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# Karnaugh Graph or K-Map

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Each fact and details of K-Map

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### Karnaugh Graph or K-Map

1. 1. KARNAU GH MAP
2. 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. 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. 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. 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. 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. 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.
8. 8. Different Types of K-maps
9. 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
10. 10. Two Variable K-map Grouping V 0 0 0 0 B A A Groups of One – 4 1 A B B
11. 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. 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. 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. 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
15. 15. 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
16. 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. 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
18. 18. Simplyfing Logic Expression by Different types of K-Map
19. 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. 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. 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. 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. 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. 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. 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. 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. 27. Implicants The group of 1s is called implicants. Two types of Implicants: Prime Implicants. Essential Prime Implicants.
28. 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. 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. 30. Besides some disadvantages like usage of limited variables K-Map is very efficient to simplify logic expression. Conclusion
31. 31. References  Wikipedia.com.  Digital Design by Morris Mano
32. 32. Thank You