This slide helps you to understand the basic concept of Karnaugh Map Technique, & its implementation.

1. THETHE
KARNAUGHKARNAUGH
MAPMAP

2. GROUP MEMBERS

3. K MAP
• MAURICE KARNAUGH INTRODUCED K-MAP IN 1953
• THE KARNAUGH MAP, ALSO KNOWN AS THE K-MAP, IS A METHOD TO SIMPLIFY
BOOLEAN ALGEBRA EXPRESSIONS.
• THE KARNAUGH MAP REDUCES THE NEED FOR EXTENSIVE CALCULATIONS BY
TAKING ADVANTAGE OF HUMANS' PATTERN-RECOGNITION CAPABILITY.
• THE REQUIRED BOOLEAN RESULTS ARE TRANSFERRED FROM A TRUTH TABLE
ONTO A TWO-DIMENSIONAL GRID WHERE THE CELLS ARE ORDERED IN GRAY
CODE, AND EACH CELL POSITION REPRESENTS ONE COMBINATION OF INPUT
CONDITIONS, WHILE EACH CELL VALUE REPRESENTS THE CORRESPONDING
OUTPUT VALUE. OPTIMAL GROUPS OF 1S OR 0S ARE IDENTIFIED.
• THESE TERMS CAN BE USED TO WRITE A MINIMAL BOOLEAN EXPRESSION
REPRESENTING THE REQUIRED LOGIC.

4. • ADVANTAGES
• REDUCES EXTENSIVE
CALC.
• REDUCES EXPRESSION
WITHOUT BOOLEAN
THEOREMS
• USED FOR MINIMIZING
CIRCUITS
• LESS TIME CONSUMING
• LESS SPACE
CONSUMING
• DISADVANTAGES
• TEDIOUS FOR MORE
THAN 5 VARIABLES
• SOME EXAMPLES
ARE SOLVED IN FEW
SECONDS BY
BOOLEAN
THEOREMS EASILY

5. THE KARNAUGH MAPTHE KARNAUGH MAP
• NUMBER OF SQUARES = NUMBER OF COMBINATIONS
• EACH SQUARE REPRESENTS A MINTERM
• 2 VARIABLES  4 SQUARES
• 3 VARIABLES  8 SQUARES
• 4 VARIABLES  16 SQUARES

6. THE 3 VARIABLE K-MAP
• THERE ARE 8 CELLS AS SHOWN:
CC
ABAB
00 11
0000
0101
1111
1010
CBA CBA
CBA BCA
CAB ABC
CBA CBA

7. THE 4-VARIABLE K-MAP
CDCD
ABAB
0000 0101 1111 1010
0000
0101
1111
1010 DCBA
DCAB
DCBA
DCBA
DCBA
DCAB
DCBA
DCBA
CDBA
ABCD
BCDA
CDBA
DCBA
DABC
DBCA
DCBA

8. ADJACENT SQUARES
• A LARGER NUMBER OF ADJACENT SQUARES
ARE COMBINED, WE OBTAIN A PRODUCT
TERM WITH FEWER LITERALS.
1 SQUARE = 1 MINTERM = THREE LITERALS.
2 ADJACENT SQUARES = 1 TERM = TWO LITERALS.
4 ADJACENT SQUARES = 1 TERM = ONE LITERAL.
8 ADJACENT SQUARES ENCOMPASS THE ENTIRE MAP AND
PRODUCE A FUNCTION THAT IS ALWAYS EQUAL TO 1.
• IT IS OBVIOUSLY TO KNOW THE NUMBER OF ADJACENT
SQUARES IS COMBINED IN A POWER OF TWO SUCH AS 1,2,4,
AND 8.
8

9. CELL ADJACENCY
CDCD
ABAB
0000 0101 1111 1010
0000
0101
1111
1010

10. K-MAP SOP MINIMIZATION
• THE K-MAP IS USED FOR SIMPLIFYING BOOLEAN
EXPRESSIONS TO THEIR MINIMAL FORM.
• A MINIMIZED SOP EXPRESSION CONTAINS THE FEWEST
POSSIBLE TERMS WITH FEWEST POSSIBLE VARIABLES
PER TERM.
• GENERALLY, A MINIMUM SOP EXPRESSION CAN BE
IMPLEMENTED WITH FEWER LOGIC GATES THAN A
STANDARD EXPRESSION.

11. MAPPING A STANDARD SOP
EXPRESSION
FOR AN SOP
EXPRESSION IN
STANDARD FORM:
A 1 IS PLACED ON THE
K-MAP FOR EACH
PRODUCT TERM IN
THE EXPRESSION.
EACH 1 IS PLACED IN A
CELL
CORRESPONDING TO
THE VALUE OF A
PRODUCT TERM.
EXAMPLE: FOR THE
PRODUCT TERM , A
1 GOES IN THE 101
CELL ON A 3-VARIABLE
CBA
CC
ABAB
00 11
0000
0101
1111
1010
CBA CBA
CBA BCA
CAB ABC
CBA CBA1

12. CC
ABAB
00 11
0000
0101
1111
1010
MAPPING A STANDARD SOP
EXPRESSION (FULL EXAMPLE)
THE EXPRESSION:
CBACABCBACBA +++
000 001 110 100
1 1
1
1

13. MAPPING A NONSTANDARD
SOP EXPRESSION
• A BOOLEAN EXPRESSION MUST BE IN STANDARD FORM
BEFORE YOU USE A K-MAP.
• IF ONE IS NOT IN STANDARD FORM, IT MUST BE CONVERTED.
• YOU MAY USE THE PROCEDURE MENTIONED EARLIER OR
USE NUMERICAL EXPANSION.

14. MAPPING A NONSTANDARD SOP
EXPRESSION
NUMERICAL EXPANSION OF A NONSTANDARD
PRODUCT TERM
ASSUME THAT ONE OF THE PRODUCT TERMS IN A
CERTAIN 3-VARIABLE SOP EXPRESSION IS .
IT CAN BE EXPANDED NUMERICALLY TO STANDARD
FORM AS FOLLOWS:
STEP 1: WRITE THE BINARY VALUE OF THE TWO
VARIABLES AND ATTACH A 0 FOR THE MISSING VARIABLE
: 100.
STEP 2: WRITE THE BINARY VALUE OF THE TWO
VARIABLES AND ATTACH A 1 FOR THE MISSING VARIABLE
: 101.
THE TWO RESULTING BINARY NUMBERS ARE THE
C
C
BA
CBA CBA

15. K-MAP SIMPLIFICATION OF
SOP EXPRESSIONS
• AFTER AN SOP EXPRESSION HAS BEEN MAPPED, WE CAN
DO THE PROCESS OF MINIMIZATIO N:
• GROUPING THE 1S
• DETERMINING THE MINIMUM SOP EXPRESSION FROM THE
MAP

16. GROUPING THE 1S
• YOU CAN GROUP 1S ON THE K-MAP ACCORDING TO
THE FOLLOWING RULES BY ENCLOSING THOSE
ADJACENT CELLS CONTAINING 1S.
• THE GOAL IS TO MAXIMIZE THE SIZE OF THE GROUPS
AND TO MINIMIZE THE NUMBER OF GROUPS.

17. GROUPING THE 1S (EXAMPLE)
CC
ABAB 00 11
0000 11
0101 11
1111 11 11
1010
CC
ABAB 00 11
0000 11 11
0101 11
1111 11
1010 11 11

18. GROUPING THE 1S (EXAMPLE)
CDCD
ABAB 0000 0101 1111 1010
0000 11 11
0101 11 11 11 11
1111
1010 11 11
CDCD
ABAB 0000 0101 1111 1010
0000 11 11
0101 11 11 11
1111 11 11 11
1010 11 11 11

19. DETERMINING THE MINIMUM
SOP EXPRESSION FROM THE
MAP
• THE FOLLOWING RULES ARE APPLIED TO FIND THE
MINIMUM PRODUCT TERMS AND THE MINIMUM SOP
EXPRESSION:
1. GROUP THE CELLS THAT HAVE 1S. EACH
GROUP OF CELL CONTAINING 1S CREATES ONE
PRODUCT TERM COMPOSED OF ALL VARIABLES
THAT OCCUR IN ONLY ONE FORM (EITHER
COMPLEMENTED OR COMPLEMENTED) WITHIN
THE GROUP. VARIABLES THAT OCCUR BOTH
COMPLEMENTED AND UNCOMPLEMENTED
WITHIN THE GROUP ARE ELIMINATED  CALLED
CO NTRADICTO RY VARIABLES.

20. DETERMINING THE MINIMUM
SOP EXPRESSION FROM THE
MAP2. DETERMINE THE MINIMUM PRODUCT TERM FOR EACH GROUP.
• FOR A 3-VARIABLE MAP:
1. A 1-CELL GROUP YIELDS A 3-VARIABLE PRODUCT TERM
2. A 2-CELL GROUP YIELDS A 2-VARIABLE PRODUCT TERM
3. A 4-CELL GROUP YIELDS A 1-VARIABLE PRODUCT TERM
4. AN 8-CELL GROUP YIELDS A VALUE OF 1 FOR THE EXPRESSION.
• FOR A 4-VARIABLE MAP:
1. A 1-CELL GROUP YIELDS A 4-VARIABLE PRODUCT TERM
2. A 2-CELL GROUP YIELDS A 3-VARIABLE PRODUCT TERM
3. A 4-CELL GROUP YIELDS A 2-VARIABLE PRODUCT TERM
4. AN 8-CELL GROUP YIELDS A A 1-VARIABLE PRODUCT TERM
5. A 16-CELL GROUP YIELDS A VALUE OF 1 FOR THE EXPRESSION.

21. DETERMINING THE MINIMUM
SOP EXPRESSION FROM THE
MAP
3. WHEN ALL THE MINIMUM PRODUCT TERMS ARE DERIVED
FROM THE K-MAP, THEY ARE SUMMED TO FORM THE MINIMUM
SOP EXPRESSION.

22. DETERMINING THE MINIMUM SOP
EXPRESSION FROM THE MAP (EXAMPLE)
CDCD
ABAB
0000 0101 1111 1010
0000 11 11
0101 11 11 11 11
1111 11 11 11 11
1010 11
B
CA
DCA
DCACAB ++

23. DETERMINING THE MINIMUM SOP
EXPRESSION FROM THE MAP (EXERCISES)
CBABCAB ++ ACCAB ++
CC
ABAB 00 11
0000 11
0101 11
1111 11 11
1010
CC
ABAB 00 11
0000 11 11
0101 11
1111 11
1010 11 11

24. DETERMINING THE MINIMUM SOP
EXPRESSION FROM THE MAP
(EXERCISES)
DBACABA ++ CBCBAD ++
CDCD
ABAB 0000 0101 1111 1010
0000 11 11
0101 11 11 11 11
1111
1010 11 11
CDCD
ABAB 0000 0101 1111 1010
0000 11 11
0101 11 11 11
1111 11 11 11
1010 11 11 11

25. MAPPING DIRECTLY FROM A
TRUTH TABLEI/PI/P O/PO/P
AA BB CC XX
00 00 00 11
00 00 11 00
00 11 00 00
00 11 11 00
11 00 00 11
11 00 11 00
11 11 00 11
11 11 11 11
CC
ABAB
00 11
0000
0101
1111
1010
1
1
1
1