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• The “operator” keyword
Constructors as conversion routines
Converting between basic and user-defined types
• Unary operators require no arguments because they automatically refer to the object that calls them.
For ageClass, the ++ and -- operators would intuitively increment and decrement the age data member.
• Unary operators require no arguments because they automatically refer to the object that calls them.
For ageClass, the ++ and -- operators would intuitively increment and decrement the age data member.
• KMAP

1. 1. 01/30/15 Karnaugh Maps 1 Subject : Computer Science (083) Unit : Boolean Algebra Topic : Minimization of Boolean Expressions Using Karnaugh Maps (K-Maps)
2. 2. 01/30/15 Karnaugh Maps 2 Learning Objectives : 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
3. 3. 01/30/15 Karnaugh Maps 3 Previous Knowledge : The students should be familiar with the following terms in Boolean Algebra before going through this module on K-MAPS xx yy x+yx+y Boolean variable, Constants and Operators Postulates of Boolean Algebra Theorems of Boolean Algebra Logic Gates- AND, OR, NOT, NAND, NOR Boolean Expressions and related terms MINTERM (Product Term) MAXTERM (Sum Term) Canonical Form of Expressions
4. 4. 01/30/15 Karnaugh Maps 4 Minimization Of Boolean Expressions Who Developed it NEED For Minimization Different Methods What is K-Map Drawing a K-Map Minimization Steps Important Links Recap. K-Map Rules (SOP Exp.) K-Map Quiz EXIT INDEX
5. 5. 01/30/15 Karnaugh Maps 5 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
6. 6. 01/30/15 Karnaugh Maps 6 The End
7. 7. 01/30/15 Karnaugh Maps 7 Boolean expressions are practically implemented in the form of GATES (Circuits). A minimized Boolean expression means less number of gates which means Simplified Circuit MINIMIZATION OF BOOLEAN EXPRESSION WHY we Need to simplify (minimize) expressions?
8. 8. 01/30/15 Karnaugh Maps 8 MINIMIZATION OF BOOLEAN EXPRESSION Different methods Karnaugh Maps Algebraic Method
9. 9. 01/30/15 Karnaugh Maps 9 Karnaugh Maps 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. 10. 01/30/15 Karnaugh Maps 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 Karnaugh Maps……… (Contd.) NEXT
11. 11. 01/30/15 Karnaugh Maps 11 Correspondence between the Karnaugh Map and the Truth Table for the general case of a two Variable Problem 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 Truth Table 2 Variable K-Map Karnaugh Maps……… (Contd.)
12. 12. 01/30/15 Karnaugh Maps 12 Drawing a Karnaugh Map (K-Map) K-map is a rectangle made up of certain number of SQUARES For a given Boolean function there are 2N 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 22 =4 squares Each square is different from its neighbour by ONE Literal Each SQUARE represents a MAXTERM or MINTERM NEXT
13. 13. 01/30/15 Karnaugh Maps 13 Karnaugh maps consist of a set of 22 squares where 2 is the number of variables in the Boolean expression being minimized. Truth Table 2 Variable K-Map Karnaugh Maps……… (Contd.) A B 0 1 0 0 1 1 1 11 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
14. 14. 01/30/15 Karnaugh Maps 14  For three and four variable expressions Maps with 23 = 8 and 24 = 16 cells are used. Each cell represents a MINTERM or a MAXTERM 4 Variable K-Map 24 = 16 Cells Karnaugh Maps……… (Contd.) BC A 00 01 11 10 0 1 A B C D 00 01 11 10 00 01 11 10 3 Variable K-Map 23 = 8 Cells
15. 15. 01/30/15 Karnaugh Maps 15 Minimization Steps (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 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 1 1 1 1 NEXT
16. 16. 01/30/15 Karnaugh Maps 16 Minimization Steps (SOP Expression) Form groups of adjacent 1's. Make groups as large as possible. Group size must be a power of two. i.e. Group of • 8 (OCTET), • 4 (QUAD), • 2 (PAIR) or • 1 (Single) 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
17. 17. 01/30/15 Karnaugh Maps 17 Minimization Steps (SOP Expression) Select the least number of groups that cover all the 1's. 1100 1101 0111 0110 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 Ensure that every 1 is in a group. 1's can be part of more than one group. Eliminate Redundant Groups NEXT
18. 18. 01/30/15 Karnaugh Maps 18 Example: Reduce f(wxyz)=Σ(1,3,4,5,7,10,11,12,14,15) PAIR (m4,m5) REDUNDANTGROUP 1100 1101 0111 0110 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
19. 19. 01/30/15 Karnaugh Maps 19 OCTET REDUCTION ( Group of 8:) 0011 0011 0011 0011 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 OCTET (m0,m1,m4,m5,m8, m9, m12,m13) •The term gets reduced by 3 literals i.e. 3 variables change within the group of 8 ( Octets ) NEXT
20. 20. 01/30/15 Karnaugh Maps 20 OCTET REDUCTION ( Group of 8:) 0110 0110 0110 0110 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 OCTET (m1,m3,m5,m7,m9, m11, m13,m15) NEXT
21. 21. 01/30/15 Karnaugh Maps 21 OCTET REDUCTION ( Group of 8:) MAP ROLLING OCTET (m0,m2,m4,m6, m8, m10, m12,m14) 1001 1001 1001 1001 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
22. 22. 01/30/15 Karnaugh Maps 22 OCTET REDUCTION ( Group of 8:) 0000 1111 1111 0000 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 OCTET (m4,m5,m6,m7,m12, m13, m14,m15) NEXT
23. 23. 01/30/15 Karnaugh Maps 23 OCTET REDUCTION ( Group of 8:) MAP ROLLING OCTET (m0,m1,m2,m3 M8,m9,m10,m11) 1111 0000 0000 1111 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
24. 24. 01/30/15 Karnaugh Maps 24 QUAD REDUCTION ( Group of 4) 1100 1111 0111 0110 0 WX YZ 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 (m4,m5,m12,m13) 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 •The term gets reduced by 2 literals i.e. 2 variables change within the group of 4( QUAD ) NEXT
25. 25. 01/30/15 Karnaugh Maps 25 QUAD REDUCTION ( Group of 4) MAP ROLLING QUAD (m1,m3,m9,m11) QUAD (m4,m6,m12,m14) 1110 1111 1111 0110 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
26. 26. 01/30/15 Karnaugh Maps 26 QUAD REDUCTION ( Group of 4) QUAD (m0,m2,m8,m10) 1001 0000 0000 1001 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 CORNER ROLLING
27. 27. 01/30/15 Karnaugh Maps 27 SINGLE CELL REDUCTION 1100 1101 0000 0010 wx yz 00 01 11 10 00 01 11 10 SINGLE CELL (m1) SINGLE CELL (m12) QUAD (m10,m11,m14,m15) •The term is not reduced in a single cell
28. 28. 01/30/15 Karnaugh Maps 28 PAIR REDUCTION ( Group of 2) YZ MAP ROLLING PAIR (m0,m2) 0000 0000 0110 1001 0 WX 3 2 4 5 7 6 1 12 13 15 14 8 9 11 10 PAIR (m5,m7) 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 •The term gets reduced by 1 literals i.e. 1 variables change within the group of 2( PAIR )
29. 29. 01/30/15 Karnaugh Maps 29 • Groups may not include any cell containing a zero NEXT Karnaugh Maps - Rules of Simplification (SOP Expression)
30. 30. 01/30/15 Karnaugh Maps 30 •Groups may be horizontal or vertical, but not diagonal. NEXT Karnaugh Maps - Rules of Simplification (SOP Expression)
31. 31. 01/30/15 Karnaugh Maps 31 • Groups must contain 1, 2, 4, 8, or in general 2n cells. • That is if n = 1, a group will contain two 1's since 21 = 2. • If n = 2, a group will contain four 1's since 22 = 4.                                                                                    NEXT Karnaugh Maps - Rules of Simplification (SOP Expression)
32. 32. 01/30/15 Karnaugh Maps 32 •Each group should be as large as possible.                                                                           NEXT Karnaugh Maps - Rules of Simplification (SOP Expression)
33. 33. 01/30/15 Karnaugh Maps 33 •Each cell containing a 1 must be in at least one group.                                                                           NEXT Karnaugh Maps - Rules of Simplification (SOP Expression)
34. 34. 01/30/15 Karnaugh Maps 34 •Groups may overlap.                                                                     NEXT Karnaugh Maps - Rules of Simplification (SOP Expression)
35. 35. 01/30/15 Karnaugh Maps 35 • Groups may wrap around the table. • The leftmost cell in a row may be grouped with the rightmost cell and • The top cell in a column may be grouped with the bottom cell.                                                             NEXT Karnaugh Maps - Rules of Simplification (SOP Expression)
36. 36. 01/30/15 Karnaugh Maps 36 • There should be as few groups as possible, as long as this does not contradict any of the previous rules.                                                                    NEXT Karnaugh Maps - Rules of Simplification (SOP Expression)
37. 37. 01/30/15 Karnaugh Maps 37 1. No 0’s allowed in the groups. 2. No diagonal grouping allowed. 3. Groups should be as large as possible. 4. Only power of 2 number of cells in each group. 5. Every 1 must be in at least one group. 6. Overlapping allowed. 7. Wrap around allowed. 8. Fewest number of groups are considered. 9. Redundant groups ignored Karnaugh Maps - Rules of Simplification (SOP Expression)
38. 38. 01/30/15 Karnaugh Maps 38 • Minimalization logic function with 3-10inputs. • Draw karnaugh map • Draw shema • Cońvert to NOR and NANDS. Karnaugh map minimalization software is freeware. Karnaugh Minimizer is a tool for developers of small digital devices and radio amateurs, also for those who is familiar with Boolean algebra, mostly for electrical engineering students. Important Links… K-Min
39. 39. 01/30/15 Karnaugh Maps 39 Who Developed K-Maps… • Name: Maurice Karnaugh, a telecommunications engineer at Bell Labs. While designing digital logic based telephone switching circuits he developed a method for Boolean expression minimization. • Year : 1950 same year that Charles M. Schulz published his first Peanuts comic.