Successfully reported this slideshow.
We use your LinkedIn profile and activity data to personalize ads and to show you more relevant ads. You can change your ad preferences anytime.
By  Ms. Nita Arora, PGT Computer Science Kulachi Hansraj Model  School e -Lesson
Subject  : Computer Science (083)   Unit  : Boolean Algebra Topic  :  Minimization of Boolean  Expressions Using       Kar...
L earning  O bjectives :  <ul><li>After successfully completing this module students should be able to: </li></ul><ul><ul>...
P revious  K nowledge :  <ul><ul><li>The students should be familiar with the following terms in  Boolean Algebra  before ...
Minimization Of Boolean Expressions <ul><li>Who Developed it  </li></ul><ul><li>NEED For Minimization </li></ul><ul><li>Di...
References  For K-Map Minimizer Download http:// karnaugh.shuriksoft.com Thomas C. Bartee, DIGITAL COMPUTER FUNDAMENTALS, ...
The End
<ul><li>Boolean expressions  are practically implemented in the form of GATES (Circuits). </li></ul><ul><li>A minimized Bo...
M INIMIZATION  OF  B OOLEAN  E XPRESSION Different methods Karnaugh  Maps Algebraic  Method
K arnaugh  M aps WHAT  is Karnaugh Map (K-Map)? A special version of a truth table Karnaugh Map (K-Map) is a  GRAPHICAL  d...
K-maps provide an alternate way of simplifying logic circuits. One can transfer logic values from a Truth Table into a K-M...
Correspondence between the  Karnaugh Map and the Truth Table   for the general case of a two Variable Problem  Truth Table...
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 fu...
Karnaugh maps consist of a set of  2 2  squares where  2  is the number of variables  in the Boolean expression being mini...
<ul><li>For three and four variable expressions  Maps with 2 3  = 8 and 2 4  = 16 cells are used. Each cell  represents a ...
M inimization  S teps  (SOP Expression with 4 var.) The process has following steps:  Draw the K-Map for given function  a...
M inimization  S teps  (SOP Expression) <ul><li>Form groups  of adjacent  1's .  Make groups as large as possible. </li></...
M inimization  S teps  (SOP Expression) Select the  least  number of groups that cover all the 1's.  Ensure that every  1 ...
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 ...
OCTET REDUCTION ( Group of 8:) OCTET (m0,m1,m4,m5,m8, m9, m12,m13) <ul><li>The term gets reduced by 3 literals i.e. 3 vari...
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  ...
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 ...
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 ...
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 ...
QUAD REDUCTION ( Group of 4) 1 1 0 0 1 1 1 1 0 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 QUAD  (m1,m3,m5,...
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 ...
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...
SINGLE CELL REDUCTION  SINGLE CELL (m1) SINGLE CELL (m12) QUAD (m10,m11,m14,m15) <ul><li>The term is not reduced in a sing...
PAIR REDUCTION ( Group of 2) YZ MAP ROLLING PAIR (m0,m2) <ul><li>The term gets reduced by 1 literals i.e. 1 variables chan...
<ul><ul><li>Groups may not include any cell containing a zero                                                             ...
<ul><ul><li>Groups may be horizontal or vertical, but not diagonal.                                       </li></ul></ul>K...
<ul><ul><li>Groups must contain 1, 2, 4, 8, or in general 2 n  cells.  </li></ul></ul><ul><ul><li>That is if n = 1, a grou...
<ul><ul><li>Each group should be as large as possible.                                                                    ...
<ul><ul><li>Each cell containing a  1  must be in at least one group.                                                     ...
<ul><ul><li>Groups may overlap.                                                                        </li></ul></ul>Karn...
<ul><li>Groups may wrap around the table.  </li></ul><ul><li>The  leftmost  cell in a row may be grouped with </li></ul><u...
<ul><ul><li>There should be as  few groups as possible , as long as this does not contradict any of the previous rules.   ...
<ul><ul><li>No 0’s allowed in the groups.  </li></ul></ul><ul><ul><li>No diagonal grouping allowed. </li></ul></ul><ul><ul...
<ul><li>Minimalization logic function with 3-10inputs. </li></ul><ul><li>Draw karnaugh map </li></ul><ul><li>Draw shema </...
Who Developed K-Maps… <ul><li>Name :  Maurice Karnaugh, a telecommunications engineer at Bell Labs.  While designing digit...
Upcoming SlideShare
Loading in …5
×

Kmaps By Ms Nita Arora

2,669 views

Published on

To learn Kmaps of Boolean algebra.

Published in: Technology, Education
  • Be the first to like this

Kmaps By Ms Nita Arora

  1. 1. By Ms. Nita Arora, PGT Computer Science Kulachi Hansraj Model School e -Lesson
  2. 2. Subject : Computer Science (083) Unit : Boolean Algebra Topic : Minimization of Boolean Expressions Using Karnaugh Maps (K-Maps) Category : Senior Secondary Class : XII
  3. 3. L earning O bjectives : <ul><li>After successfully completing this module students should be able to: </li></ul><ul><ul><li>Understand the Need to simplify (minimize) expressions </li></ul></ul><ul><ul><li>List Different Methods for Minimization </li></ul></ul><ul><ul><ul><ul><ul><li>Karnaugh Maps </li></ul></ul></ul></ul></ul><ul><ul><ul><ul><ul><li>Algebraic method </li></ul></ul></ul></ul></ul><ul><ul><li>Use Karnaugh Map method to minimize the Boolean expression </li></ul></ul>
  4. 4. P revious K nowledge : <ul><ul><li>The students should be familiar with the following terms in Boolean Algebra before going through this module on K-MAPS </li></ul></ul><ul><li>Boolean variable, Constants and Operators </li></ul><ul><li>Postulates of Boolean Algebra </li></ul><ul><li>Theorems of Boolean Algebra </li></ul><ul><li>Logic Gates- AND, OR, NOT, NAND, NOR </li></ul><ul><li>Boolean Expressions and related terms </li></ul><ul><ul><li>MINTERM (Product Term) </li></ul></ul><ul><ul><li>MAXTERM (Sum Term) </li></ul></ul><ul><ul><li>Canonical Form of Expressions </li></ul></ul>x y x+y
  5. 5. Minimization Of Boolean Expressions <ul><li>Who Developed it </li></ul><ul><li>NEED For Minimization </li></ul><ul><li>Different Methods </li></ul><ul><li>What is K-Map </li></ul><ul><li>Drawing a K-Map </li></ul><ul><li>Minimization Steps </li></ul><ul><li> Important Links </li></ul><ul><li>Recap . K-Map Rules </li></ul><ul><li>(SOP Exp.) </li></ul><ul><li>K-Map Quiz </li></ul>EXIT Karnaugh Maps INDEX
  6. 6. 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
  7. 7. The End
  8. 8. <ul><li>Boolean expressions are practically implemented in the form of GATES (Circuits). </li></ul><ul><li>A minimized Boolean expression means less number of gates which means </li></ul><ul><li>Simplified Circuit </li></ul>M INIMIZATION OF B OOLEAN E XPRESSION WHY we Need to simplify (minimize) expressions?
  9. 9. M INIMIZATION OF B OOLEAN E XPRESSION Different methods Karnaugh Maps Algebraic Method
  10. 10. 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
  11. 11. 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
  12. 12. 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 The diagram below illustrates the correspondence between the Karnaugh map and the truth table for the general case of a two variable problem. Truth Table Karnaugh Map A B 0 1 0 a b 1 c d
  13. 13. 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
  14. 14. 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 The diagram below illustrates the correspondence between the Karnaugh map and the truth table for the general case of a two variable problem. Truth Table Karnaugh Map 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. 15. <ul><li>For three and four variable expressions Maps with 2 3 = 8 and 2 4 = 16 cells are used. Each cell represents a MINTERM or a MAXTERM </li></ul>4 Variable K-Map 2 4 = 16 Cells K arnaugh M aps……… (Contd.) 3 Variable K-Map 2 3 = 8 Cells The diagram below illustrates the correspondence between the Karnaugh map and the truth table for the general case of a two variable problem. Truth Table Karnaugh Map BC A 00 01 11 10 0 1 A B C D 00 01 11 10 00 01 11 10
  16. 16. 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
  17. 17. M inimization S teps (SOP Expression) <ul><li>Form groups of adjacent 1's . Make groups as large as possible. </li></ul><ul><li>Group size must be a power of two. i.e. Group of </li></ul><ul><ul><li>8 (OCTET), </li></ul></ul><ul><ul><li>4 (QUAD), </li></ul></ul><ul><ul><li> 2 (PAIR) or </li></ul></ul><ul><ul><li> 1 (Single) </li></ul></ul>NEXT 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
  18. 18. 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
  19. 19. 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
  20. 20. OCTET REDUCTION ( Group of 8:) OCTET (m0,m1,m4,m5,m8, m9, m12,m13) <ul><li>The term gets reduced by 3 literals i.e. 3 variables change within the group of 8 ( Octets ) </li></ul>0 0 1 1 0 0 1 1 0 0 1 1 0 0 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 NEXT
  21. 21. 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
  22. 22. 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
  23. 23. 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
  24. 24. 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
  25. 25. QUAD REDUCTION ( Group of 4) 1 1 0 0 1 1 1 1 0 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 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 <ul><li>The term gets reduced by 2 literals i.e. 2 variables change within the group of 4( QUAD ) </li></ul>NEXT
  26. 26. 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
  27. 27. 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
  28. 28. SINGLE CELL REDUCTION SINGLE CELL (m1) SINGLE CELL (m12) QUAD (m10,m11,m14,m15) <ul><li>The term is not reduced in a single cell </li></ul>1 1 0 0 1 1 0 1 0 0 0 0 0 0 1 0 wx yz 00 01 11 10 00 01 11 10
  29. 29. PAIR REDUCTION ( Group of 2) YZ MAP ROLLING PAIR (m0,m2) <ul><li>The term gets reduced by 1 literals i.e. 1 variables change within the group of 2( PAIR ) </li></ul>0 0 0 0 0 0 0 0 0 1 1 0 1 0 0 1 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
  30. 30. <ul><ul><li>Groups may not include any cell containing a zero                                                              </li></ul></ul>Karnaugh Maps - Rules of Simplification (SOP Expression) NEXT
  31. 31. <ul><ul><li>Groups may be horizontal or vertical, but not diagonal.                                     </li></ul></ul>Karnaugh Maps - Rules of Simplification (SOP Expression) NEXT
  32. 32. <ul><ul><li>Groups must contain 1, 2, 4, 8, or in general 2 n cells. </li></ul></ul><ul><ul><li>That is if n = 1, a group will contain two 1's since 2 1 = 2. </li></ul></ul><ul><ul><li>If n = 2, a group will contain four 1's since 2 2 = 4.                                                                                    </li></ul></ul>Karnaugh Maps - Rules of Simplification (SOP Expression) NEXT
  33. 33. <ul><ul><li>Each group should be as large as possible.                                                                           </li></ul></ul>Karnaugh Maps - Rules of Simplification (SOP Expression) NEXT
  34. 34. <ul><ul><li>Each cell containing a 1 must be in at least one group.                                                                            </li></ul></ul>Karnaugh Maps - Rules of Simplification (SOP Expression) NEXT
  35. 35. <ul><ul><li>Groups may overlap.                                                                     </li></ul></ul>Karnaugh Maps - Rules of Simplification (SOP Expression) NEXT
  36. 36. <ul><li>Groups may wrap around the table. </li></ul><ul><li>The leftmost cell in a row may be grouped with </li></ul><ul><li>the rightmost cell and </li></ul><ul><li>The top cell in a column may be grouped with the </li></ul><ul><li>bottom cell .                                                             </li></ul>Karnaugh Maps - Rules of Simplification (SOP Expression) NEXT
  37. 37. <ul><ul><li>There should be as few groups as possible , as long as this does not contradict any of the previous rules.                                                                    </li></ul></ul>Karnaugh Maps - Rules of Simplification (SOP Expression) NEXT
  38. 38. <ul><ul><li>No 0’s allowed in the groups. </li></ul></ul><ul><ul><li>No diagonal grouping allowed. </li></ul></ul><ul><ul><li>Groups should be as large as possible. </li></ul></ul><ul><ul><li>Only power of 2 number of cells in each group. </li></ul></ul><ul><ul><li>Every 1 must be in at least one group. </li></ul></ul><ul><ul><li>Overlapping allowed. </li></ul></ul><ul><ul><li>Wrap around allowed. </li></ul></ul><ul><ul><li>Fewest number of groups are considered. </li></ul></ul><ul><ul><li>Redundant groups ignored </li></ul></ul>Karnaugh Maps - Rules of Simplification (SOP Expression)
  39. 39. <ul><li>Minimalization logic function with 3-10inputs. </li></ul><ul><li>Draw karnaugh map </li></ul><ul><li>Draw shema </li></ul><ul><li>Cońvert to NOR and NANDS. </li></ul>Karnaugh map minimalization software is freeware. Important Links… K-Min 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.
  40. 40. Who Developed K-Maps… <ul><li>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. </li></ul><ul><li>Year : 1950 same year that Charles M. Schulz published his first Peanuts comic. </li></ul>

×