2. 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.
3. 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
4. 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.
5. 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)
6. 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 .
7. 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.
9. 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
11. 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
12. 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
13. 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
14. 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
16. 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
17. 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
19. 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
20. 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
21. 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
22. Four Variable K-Map
Simplified sum-of-products (SOP) logic expression for the logic
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
23. 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
24. 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.
25. 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.
26. Don’t Care Conditions
Simplified sum-of-products (SOP) logic expression for the logic
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
27. Implicants
The group of 1s is called implicants.
Two types of Implicants:
Prime Implicants.
Essential Prime Implicants.
28. Prime and Essential Prime
Implicants
Chapter 2 - Part 2 28
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
29. Example with don’t Care
Chapter 2 - Part 2 29
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
30. Besides some disadvantages like usage of
limited variables K-Map is very efficient
to simplify logic expression.
Conclusion