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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 )
Ch05L1-"Digital Principles and Design", Raj Kamal, Pearson Education, 2006 7
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 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
Ch05L1-"Digital Principles and Design", Raj Kamal, Pearson Education, 2006 30
End of Lesson 1
THREE VARIABLE KARNAUGH MAP