Mind Mangler - A Karnaugh Map Example

8/6/2019 Mind Mangler - A Karnaugh Map Example

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5/4/11 2:Mind Mangler - a Karnaugh Map Example

Page ttp://www.generalnumbers.com/karnaugh_application1.html

A Worked Example

Problem:Given a Boolean expression:

F=AB +B'C

use a Karnaugh map (K-map) to

minimize.

Wonderinghow to

pronounce

"Karnaughmap?" Click

here.

Approach: Determine the size (number of cells) forthe Karnaugh map. To do this, count the

number of unique variables in the

expression. Do not countB' (thecomplement ofB) as a separate variable

fromB. Hence, in AB +B'Cthere

are 3 variables:A,B, and C.

Forkvariables, each of which can take

one of two values (e.g., 1 or 0, true orfalse, high or low voltage), there are 2k

possible combinations of variable values.

Here, 2 is the base of the numbersystem, since there are only 2 possible

values. Hence, for the 3 variables in this

Side Note: Aleft, the text

correspondin

to the numbeof unique

variables ishighlighted in

yellow, and

the textcorrespondin

to the base o

the numbersystem is

highlighted inturquoise.

Note that
http://www.generalnumbers.com/karnaugh/karnaugh_map.wavhttp://www.generalnumbers.com/karnaugh/karnaugh_map.wavhttp://www.generalnumbers.com/karnaugh/karnaugh_map.wav


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problem, we must have 23 = 2 x 2 x2 = 8cells in the Karnaugh map to holdall the possible combinations.

Number of cells in map = 23 =8

Just in case you are curious, here is atruth table of all 8 combinations, color-

coded so we can match them up easily

with the Karnaugh map that we are doingnext:

Possible Combinations:

A B C

1 1 1

1 1 0

1 0 1

1 0 0

0 1 1

0 1 0

0 0 1

color is not

used inKarnaugh-

map-related

homework,

although withmore complemaps

sometimes

differentcolors are

used around

the primeimplicants, fo

clarity.

Color is used

here only as

aneducational

aid.


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0 0 0

Next we examine an empty Karnaugh

map with 8 cells, this time colored (for

educational purposes only) to showwhichABCvalue from the above tablecorresponds to which cell. Here the four

rows represent all the combinations of

the variablesBC, and the two columnsrepresent all the values that can be taken

by the remaining variable,A. Note thatthe values ofB and Care in Gray codeorder in the rows.

We can see that the upper left cell

corresponds to anABCvalue of 000,

and the lower right to anABCvalue of110. Inside each cell will go the value of


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F=AB +B'Cfor that cell'sABC

value.

Here's an untinted version of the same

empty 3-variable Karnaugh map. Havingan empty version can be handy, so thatyou can quickly make a new 3-variable

Karnaugh map without having to redraw

one each time. To save the image toyour hard drive, right-click on it, and

depending on your browser, choose"Save Picture As" or an equivalent

command.

Now back to the problem. Let's populate

the empty Karnaugh map with values ofF=AB +B'C, one for each possible

value ofABC. First make a truth table

forF, so we can know what values toput in the cells. The leftmost 3 columns


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of the truth table forFare the same asthe truth table above showing all possible

value combinations forABC.

Truth Table forF=AB +B'C

A B C B' AB B'C F=AB +B'C

1 1 1 0 1 0 11 1 0 0 1 0 1

1 0 1 1 0 1 1

1 0 0 1 0 0 0

0 1 1 0 0 0 00 1 0 0 0 0 0

0 0 1 1 0 1 1

0 0 0 1 0 0 0

Continuing

on:

Here is the 8-cell Karnaugh map, filled with

the values from the truth table:


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Next, combine prime implicants, which are

maximal groupings of 1's of sizes that arepowers of 2, such as 1 one, 2 ones, 4 ones,

8 ones, etc. Here the largest number of 1'sthat is a power of two that we can find in a

block or straight line is 2. We do have onecolumn of three 1's, but 3 is not an integral

power of 2, so we can't use that. Twogroupings of 2 ones each is the best we can

manage.


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The lower circle representsA=1 and

B=1, with no reference to Csince Ccanbe either 0 or 1. Hence the prime implicant

represented by the lower circle is AB.

The upper circle representsB=0 and

C=1, with no reference toA sinceA canbe either 0 or 1. Hence the prime implicant

represented by the upper circle isB'C.

Since there are the only two primeimplicants, the resulting answer for the

minimized representation of Fis:

F=AB +B'C

which is coincidentally the same

representation we had forFat the start.

This means that Fwas already in itsminimal form when we started the problem.

DONE

For the drawing Smartdraw Professional

Plus v. 6.2 was used, with the resulting

screen image captured by Snagit from


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Techsmith.com, and further embellished,

then optimized for the Web, usingPhotoshop7. The title graphic was done in

Photoshop 7.

Copyright 2003 Crystal Sloan and Dr. Yul Williams

Page Design and Original Graphics Copyright 2003 Crystal Sloan.
http://www.ertin.com/sloan.html

Mind Mangler - A Karnaugh Map Example

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