3. WWhhaatt aarree KKaarrnnaauugghh11 mmaappss??
Karnaugh maps provide an alternative way of simplifying
logic circuits.
Instead of using Boolean algebra simplification
techniques, you can transfer logic values from a Boolean
statement or a truth table into a Karnaugh map.
The arrangement of 0's and 1's within the map helps
you to visualise the logic relationships between the
variables and leads directly to a simplified Boolean
statement.
1Named for the American electrical engineer Maurice Karnaugh.
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4. KKaarrnnaauugghh mmaappss
Karnaugh maps, or K-maps, are often used to simplify logic problems with 2,
3 or 4 variables.
AB
For the case of 2 variables, we form a map consisting of 22=4 cells
as shown in Figure
A
0 1
B
0
1
Cell = 2n ,where n is a number of variables
00 10
01 11
A
B
0 1
0
1
A
B
0 1
0
1
AB
AB AB
A+ B A + B
A+ B A + B
0 2
1 3
Maxterm Minterm
4
7. KKaarrnnaauugghh mmaappss
The Karnaugh map is completed by entering a '1‘(or
‘0’) in each of the appropriate cells.
Within the map, adjacent cells containing 1's (or 0’s)
are grouped together in twos, fours, or eights.
7
8. EExxaammppllee
2-variable Karnaugh maps are trivial but can be used to introduce
the methods you need to learn. The map for a 2-input OR gate
looks like this:
A B Y
0 0 0
0 1 1
1 0 1
1 1 1
A
B
0 1
0
1
1
1 1
BB
AA
AA++BB
8
9. EExxaammppllee
AA BB CC YY
00 00 00 11
00 00 11 11
00 11 00 00
00 11 11 00
11 00 00 11
11 00 11 11
11 11 00 11
11 11 11 00
AC
AABB
CC 00 01 11 10
0
B + AC
B
1
1
1
1 1
1
9
10. TTrruutthh TTaabbllee ttoo KK--MMaapp MMaappppiinngg
10
Four Variable K-Map W X Y Z FWXYZ
0 0 0 0 1
0 0 0 1 0
0 0 1 0 0
0 0 1 1 0
0 1 0 0 1
0 1 0 1 0
0 1 1 0 0
0 1 1 1 1
1 0 0 0 1
1 0 0 1 0
1 0 1 0 0
1 0 1 1 0
1 1 0 0 1
1 1 0 1 0
1 1 1 0 0
1 1 1 1 1
V
1 0 0 0
0 1 3 2
4 5 7 6
12 13 15 14
8 9 11 10
W X
W X
W X
W X
Y Z Y Z Y Z Y Z
1 0 1 0
1 0 1 0
1 0 0 0
Only one
variable changes
for every row
change
Only one variable
changes for
every column
change
Fwxyz= Y Z + X Y Z
12. ((QQuueessttiioonn((11
A Karnaugh map is nothing more than a special form of truth table, useful for
reducing logic functions into minimal Boolean expressions.
Here is a truth table for a specific three-input logic circuit:
Out= A D + A C + B C
A
D
A
C
B
C
12
13. ((QQuueessttiioonn((22
A Karnaugh map is nothing more than a special form of truth table,
useful for reducing logic functions into minimal Boolean expressions.
Here is a truth table for a four-input logic circuit:
Out= B C
B
C
13
14. ((QQuueessttiioonn((33
A Karnaugh map is nothing more than a special form of truth table,
useful for reducing logic functions into minimal Boolean expressions.
Here is a truth table for a four-input logic circuit:
Out = A B
A
B
14
15. ((QQuueessttiioonn((44
A Karnaugh map is nothing more than a special form of truth table,
useful for reducing logic functions into minimal Boolean expressions.
Here is a truth table for a four-input logic circuit:
Out = B C D + B C D
B
C
D
B
C
D
15
