After successfully completing this module students should be able to: Understand the Need to simplify (minimize) expressions List Different Methods for Minimization Karnaugh Maps Algebraic method Use Karnaugh Map method to minimize the Boolean expression

1. By Ms. Nita Arora, PGT Computer Science Kulachi Hansraj Model School e -Lesson

2. Subject : Computer Science (083) Unit : Boolean Algebra Topic : Minimization of Boolean Expressions Using Karnaugh Maps (K-Maps) Category : Senior Secondary Class : XII

8. M INIMIZATION OF B OOLEAN E XPRESSION Different methods Karnaugh Maps Algebraic Method

9. K arnaugh M aps WHAT is Karnaugh Map (K-Map)? A special version of a truth table Karnaugh Map (K-Map) is a GRAPHICAL display of fundamental terms in a Truth Table . Don’t require the use of Boolean Algebra theorems and equation Works with 2,3,4 (even more) input variables (gets more and more difficult with more variables) NEXT

10. K-maps provide an alternate way of simplifying logic circuits. One can transfer logic values from a Truth Table into a K-Map. The arrangement of 0’s and 1’s within a map helps in visualizing, leading directly to Simplified Boolean Expression K arnaugh M aps……… (Contd.) NEXT

11. Correspondence between the Karnaugh Map and the Truth Table for the general case of a two Variable Problem Truth Table 2 Variable K-Map K arnaugh M aps……… (Contd.) A B 0 0 0 1 1 0 1 1 F a b c d A B 0 1 0 a b 1 c d

12. D rawing a K arnaugh M ap (K-Map) K-map is a rectangle made up of certain number of SQUARES For a given Boolean function there are 2 N squares where N is the number of variables (inputs) In a K-Map for a Boolean Function with 2 Variables f(a,b) there will be 2 2 =4 squares Each square is different from its neighbour by ONE Literal Each SQUARE represents a MAXTERM or MINTERM NEXT

13. Karnaugh maps consist of a set of 2 2 squares where 2 is the number of variables in the Boolean expression being minimized. Truth Table 2 Variable K-Map K arnaugh M aps……… (Contd.) 1 A B 0 1 0 0 1 1 1 1 A B F 0 0 0 0 1 1 1 0 1 1 1 1 Minterm A’B’ A’B A B’ A B Maxterm A + B A + B’ A’ + B A’ + B’ NEXT

15. M inimization S teps (SOP Expression with 4 var.) The process has following steps: Draw the K-Map for given function as shown Enter the function values into the K-Map by placing 1's and 0's into the appropriate Cells 1 1 1 1 A B C D 00 01 11 10 00 0 0 0 1 0 3 0 2 01 0 0 0 0 11 1 1 0 0 10 1 1 0 0 0 5 0 4 0 7 0 6 0 0 12 13 15 14 8 9 11 10 NEXT

18. OCTET REDUCTION ( Group of 8:) OCTET (m1,m3,m5,m7,m9, m11, m13,m15) 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 W X YZ 0 0 0 1 1 1 1 0 Y.Z Y.Z Y. Z Y. Z 0 0 W.X 0 1 W.X 1 1 W.X 1 0 W.X NEXT

19. OCTET REDUCTION ( Group of 8:) MAP ROLLING OCTET (m0,m2,m4,m6, m8, m10, m12,m14) 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 W X YZ 0 0 0 1 1 1 1 0 Y.Z Y.Z Y. Z Y. Z 0 0 W.X 0 1 W.X 1 1 W.X 1 0 W.X 0 1 3 2 4 5 7 6 12 13 15 14 8 9 11 10 NEXT

20. OCTET REDUCTION ( Group of 8:) OCTET (m4,m5,m6,m7,m12, m13, m14,m15) 0 0 0 0 1 1 1 1 1 1 1 1 0 0 0 0 W X YZ 0 0 0 1 1 1 1 0 Y.Z Y.Z Y. Z Y. Z 0 0 W.X 0 1 W.X 1 1 W.X 1 0 W.X 0 1 3 2 4 5 7 6 12 13 15 14 8 9 11 10 NEXT

21. OCTET REDUCTION ( Group of 8:) MAP ROLLING OCTET (m0,m1,m2,m3 M8,m9,m10,m11) 1 1 1 1 0 0 0 0 0 0 0 0 1 1 1 1 W X YZ 0 0 0 1 1 1 1 0 Y.Z Y.Z Y. Z Y. Z 0 0 W.X 0 1 W.X 1 1 W.X 1 0 W.X 0 1 3 2 4 5 7 6 12 13 15 14 8 9 11 10

23. QUAD REDUCTION ( Group of 4) MAP ROLLING QUAD (m1,m3,m9,m11) QUAD (m4,m6,m12,m14) 1 1 1 0 1 1 1 1 1 1 1 1 0 1 1 0 0 WX YZ 3 2 4 5 7 6 1 12 13 15 14 8 9 11 10 0 0 0 1 1 1 1 0 Y.Z Y.Z Y. Z Y. Z 0 0 W.X 0 1 W.X 1 1 W.X 1 0 W.X NEXT

24. QUAD REDUCTION ( Group of 4) QUAD (m0,m2,m8,m10) CORNER ROLLING 1 0 0 1 0 0 0 0 0 0 0 0 1 0 0 1 0 WX YZ 3 2 4 5 7 6 1 12 13 15 14 8 9 11 10 0 0 0 1 1 1 1 0 Y.Z Y.Z Y. Z Y. Z 0 0 W.X 0 1 W.X 1 1 W.X 1 0 W.X

27. Karnaugh Maps - Rules of Simplification (SOP Expression) NEXT

37. M inimization S teps (SOP Expression) Select the least number of groups that cover all the 1's. Ensure that every 1 is in a group. 1's can be part of more than one group. Eliminate Redundant Groups 1 1 0 0 1 1 0 1 0 1 1 1 0 1 1 0 0 wx yz 00 01 11 10 00 01 11 10 3 2 4 5 7 6 1 12 13 15 14 8 9 11 10 NEXT

38. Example: Reduce f(wxyz)=Σ(1,3,4,5,7,10,11,12,14,15) PAIR (m4,m5) REDUNDANTGROUP 1 1 0 0 1 1 0 1 0 1 1 1 0 1 1 0 0 wx yz 00 01 11 10 00 01 11 10 3 2 4 5 7 6 1 12 13 15 14 8 9 11 10 QUAD (m1,m3,m5,m7) QUAD (m10,m11,m14,m15) QUAD (m3,m7,m11,m15) REDUNDANT Group PAIR (m4,m12) Minimized Expression : xy’z’ + wy + w’z

39. Happy Learning……….

40. References For K-Map Minimizer Download http:// karnaugh.shuriksoft.com Thomas C. Bartee, DIGITAL COMPUTER FUNDAMENTALS, McGraw Hill International. Computer Science (Class XII) By Sumita Arora http://www.ee.surrey.ac.uk/Projects/Labview/minimisation/karrules.html

41. The End