Karnaugh Graph or K-Map

CONTENTS
 Introduction.
 Advantages of Karnaugh Maps.
 SOP & POS.
 Properties.
 Simplification Process
 Different Types of K-maps
 Simplyfing logic expression by different types of K-Map
 Don’t care conditions
 Prime Implicants
 References.

 Also known as Veitch diagram or K-Map.
 Invented in 1953 by
Maurice Karnaugh.
 A graphical way of
minimizing Boolean
expressions.
 It consists tables of rows
and columns with entries
represent 1`s or 0`s.
Introduction

Advantages of Karnaugh Maps
 Data representation’s simplicity.
 Changes in neighboring variables are easily displayed
 Changes Easy and Convenient to implement.
 Reduces the cost and quantity of logical gates.

SOP & POS
 The SOP (Sum of Product) expression represents
1’s .
 SOP form such as (A.B)+(B.C).
 The POS (Product of Sum) expression represents the
low (0) values in the K-Map.
 POS form like (A+B).(C+D)

Properties
An n-variable K-map has 2n
cells with n-variable truth
table value.
Adjacent cells differ
in only one bit .
Each cell refers to a
minterm or maxterm.
For minterm mi ,
maxterm Mi and
don’t care of f we
place 1 , 0 , x .

Simplification Process
No diagonals.
Only 2^n cells in each group.
Groups should be as large as possible.
A group can be combined if all cells of the group have
same set of variable.
Overlapping allowed.
Fewest number of groups possible.

Two Variable K-map(continued)
The K-Map is just a different form of the truth table.
V
W X FWX
Minterm – 0 0 0 1
Minterm – 1 0 1 0
Minterm – 2 1 0 1
Minterm – 3 1 1 0
V
0 1
2 3
X
W
W
X
1 0
1 0

Two Variable K-map Grouping
V
0 0
0 0
B
A
A
Groups of One – 4
1
A B
B

Groups of Two – 2
Two Variable K-Map Groupings
Group of Four
V
0 0
0 0
B
A
A
B
1
B
1
V
1 1
1 1
B
A
A
1
B

Three Variable K-map (continued)
 K-map from truth table.
W X Y FWXY
Minterm – 0 0 0 0 1
Minterm – 1 0 0 1 0
Minterm – 2 0 1 0 0
Minterm – 3 0 1 1 0
Minterm – 4 1 0 0 0
Minterm – 5 1 0 1 1
Minterm – 6 1 1 0 1
Minterm – 7 1 1 1 0
V
0 1
2 3
6 7
4 5
Y
XW
Y
1
XW
XW
XW
0
0 0
0 1
1 0
Only one
variable changes
for every row
cnge
12

Three Variable K-Map Groupings
V
0 0
0 0
0 0
0 0
C C
BA
BA
BA
BA
BA
1 1
BA
1 1
BA
1 1
BA
1 1
1
CA
1
1
CA
1
1
CA
1
1
CB
1
1
CB
1
1
CA
11
CB
1
1
CB
1
Groups of One – 8 (not shown)
Groups of Two – 12

Three Variable K-Map Groupings
Groups of Four – 6 Group of Eight - 1
V
1 1
1 1
1 1
1 1
C C
BA
BA
BA
BA
1
V
0 0
0 0
0 0
0 0
C C
BA
BA
BA
BA
1
C
1
1
1
1
C
1
1
1
A
1 1
1 1
B
1 1
1 1
A
1 1
1 1
B
1 1
1 1

Truth Table to K-Map Mapping
Four Variable K-Map
W X Y Z FWXYZ
Minterm – 0 0 0 0 0 0
Minterm – 1 0 0 0 1 1
Minterm – 2 0 0 1 0 1
Minterm – 3 0 0 1 1 0
Minterm – 4 0 1 0 0 1
Minterm – 5 0 1 0 1 1
Minterm – 6 0 1 1 0 0
Minterm – 7 0 1 1 1 1
Minterm – 8 1 0 0 0 0
Minterm – 9 1 0 0 1 0
Minterm –
10
1 0 1 0 1
Minterm –
11
1 0 1 1 0
Minterm –
12
1 1 0 0 1
Minterm –
13
1 1 0 1 0
V
0 1 3 2
4 5 7 6
12 13 15 14
8 9 11 10
XW
XW
XW
XW
ZY ZY ZY ZY
1 01 1
1 10 1
0 10 0
0 11 0

FOUR VARIABLE K-MAP
GROUPINGS
V
0 0 0 0
0 0 0 0
0 0 0 0
0 0 0 0
BA
BA
BA
BA
DC DC DC DC
CB
1 1
1 1
DB
1 1
1 1
DA
1
1
1
1
CB
1 1
1 1
DB
1
1
1
1
DA
1
1
1
1 DB11
11

FOUR VARIABLE K-MAP
GROUPINGS
Groups of Eight – 8 (two shown)
V
0 0 0 0
0 0 0 0
0 0 0 0
0 0 0 0
BA
BA
BA
BA
DC DC DC DC
B
1 1 1 1
1 1 1 1
D
1
1
1
1
1
1
1
1
Group of Sixteen – 1
V
1 1 1 1
1 1 1 1
1 1 1 1
1 1 1 1
BA
BA
BA
BA
DC DC DC DC
1

Simplyfing Logic
Expression
by
Different types of K-Map

TWO VARIABLE K-MAP
 Differ in the value of y in
m0 and m1.
 Differ in the value of x in
m0 and m2.
y = 0 y = 1
x = 0
m 0 = m 1 =
x = 1 m 2 = m 3 =
yx yx
yx yx

Two Variable K-Map
Simplified sum-of-products (SOP) logic expression for the logic
function F1.
V
1 1
0 0
K
J
J
K
J
JF =1
J K F1
0 0 1
0 1 1
1 0 0
1 1 0
20

Three Variable Maps
 A three variable K-map :
yz=00 yz=01 yz=11 yz=10
x=0 m0
m1 m3 m2
x=1 m4 m5 m7 m6
 Where each minterm corresponds to the product terms:
yz=00 yz=01 yz=11 yz=10
x=0
x=1
zyx zyx zyx zyx
zyx zyx zyx zyx

Four Variable K-Map
function F3.
TSURUTSUSRF +++=3
R S T U F3
0 0 0 0 0
0 0 0 1 1
0 0 1 0 0
0 0 1 1 1
0 1 0 0 0
0 1 0 1 1
0 1 1 0 1
0 1 1 1 1
1 0 0 0 0
1 0 0 1 1
1 0 1 0 0
1 0 1 1 0
1 1 0 0 1
1 1 0 1 0
1 1 1 0 1
1 1 1 1 1
V
0 1 1 0
0 1 1 1
1 0 1 1
0 1 0 0
SR
SR
SR
SR
UT UT UT UT
UR
TS
USR
UTS

 Five variable K-map is formed using two connected 4-
variable maps:
Chapter 2 - Part 2 23
23
0
1 5
4
VWX
YZ
V
Z
000 001
00
13
12
011
9
8
010
X
3
2 6
7
14
15
10
11
01
11
10
Y
16
17 21
20
29
28
25
24
19
18 22
23
30
31
26
27
100 101 111 110
W W
X
Five Variable K-Map

Don’t-care condition
 Minterms that may produce either
0 or 1 for the function.
 Marked with an ‘x’
in the K-map.
 These don’t-care conditions can
be used to provide further simplification.

SOME YOU GROUP, SOME YOU
DON’T
V
X 0
1 0
0 0
X 0
C C
BA
BA
BA
BA
CA
This don’t care condition was treated as a
(1).
There was no advantage in treating
this don’t care condition as a (1),
thus it was treated as a (0) and not
grouped.

Don’t Care Conditions
function F4.
SRTRF +=4
R S T U F4
0 0 0 0 X
0 0 0 1 0
0 0 1 0 1
0 0 1 1 X
0 1 0 0 0
0 1 0 1 X
0 1 1 0 X
0 1 1 1 1
1 0 0 0 1
1 0 0 1 1
1 0 1 0 1
1 0 1 1 X
1 1 0 0 X
1 1 0 1 0
1 1 1 0 0
1 1 1 1 0
V
X 0 X 1
0 X 1 X
X 0 0 0
1 1 X 1
SR
SR
SR
SR
UT UT UT UT
TR
SR

Implicants
The group of 1s is called implicants.
Two types of Implicants:
Prime Implicants.
Essential Prime Implicants.

Prime and Essential Prime
Implicants
DB
CB
1 1
1 1
1 1
B
D
A
1 1
1 1
1
ESSENTIAL Prime
ImplicantsC
BD
CD
BD
Minterms covered by single prime implicant
DB
1 1
1 1
1 1
B
C
D
A
1 1
1 1
1
AD
BA

Example with don’t Care
x
x
1
1 1
1
1
B
D
A
C
1
1
1
x
x
1
1 1
1
1
B
D
A
C
1
1
EssentialSelected

Besides some disadvantages like usage of
limited variables K-Map is very efficient
to simplify logic expression.
Conclusion

References
 Wikipedia.com.
 Digital Design by Morris Mano

Karnaugh Graph or K-Map

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