Chapter 5

Karnaugh Map and Minimization Procedures

Ch05L1-"Digital Principles and Design", Raj Kamal, Pearson Education, 2006 2

Lesson 1

KARNAUGH MAP

Ch05L1-"Digital Principles and Design", Raj Kamal, Pearson Education, 2006 3

Outline

• Three variable Karnaugh map• Four variable Karnaugh map• Five/Six Variable Karnaugh Map

Ch05L1-"Digital Principles and Design", Raj Kamal, Pearson Education, 2006 4

Map from 3 variables Truth table or SOP form Boolean Expression

� A two-dimensional map built from a truth table or 3 variables SOP form Boolean Expression� Since number of rows in three variable (three inputs) truth table are 8, the map has 8 cells� Two cells horizontal and four cells vertical. [It can also be vice versa.]

Ch05L1-"Digital Principles and Design", Raj Kamal, Pearson Education, 2006 5

Map for F = A.B.C + A.B.C

CAB

C0

C1

AB 00

AB 01 1

AB 11 AB 10 1

ΣΣΣΣΣΣΣΣ m(3, 4)m(3, 4)

Ch05L1-"Digital Principles and Design", Raj Kamal, Pearson Education, 2006 6

Map for F = A.B.C + A.B.C +A.B.C + A.B.C + A.B.C

CAB

C0

C1

AB 00 1

AB 01 1

AB 11 1

AB 10 1 1

ΣΣΣΣΣΣΣΣ m(0, 3, m(0, 3, 4, 5, 7 )4, 5, 7 )

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1 When output is 1 for a given combination of A, B and C, we place 1 at the corresponding cell.2 Complete the step 1 for all the rows of truth table with outputs = 1.

Filling the cell with 1s

Ch05L1-"Digital Principles and Design", Raj Kamal, Pearson Education, 2006 8

CAB

C0

C1

AB 00 m0m2

m1

AB 01 m3m7AB 11 m6

AB 10 m4 m5

Corresponding miniterms of the cells

Ch05L1-"Digital Principles and Design", Raj Kamal, Pearson Education, 2006 9

Map for F = (A+ B + C) . (A+ B+ C)

CAB

C0

C1

A+B 00 0

A+B 01

A+B 11 0

A+B 10

ΠΠΠΠΠΠΠΠ M(1, 6)M(1, 6)

Ch05L1-"Digital Principles and Design", Raj Kamal, Pearson Education, 2006 10

1 When output is 0 for a given combination of A, B and C, we place 0 at the corresponding cell.2 Complete the step 1 for all 8 rows of truth table with outputs = 0.

Filling the cell with 0s from POS form

Ch05L1-"Digital Principles and Design", Raj Kamal, Pearson Education, 2006 11

CAB

C0

C1

A+B 00 M0M2

M1

A+B 01 M3M7A+B 11 M6

A+B 10 M4 M5

Corresponding Maxterms of the cells

Ch05L1-"Digital Principles and Design", Raj Kamal, Pearson Education, 2006 12

Outline

• Three variable Karnaugh map• Four variable Karnaugh map• Five/Six Variable Karnaugh Map

Ch05L1-"Digital Principles and Design", Raj Kamal, Pearson Education, 2006 13

Map from 4 variables Truth table or SOP form Boolean Expression

� A two-dimensional map built from a truth table or 4 variables SOP form Boolean Expression� Since number of rows in a four variable (three inputs) truth table are 16, the map has 16 cells� Four cells horizontal and four cells vertical.

Ch05L1-"Digital Principles and Design", Raj Kamal, Pearson Education, 2006 14

Map for F = A.B.C.D + A.B.C.D + A.B.C.D

CAB

CD00

CD01

AB 00

AB 01

1

AB 11 AB 10 1

ΣΣΣΣΣΣΣΣ m(3, 6, 8)m(3, 6, 8)CD11

CD10

1

Ch05L1-"Digital Principles and Design", Raj Kamal, Pearson Education, 2006 15

Map for F = A.B.C.D + A.B.C.D + A.B.C.D

CAB

CD00

CD01

AB 00

AB 01 1

AB 11 AB 10 1

ΣΣΣΣΣΣΣΣ m(4, 8, 10)m(4, 8, 10)CD11

CD10

1

Ch05L1-"Digital Principles and Design", Raj Kamal, Pearson Education, 2006 16

1. When output is 1 for a given combination of A, B, C and D, we place 1 at the corresponding cell.2. Complete the step 1 for all 16 rows of truth table with outputs = 1.

Filling the cell with 1s

Ch05L1-"Digital Principles and Design", Raj Kamal, Pearson Education, 2006 17

Corresponding Miniterms of the cells

CAB

CD00

CD01

AB 00 m0

m5 AB 01 m4

m3

AB 11 m12 m13m9AB 10 m8

m1

CD11

CD10

m7

m2

m6

m15 m11

m14m10

Ch05L1-"Digital Principles and Design", Raj Kamal, Pearson Education, 2006 18

1 When output is 0 for a given combination of A, B and C, we place 0 at the corresponding cell.2 Complete the step 1 for all the rows of truth table with outputs = 0.

Filling the cell with 0s from POS form

Ch05L1-"Digital Principles and Design", Raj Kamal, Pearson Education, 2006 19

Map for F = (A+B+C+D) . (A+B+C+D)

CAB

C+D00

C+D01

A+B 00 0

A+B 01

A+B 11 A+B 10

ΠΠΠΠΠΠΠΠ M(0, 10)M(0, 10)C+D

11C+D10

0

Ch05L1-"Digital Principles and Design", Raj Kamal, Pearson Education, 2006 20

Corresponding Maxterms of the cells

CAB

C+D00

C+D01

A+B 00 M0

M5A+B 01 M4

M3

A+B 11 M12

M8M13

M9A+B 10

M1

C+D

11C+D10

M7

M2

M6

M15 M11

M14

M10

Ch05L1-"Digital Principles and Design", Raj Kamal, Pearson Education, 2006 21

Outline

• Three variable Karnaugh map• Four variable Karnaugh map•• Five/Six Variable Five/Six Variable Karnaugh Karnaugh MapMap

Ch05L1-"Digital Principles and Design", Raj Kamal, Pearson Education, 2006 22

Five Variable Map - Left part A = 0

CBC

DE00

DE01

BC 00 m0

m5 BC 01 m4

m3

BC 11 m12 m13m9BC 10 m8

m1

DE11

DE10

m7

m2

m6

m15 m11

m14m10

Ch05L1-"Digital Principles and Design", Raj Kamal, Pearson Education, 2006 23

Five Variable Map - Left part A = 1

CBC

DE00

DE01

BC 00 m16

m21 BC 01 m20

m19

BC 11 m28 m29m25BC 10 m24

m17

DE11

DE10

m23

m18

m22

m31 m27

m30m26

Ch05L1-"Digital Principles and Design", Raj Kamal, Pearson Education, 2006 24

Six Variable Map - Left Top part A, B = 0, 0

CCD

EF00

EF01

CD 00 m0

m5 CD 01 m4

m3

CD 11 m12 m13m9CD 10 m8

m1

EF11

EF10

m7

m2

m6

m15 m11

m14m10

Ch05L1-"Digital Principles and Design", Raj Kamal, Pearson Education, 2006 25

Six Variable Map - Right Top part A, B = 0, 1

CCD

EF00

EF01

CD 00 m16

m21 CD 01 m20

m19

CD 11 m28 m29m25CD 10 m24

m17

EF11

EF10

m23

m18

m22

m31 m27

m30m26

Ch05L1-"Digital Principles and Design", Raj Kamal, Pearson Education, 2006 26

Six Variable Map - Left Bottom part A, B = 1, 1

CCD

EF00

EF01

CD 00

CD 01

CD 11 m57CD 10 m56

EF11

EF10

m63 m59

m62

m58

Ch05L1-"Digital Principles and Design", Raj Kamal, Pearson Education, 2006 27

Six Variable Map - Right Bottom part A, B = 1, 0

CCD

EF00

EF01

CD 00 m32

CD 01

CD 11 CD 10 m40

EF11

EF10

m34

m42

Ch05L1-"Digital Principles and Design", Raj Kamal, Pearson Education, 2006 28

Summary

Ch05L1-"Digital Principles and Design", Raj Kamal, Pearson Education, 2006 29

• Karnaugh Map has cells. On moving from one cell to nearby cell, a variable complements

• First column and last column adjacent• First row and last row adjacent • Each cell represent one miniterm or

one Maxterm• Map reflect the truth table• Map reflects the Boolean expression

for output in SOP or POS form

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End of Lesson 1

THREE VARIABLE KARNAUGH MAP

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