Use Kmap to simplify the following functions in SOP forms as

Use K-map to simplify the following functions in SOP forms as much as possible: (a) (10 points) f(a) = m(0,1) (b) (30 points) f(a,b,c) = a\'bc\' + a\'b\'c + a\'bc + ab\'c + abc (c) (60 points) f(a,b,c,d) = m(0,2,5,8,9,10,12,13,14,15) (d) (60 points) f(a,b,c,d) = m(1,7,9,10,11,12,13,15)

Solution

(a) draw a table with two rows and two clumns

i.e

place the value \' 1\' in table i.e in the index \'0\' and \'1\'

Hence the expresssion using the k_Map in sop form i.e = a\'b\' +a\'b

=a\'(b\'+b)

=a\'(1)   

=a\' [Identity b\'+b =1]

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(B) f(a,b,c)=a\'bc\'+a\'b\'c+a\'bc+ab\'c+abc

I.e f(a,b,c) =(2,1,3,5,7)

Now draw a table with 2 rows and four columns

place \'1\' in every place as per the given question i.e \'1\' in table 2,1,3,5,7 places

Hence the k_map obtained in SOP Form as

f =c+bc+bc\'

=c+b(c+c\')

=c+b [identity c+c\'=1]

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(c) draw a table with 4 rows and 4 columns

place \'1\' in their corresponding places in the table (in 0,2,5,8,9,10,11,12,13,14,15)

hence by using the k_map in sop form as below

f= ac\'+ab+acd\'+ab+a\'b\'c\'d\'+a\'b\'cd\'+bc\'d

= ac\'+ab+acd\'+a\'b\'c\'d\'+a\'b\'cd\'+bc\'d [identity ab+ab= ab]

=ac\'+ab+acd\'+a\'b\'d\'(c\'+c)+bc\'d

=ac\'+ab+acd\'+a\'b\'d\'(1)+bc\'d [identity c+c\' =1]

=ac\'+ab+acd\'+ a\'b\'d\'+bc\'d

(d) Draw a table with 4 rows and 4 columns

place the value \'1\' in a table as per the given places (1,7,9,10,11,12,13,15)

by using k_map f= ad+ab\'c+a\'b\'c\'d+abc\'+bcd