KARNAUGH MAP

1. KARNAUGH MAP • 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. • K Map is a grid –like representation of a truth table. It is really just another way of presenting truth table.

2. HISTORY • The Karnaugh map was invented in 1952 by Edward W. Veitch and developed further in 1953 by Maurice Karnaugh, a telecommunications engineer at bell It is a method to simplify Boolean algebra expressions. The Karnaugh map reduces the need for extensive calculations by taking advantage of humans' pattern-recognition capability, permitting the rapid identification and elimination of potential race hazards

3. PURPOSE • Normally, extensive calculations are required to obtain the minimal expression of a Boolean function, however Karnaugh mapping reduces the need for such calculations by: • Taking advantage of the human brain's pattern-matching capability to decide which terms should be combined to obtain the simplest expression. • Permitting the rapid identification and elimination of potential race hazards, which is unavoidable in boolean equations. • Providing an excellent aid for simplification of up to six variables, however with more variables it becomes more difficult to discern optimal patterns. • Helping to teach about Boolean functions and minimization.

4. PROCEDURE An significant about the construction of K-map is the arrangement of the cells. Two cells are physically adjacent within the configuration if and only if their respective n-tuples differ in exactly by one element. So that the Boolean law x+x=1 cab be applied to adjacent cells. Ex. Two 3- tuples (0,1,1) and (0,1,0) are physically a djacent since these tuples vary by one element. When the Karnaugh map has been completed, to derive a minimized function the "1s" or desired outputs are grouped into the largest possible rectangular groups in which the number of grid boxes (output possibilities) in the groups must be equal to a power of 2.[1] For example, the groups may be 4 boxes in a line, 2 boxes high by 4 boxes long, 2 boxes by 2 boxes, and so on. "Don't care(s)" possibilities (generally represented by a "X") are grouped only if the group created is larger than the group with "Don't care" is excluded. The boxes can be used more than once only if it generates the least number of groups. Each "1" or desired output possibilities must be contained within at least one grouping.

5. OBJECTIVES • This section presents a technique for simplifying logical expressions. It will: • Define Karnaugh and establish the correspondence between Karnaugh maps and truth tables and logical expressions. • Show how to use Karnaugh maps to derive minimal sum of-products and product-of-sums expressions. • Introduce the concept of "don't care“ entries and show how to extend Karnaugh map techniques to include maps with don't care entries.

6. 2 VARIABLE K-MAP 2 variable needs 2^2= 4cells B 0 1 0 1 m0 m1 0 1 0 A m2 m3 1 S= a b’+ a’ b

7. 3 VARIABLE K-MAP 3 variables need 2^3 =8 cells F=Σm(0,1,2,4,5,6) A AB 00 01 11 10 C 0 1 c F=C’+B’

8. 4-variable K-maps

9. 5 VARIABLE K-MAP 5 variables need 2^5=32 cells y y YZ 00 01 11 10 YZ 00 01 11 10 WX WX 00 01 11 10 00 01 11 10 x X W W A’ A Z Z

10. No diagonals. No zeros allowed. 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 SIMPLIFICATION

11. SUM OF PRODUCT (SOP) AB AB’D’ CD 00 01 11 10 A'BC’ 00 01 11 10 Simplified SOP expression can be obtained by grouping the minterms (i.e. 1s)of given function. example: F=Σm(2,3,4,5,7,8,10,13,15) A’B’C

12. PRODUCT OF SUM (POS) (A’+B’) AB 00 01 11 10 CD Simplified POS expression can be obtained by grouping the maxterms (i.e. 0s) of given function. Example: F=ΣM(0,1,2,3,5,7,8,9,10,11) 00 01 11 10 (B’+D)

13. DON’T CARE CONDITION They could be chosen to be either ‘1’ or ‘0’ depending on which gives the simpler expression. Don’t care conditions can be used to help simple boolean expressions further in K-Maps. Don’t care conditions are denoted by ‘X’.

14. DON’T CARE CONDITION C CD 00 01 11 10 AB 00 01 11 10 B A D

15. USE OF BOOLEAN ALGEBRA IN SIMPLIFICATION Commutative law : x + y = y + x x . y = y . X Distributive law : x + (y . z) = (x + y) . (x + z) x . (y + z) = (x . y) + (x . z) Associative law : (x + y) + z = x + (y + z) (x . y) . z = x. (y . z) Complement : x + x’ = 1 x . x’ = 0 De Morgan’s law : ( x+ y )’ = x’ . y’ ( x . y )’ = x’+ y’

16. SIMPLIFICATION USING K-MAP GROUP OF MINTERMS MUST BE: Rectangular,and Have a size in power of 2's

17. SIMPLICATION USING K-MAP 1 2 3

18. ADVANTAGE ADVANTAGE: The advantage of K Map is that it is designed to present the information in a way that allows easy grouping of terms that can be combined.

19. LIMITATIONS • It is difficult to make judgements about which combinations from the minimum expression. • The K Map simplification is manual technique and simplification process is heavily depends on human abilities.

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