Presentech'10
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This document provides information about Karnaugh maps (k-maps), including: - K-maps are a pictorial representation of variables used to identify Boolean expressions or truth tables from grouping of minterms. - Basic k-maps include two, three, four, and five variable maps. Rules for mapping include grouping horizontally or vertically but not diagonally, and wrapping is only allowed for three variable maps. - Don't care conditions represented by 'x' can be used but all don't cares cannot be grouped as a single term. - K-maps are used to simplify Boolean expressions and design combinational and sequential logic circuits like adders. They provide an alternative to tabular methods
![THE MAP
Straight procedure for minimizing Boolean
functions
Considered as pictorial form of a truth
table
It is a diagram made of squares
It consists of minterms[0] & maxterms[1]
In Some cases dont care conditions are
used [represented by x] .](https://image.slidesharecdn.com/presentech10-130129065110-phpapp01/85/Presentech-10-5-320.jpg)
Presentech'10
- 1. INFANT JESUS COLLEGE OF ENGINEERING COMPUTER SCIENCE & ENGINEERING DEPARTMENT PRESENTECH10
- 2. Guided by: NAGALINGARAJAN Presented by: ABIRAM.A BEJOY.S.P BHARATHI RAJA.M SUSHEENTHIRAN.A
- 3. Index The map method Rules for map Present technique. Types of variables Uses of k-map Application of k-map.
- 4. WHAT IS K-MAP? The pictorial representation of variables from which the Boolean expression or truth table can be identified. It avoids ambiguity during logic designing. Since there is number of probability in mapping the answer would be same and avoid confusion. It is denoted as k-map since it is found by the person karnaugh.
- 5. THE MAP Straight procedure for minimizing Boolean functions Considered as pictorial form of a truth table It is a diagram made of squares It consists of minterms[0] & maxterms[1] In Some cases dont care conditions are used [represented by x] .
- 6. Rules Mapping can be done horizontally or vertically but not in diagonally. Wrapping technique can be done only form three variable map. Only the maxterms can be mapped to form a Boolean expression. Overlapping of maxterms is possible, utmost once.
- 7. present technique Maxterms are grouped together to derive the expression. Dont care conditions alone cannot be mapped to form a group. The mapping can be selected depending on the variable used Corresponding variables should be written for the mapped values. The grouping is done in even numbers.
- 8. Basics The frequently used variable mappings are Two variable map Three variable map Four variable map Five variable map Lets see this in detail
- 9. Two variable-map Four minterms for two variables It consists of four squares Y The 0 and 1 marked in X row and column designate the values of 0 0 variables A two variable is shown And the expression for the group is 1 1 =X+Y
- 10. Three variable map It consists of eight minterms for three binary variables It has eight squares Minterms are not arranged in binary sequence but similar to gray code Wrapping technique can start using only from the three variable mapping Consider the following truth table for which the mapping can be done.
- 11. Truth table D.NO A B C S C 0 0 0 0 0 0 1 0 0 1 1 0 2 0 1 0 1 0 3 0 1 1 0 1 4 1 0 0 1 0 5 1 0 1 0 1 6 1 1 0 0 1 7 1 1 1 1 1
- 12. Representation of three variable map YZ 00 01 11 10 X m0 m1 m3 M2 0 m4 m5 m7 m6 1
- 13. FOUR VARIABLE MAP In this method four variables are used. The map contains 16 squares. Grouping can be done only for the maxterms Wrapping can be done by folding the map. Adjacent groups cannot be formed since it contains single map. The representation of four variable map is.
- 14. Four variable map C CD AB 00 01 10 11 Adjacent 1 1 1 groups 00 cant be formed 01 1 B 10 A Wrapping can be 11 1 1 1 cone
- 15. Five variable map It consists of five variables (A,B,C,D,E) Left hand four variable consists of 16 squares and right consists of 16 squares Actually it is the combination of two four variable map The left table represents the variable value A=0 and right table represents the A value as 1. Adjacent group in the tables can be represented once Consider the following truth table for which the mapping is done.
- 16. Five variable map A=0 A=1 D D DE BC 0 2 2 3 16 17 19 18 4 5 7 6 20 21 23 22 C C 12 13 15 14 28 29 31 30 B B 8 9 11 10 24 25 27 26
- 17. Representation of five variable map without using two four variable map Adjacent groups cant be ABC 000 001 010 011 110 111 100 101 formed DE but it 00 0 1 2 3 6 7 4 5 gives the same 01 8 9 10 11 14 15 12 13 answer 24 25 26 27 30 31 28 29 10 11 16 17 18 19 22 23 20 21
- 18. DONT CARE CONDITIONS Y CD 00 01 11 11 AB 00 x 1 1 X 01 0 x 1 0 X 10 0 0 1 0 W 11 0 0 1 0
- 19. DONT CARE CONDITION The x represents the position of dont care conditions. The value of x may 1 or 0 it depends on grouping. The expression for the previous k-map is: A=c +a +bd +bd. Note that all the dont care cannot be mapped as a single group.
- 20. OUR METHOD Check the maxterms in the truth table itself. Write the corresponding expression. Simplify the expression with the help of DEMORGANS theory. The is no need of k-map in this method.
- 21. uses K-map is used to derive any form of logical circuit It is used in design and analysis procedure It is used in Quine-McCluskey method Simplification of Boolean expression is easier It is used in sum of product and in product of sum method.
- 22. Application The k-map can be used in combinational circuit and in sequential circuit. Full adder and half adder can be easily analyzed with the help of k-map. It is used in the in tabular method to simplify the Boolean expression even simpler than that.
- 23. conclusion Thus k-map is a major topic in electronics It is used in simplification of Boolean expression & logic gates It is used in the conversion of truth table to Boolean expression and vice versa. It is better to use a k-map than the tabular method.
- 24. THANK YOU