minterm,maxterm & k-map

8/16/2019 Minterm,Maxterm & K-Map

1/15

Minterm,Maxterm & K-Map


2/15

Minterms MintermsMinterms are AND terms with every variable present

in either true or complemented form.

Given that each binary variable may appear normal(e.g., x or complemented (e.g., , there are !n

minterms for n variables."xample# $wo variables (% and &produce

! x ! ' combinations# (both normal

(% normal, & complemented (% complemented, & normal

(both complemented

$hus there are four minterms of two variables.

Y X

XY

Y X Y X

x


3/15


4/15

MaxtermsMaxterms are OR terms with every

variable in true or complemented form.Given that each binary variable may

appear normal (e.g. x! or complemented

(e.g. x! there are "n

maxterms forn

variables. #xample$ %wo variables (X and Y! produce

" x " & ' combinations$ (both normal!

(x normal y complemented! (x complemented y normal!

(both complemented!

Y X +

Y X +

Y X +

Y X +


5/15

)um term

in *+)


6/15


7/15

oolean expansion in )+* form

#xample$%here are three variables ) and * which

we ta+e to be the standard order.#xpanding the terms with missing variables$

F = A(B + B’)(C + C’) + (A + A’) B’ C= ABC + ABC’ + AB’C + AB’C’ + AB’C + A’B’C

*ollect terms (removing all but one ofduplicate terms!$

= ABC + ABC’ + AB’C + AB’C’ + A’B’C

#xpress as ,Op$

= m7 + m6 + m5 + m4 + m1

= m1 + m4 + m5 + m6 + m7

*)- +


8/15

oolean expansion in *+) form

ny )oolean -unction can be expressed as aroduct of Maxterms (OM!.-or the function table the maxterms used are the

terms corresponding to the /0s.-or an expression expand all terms 1rst to

explicitly list all maxterms. 2o this by 1rst

applying the second distributive law 3ORing4terms missing variable v with a term e5ual to andthen applying the distributive law again.

#xample$ *onvert to product of maxterms$

pply the distributive law$

dd missing variable 6$

#xpress as OM$ f & M" 7 M8

yxx!6yx(f +

yx!y(x9!y!(xx(xyxx +

6yx!6yx(66yx +

vv⋅

A+BC = (A+B)(A+C)


9/15

Simplification of Boolean Functions:

Two Methods

$he algebraic method by using -dentities $he graphical method by

using arnaugh Map method

$he /map method is easy and straightforward. A /map for a function of n variables

consists of !n cells, and,

in every row and column, two ad0acent cells should differ in

the value of only one of the logic variables.


10/15

Examples of K-Maps:

"xamples#

1ell numbers are written in the cells.

!/variable /map

0 12 3

0

1

0 1 A

B


11/15

Re-arranging the truth table A two/variable function has four possible minterms. 2e can re/arrange these

minterms into a arnaugh map.

Now we can easily see which minterms contain common literals.

Minterms on the left and right sides contain y3 and y respectively.

Minterms in the top and bottom rows contain x3 and x respectively.

x y minterm

0 0 x’y’

0 1 x’y

1 0 xy’1 1 xy

Y

0 1

0 x’y’ x’yX

1 xy’ xy

Y

0 1

0 x’ y’ x’ yX

1 x y’ x y

Y’ Y

X’ x’ y’ x’ y

X x y’ x y


12/15

Two ariable K-MapsTwo ariable K-Maps

":variable ;arnaugh Map$

,imilar to Gray *oded


13/15

K-Map and Truth Tables K-Map and Truth Tables

%he ;:Map is %wo variable function$

?e choose abc and d from the set

@/9A to implement a particularfunction -(xy!.

FunctionTable

K-Map

Values

(x,y

Function

Value

F(x,y

0 0 a

0 1 b

1 0 c

1 1 !

B = 0 B = 1

A = 0 a b A = 1 c !


14/15


15/15

minterm,maxterm & k-map

Documents