

1

0

1

0

X Y

Y

Y

X X

1

0

1

0

Y = Z

Z = X Y + X Y = Y ( X + X ) = Y

1

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Groupings

Grouping a pair of adjacent 1s eliminates thevariable that appears in complemented and

uncomplemented form.

Grouping a quad of 1s eliminates the twovariables that appear in both complemented

and uncomplemented form.

Grouping an octet of 1s eliminates the threevariables that appear in both complemented

and uncomplemented form, etc..

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Y

Y

X X

1

1

0

0

X

X

Y

Y

X X

1

0

1

0

Y

Y

2 Variable K-Map : Groups of Two

Y

Y

X X

0

1

0

1

Y

Y

X X

0

0

1

1

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Y

Y

X X

1

1

1

1

1

2 Variable K-Map : Group of Four

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3 Variable K-Map : Vertical

minterm 0

minterm 1

minterm 2

minterm 3

minterm 4

minterm 5

minterm 6

minterm 7

C

0

1

0

1

0

10

1

Y

1

0

1

1

0

01

0

B

0

0

1

1

0

01

1

A

0

0

0

0

1

11

1

1

0

0

0

1

1

0

1

A A

B C

B C

B C

B C

0

1

4

5

3

2

7

6

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3 Variable K-Map : Horizontal

C

C

A B A B A BA B

minterm 0

minterm 1

minterm 2

minterm 3

minterm 4

minterm 5

minterm 6

minterm 7

C

0

1

0

1

0

10

1

Y

1

0

1

1

0

01

0

B

0

0

1

1

0

01

1

A

0

0

0

0

1

11

1

1

0

1

1

1

0

0

0

0

1

2

3

6

7

4

5

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3 Variable K-Map : Groups of Two

C

C

A B A B A BA B

1

0

1

0

0

0

0

0

A C0

1

0

1

0

0

0

0

A C0

0

0

0

1

0

1

0

A C0

0

0

0

0

1

0

1

A C0

0

1

0

1

0

0

0

B C0

0

0

1

0

1

0

0

B C1

0

0

0

0

0

1

0

B C0

1

0

0

0

0

0

1

B C1

1

0

0

0

0

0

0

A B0

0

1

1

0

0

0

0

A B0

0

0

0

1

1

0

0

A B0

0

0

0

0

0

1

1

A B

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3 Variable K-Map : Groups of Four

C

C

A B A B A BA B

1

1

1

1

0

0

0

0

A0

0

0

0

1

1

1

1

A0

0

1

1

1

1

0

0

B1

1

0

0

0

0

1

1

B1

0

1

0

1

0

1

0

C0

1

0

1

0

1

0

1

C

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3 Variable K-Map : Group of Eight

C

C

A B A B A BA B

1

1

1

1

1

1

1

1

1

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Three Variable Design Example #1

L

0

1

01

0

1

0

1

M

1

0

11

0

1

0

0

K

0

0

11

0

0

1

1

J

0

0

00

1

1

1

1

1

0

1

1

0

0

0

1

L

L

J K J K J KJ K

0

1

2

3

6

7

4

5

J L

J K J K L

M = F(J,K,L) = J L + J K + J K L

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Three Variable Design Example #2

C0

1

0

1

0

1

01

Z1

0

0

0

1

1

01

B0

0

1

1

0

0

11

A0

0

0

0

1

1

11

1

0

0

0

0

1

1

1

C

C

A B A B A BA B

0

1

2

3

6

7

4

5

B C

A C

Z = F(A,B,C) = A C + B C

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Three Variable Design Example #3

C0

1

0

1

0

1

01

F21

0

0

1

1

1

01

B0

0

1

1

0

0

11

A0

0

0

0

1

1

11

1

1

0

1

1

1

0

0

A

A

B C B C B CB C

0 1 23

674 5

B C B C

A B

A C

F2 = F(A,B,C) = B C + B C + A B

F2 = F(A,B,C) = B C + B C + A C

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Four Variable K-Map

minterm 0

minterm 1

minterm 2

minterm 3

minterm 4

minterm 5

minterm 6

minterm 7

minterm 8

minterm 9

minterm 10

minterm 11

minterm 12

minterm 13

minterm 14

minterm 15

Z

0

1

0

1

0

1

0

1

0

1

0

1

0

1

0

1

F1

1

0

0

0

1

1

0

1

1

1

00

0

1

1

1

Y

0

0

1

1

0

0

1

1

0

0

1

1

0

0

1

1

X

0

0

0

0

1

1

1

1

0

0

0

0

1

1

1

1

W

0

0

0

0

0

0

0

0

1

1

1

1

1

1

1

1

0

1

4

5

12

13

8

9

3

2

7

6

15

14

11

10

W X W X W XW X

Y Z

Y Z

Y Z

Y Z

0

0

1

0

1

1

0

0

1

0

1

1

0

1

1

1

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Four Variable K-Map : Groups of Four

W X W X W XW X

Y Z

Y Z

Y Z

Y Z

1

0

0

0

0

0

1

0

0

1

0

0

0

0

0

1

X Z

0

0

0

1

0

1

0

0

0

0

1

0

1

0

0

0

X ZX Z

0

1

0

0

0

0

0

1

1

0

0

0

0

0

1

0

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Four Variable Design Example #1

Z

0

1

0

1

0

1

0

1

0

1

0

1

0

1

0

1

F1

1

0

1

0

1

0

1

0

0

0

10

1

1

0

0

Y

0

0

1

1

0

0

1

1

0

0

1

1

0

0

1

1

X

0

0

0

0

1

1

1

1

0

0

0

0

1

1

1

1

W

0

0

0

0

0

0

0

0

1

1

1

1

1

1

1

1

0

1

4

5

12

13

8

9

3

2

7

6

15

14

11

10

W X W X W XW X

Y Z

Y Z

Y Z

Y Z

0

1

0

1

0

0

0

1

1

0

1

0

1

1

0

0

W X Y

X Y ZW Z

F1 = F(w,x,y,z) = W X Y + W Z + X Y Z

min 0

min 15

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Dont Care Conditions

It is not always true that cell not containing1s will contain 0s, because some

combination of input variable do not occur.

In such situations we have a freedom toassume 0 or 1 as output for each of these

combination. This is dont care condition.

In K-map it is represented as X (Cross mark)in corresponding cell.

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Four Variable Design Example

Z

0

1

0

1

0

1

0

1

0

1

0

1

0

1

0

1

F2

1

x

1

0

0

x

0

x

x

1

0

1

x

1

1

1

Y

0

0

1

1

0

0

1

1

0

0

1

1

0

0

1

1

X

0

0

0

0

1

1

1

1

0

0

0

0

1

1

1

1

W

0

0

0

0

0

0

0

0

1

1

1

1

1

1

1

1

0

1

4

5

12

13

8

9

3

2

7

6

15

14

11

10

W X W X W XW X

Y Z

Y Z

Y Z

Y Z

X

X

1

1

1

1

1

0

1

0

X

X

0

X

1

0

Y Z

F2 = F(w,x,y,z) = X Y Z + Y Z + X Y

X Y Z

X Y

min 0

min 15

Si lif th i i b l

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Simplify the expression given below

using K-map.

Y=m (1, 3, 7, 11, 15) + d(0, 2, 5)

Solution:

Given Equation is,

Y= m1+ m3+ m7+ m11+ m15 + d(0, 2, 5)

Regular minterms so

enter 1s

Dont Care

condition soenter X

Si lif th i i b l

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Simplify the expression given below

using K-map.

Y=m (1, 3, 7, 11, 15) + d(0, 2, 5)

0

4

1

5

3

7

2

6

12

8

13

9

15

11

14

10

C D C D C DC D

A B

A B

A B

A B

0

0

0

0

1

1

0

0

X

0

1

X

1

1

x

0

A B

C D

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Simplify expression by once considering

dont care condition and once by

ignoring dont care condition.

Y=m (1, 4, 8, 12, 13, 15) + d(3, 14)

Sol:

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Part1: Without dont care condition

0

4

1

5

3

7

2

6

12

8

13

9

15

11

14

10

C D C D C DC D

A B

A B

A B

A B

1

1

1

0

1

0

0

0

0

1

1

0

0

0

0

0

ABD

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Simplified expression without dont care is

Y= + + + ABD

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Part 2: With dont care condition

0

4

1

5

3

7

2

6

12

8

13

9

15

11

14

10

C D C D C DC D

A B

A B

A B

A B

1

1

1

0

1

0

X

0

0

1

1

0

X

0

0

0

AB

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Simplified expression with dont care is

Y= + + + AB

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Minimize the function using K-map and

implement using NAND Gate only.

F(A, B, C, D)= M(1, 3, 5, 8, 9, 11, 15)+

d(2, 13)

0

4

1

5

3

7

2

6

12

8

13

9

15

11

14

10

C D C D C DC D

A B

A B

A B

A B

0

1

X

1

1

1

0

0

0

0

1

1

1

0

X

0

AD

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Simplified expression

F(A,B,C,D)= +AD+ +