Rules of Karnaugh Map

Rules of Karnaugh Map

1. Rules of Karnaugh Map Submitted by Tanjarul Islam Mishu Roll #: 11102026 Dept. of CSE JKKNIU Submitted to Uzzal Kumar Prodhan Associate Professor Dept. of CSE JKKNIU

2. Karnaugh Map introduction

3. A Karnaugh map (K-map) is a pictorial method used to minimize Boolean expressions without having to use Boolean algebra theorems and equation manipulations. A K-map can be thought of as a special version of a truth table. Using a K-map, expressions with two to four variables are easily minimized. Expressions with five to six variables are more difficult but achievable, and expressions with seven or more variables are extremely difficult to minimize using a K-map. What is Karnaugh Map

4. The Karnaugh map uses the following rules for the simplification of expressions by grouping together adjacent cells containing ones. Groups can not include any cell containing a zero   . Rules of Karnaugh Map

5. Groups may be horizontal or vertical , but not diagonal . Rules of Karnaugh Map

6. Groups must contain 1 , 2 , 4 , 8 , or in general 2 n cells. That is if n = 1, a group will contain two 1's since 2 1 = 2. If n = 2, a group will contain four 1's since 2 2 = 4. Rules of Karnaugh Map

7. Each group should be as large as possible.  Rules of Karnaugh Map

8. Each cell containing a one  must be in at least one group.  Rules of Karnaugh Map

9. Groups may overlap each other. Rules of Karnaugh Map

10. 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. Rules of Karnaugh Map

11. There should be as few groups as possible, as long as this does not contradict any of the previous rules. Rules of Karnaugh Map

12. Rules of Karnaugh Map Summary

13. No zeros allowed. No diagonals.   Only power of 2 number of cells in each group. Groups should be as large as possible.   Every one must be in at least one group. Overlapping allowed.   Wrap around allowed. Fewest number of groups possible. Rules of Karnaugh Map

14. Rules of Karnaugh Map Examples

15. Examples 1 1 1 1 1 00 01 11 10 1 ab c F=a’ bc ’+ a’bc+a’b’c+ab’c F= a’b+b’c Gates : 10 Gates : 5

16. Examples 2 1 1 1 1 1 00 01 11 10 1 ab c F=a’ bc ’+a’ bc+abc ’+ abc+a’b’c F= b+a’c Gates : 12 Gates : 3

17. Examples 3 1 1 1 1 1 1 00 01 11 10 00 01 ab cd 11 10 F= a’bc’d+a’bcd+abc’d+abcd+a’b’c’d+abcd ’ F= bd+a’c’d+abc Gates : 15 Gates : 7

18. Examples 4 Y 8 9 10 11 12 13 14 15 1 3 2 5 6 4 7 1 1 1 1 1 1 1 1 1 1 XZ W’X 00 X’Z’ F(W,X,Y,Z) = XZ + X’Z’ + W’X 3,14,15 ) (3,4,5,7,9,1 Z) Y, X, F(W, m S = 11 11 10 1 01 10 00 WX YZ

19. http://books.google.com http :// en.wikipedia.org/wiki/Karnaugh_map http :// www.ustudy.in http://www.ee.surrey.ac.uk http://www.ece.rice.edu References

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