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...
Who Developed K-Maps… <ul><li>Name :  Maurice Karnaugh, a telecommunications engineer at Bell Labs.  While designing digit...
<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></...
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...
PAIR REDUCTION ( Group of 2) YZ MAP ROLLING PAIR (m0,m2) 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...
SINGLE CELL REDUCTION  SINGLE CELL (m1) SINGLE CELL (m12) QUAD (m10,m11,m14,m15) <ul><li>The term is not reduced in a sing...
Karnaugh Maps - Rules of Simplification (SOP Expression) NEXT
<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...
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 ...
Happy Learning……….
References  For K-Map Minimizer Download http:// karnaugh.shuriksoft.com Thomas C. Bartee, DIGITAL COMPUTER FUNDAMENTALS, ...
The End
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  • 1 1 The “operator” keyword Overloading Unary operators Overloading Binary operators Constructors as conversion routines Converting between basic and user-defined types LEAD IN: Overloaded Unary Operators
  • Kmap Slideshare

    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>Karnaugh Maps INDEX
    6. 6. 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>
    7. 7. <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?
    8. 8. M INIMIZATION OF B OOLEAN E XPRESSION Different methods Karnaugh Maps Algebraic Method
    9. 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. 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. 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. 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. 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
    14. 14. <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 BC A 00 01 11 10 0 1 A B C D 00 01 11 10 00 01 11 10
    15. 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
    16. 16. 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
    17. 17. 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
    18. 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. 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. 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. 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
    22. 22. 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
    23. 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. 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
    25. 25. PAIR REDUCTION ( Group of 2) YZ MAP ROLLING PAIR (m0,m2) 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 <ul><li>The term gets reduced by 1 literals i.e. 1 variables change within the group of 2( PAIR ) </li></ul>
    26. 26. 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
    27. 27. Karnaugh Maps - Rules of Simplification (SOP Expression) NEXT
    28. 28. <ul><ul><li>Groups may not include any cell containing a zero                                                              </li></ul></ul>Karnaugh Maps - Rules of Simplification (SOP Expression) NEXT
    29. 29. <ul><ul><li>Groups may be horizontal or vertical, but not diagonal.                                     </li></ul></ul>Karnaugh Maps - Rules of Simplification (SOP Expression) NEXT
    30. 30. <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
    31. 31. <ul><ul><li>Each group should be as large as possible.                                                                           </li></ul></ul>Karnaugh Maps - Rules of Simplification (SOP Expression) NEXT
    32. 32. <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
    33. 33. <ul><ul><li>Groups may overlap.                                                                     </li></ul></ul>Karnaugh Maps - Rules of Simplification (SOP Expression) NEXT
    34. 34. <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
    35. 35. <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
    36. 36. <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)
    37. 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. 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. 39. Happy Learning……….
    40. 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. 41. The End

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