problem 4.5 sop form

Problem 4.5 SOP form

1

F= +

x 1 x 2 x 3 x 4

0

00 01 11 10

0 1 0

1 1 1 1

0 0 0 0

1 1 1 0

00

01

11

10

(b) =1

Primary implicants:

, x2

x1x2x 1 x 2 x 3 x 4

1

00 01 11 10

1 1 1

1 1 1 1

0 1 1 0

0 0 0 0

00

01

11

10

(b) =0

POS form

2

x 1 x 2 x 3 x 4

1

00 01 11 10

1 1 1

1 1 1 1

0 1 1 0

0 0 0 0

00

01

11

10

(a)

F= () () ()() ()

x 1 x 2 x 3 x 4

0

00 01 11 10

0 1 0

1 1 1 1

0 0 0 0

1 1 1 0

00

01

11

10

(a)

Problem 4.6

3

x 1 x 2 x 3 x 4

1

00 01 11 10

1 1 1

0 0 0 0

1 1 0 1

1 d 0 1

00

01

11

10

(a)

x 1 x 2 x 3 x 4

1

00 01 11 10

1 1 d

1 1 0 1

d d 0 d

d 1 0 1

00

01

11

10

(b)

+ ++ +

4

x 1 x 2 x 3 x 4

1

00 01 11 10

1 1 1

0 0 0 0

1 1 0 1

1 d 0 1

00

01

11

10

(a)

x 1 x 2 x 3 x 4

1

00 01 11 10

1 1 d

1 1 0 1

d d 0 d

d 1 0 1

00

01

11

10

(b)

+ + +

Problem 4.7

5

x 1 x 2 x 3 x 4

0

00 01 11 10

d 0 d

0 1 1 0

1` 0 1 d

1 1 1 1

00

01

11

10

(a)

x 1 x 2 x 3 x 4

1

00 01 11 10

1 1 1

0 0 1 1

1 1 1 1

0 0 0 d

00

01

11

10

(b)

6

x 1 x 2 x 3 x 4

1

00 01 11 10

0 0 1

d 0 0 1

d d 1 0

1 d 0 1

00

01

11

10

(a) f

f= +

x 1 x 2 x 3 x 4

1

00 01 11 10

0 0 1

d 0 0 1

d d 1 0

1 d 0 1

00

01

11

10

(a) f

f=

Problem 4.12

7

x 1 x 2 x 3 x 4

1

00 01 11 10

1 0 0

0 0 0 1

0 1 0 d

1 1 0 d

00

01

11

10

(a) f

x 1 x 2 x 3 x 4

d

00 01 11 10

1 0 0

0 0 d 1

0 0 1 0

1 0 d 1

00

01

11

10

(a) g

f= + g= +

f= g=

f= g=

f= + g= +

Problem 4.12

8

x 1 x 2 x 3 x 4

1

00 01 11 10

1 0 0

0 0 0 1

0 1 0 d

1 1 0 d

00

01

11

10

(a) f

x 1 x 2 x 3 x 4

d

00 01 11 10

1 0 0

0 0 d 1

0 0 1 0

1 0 d 1

00

01

11

10

(a) g

f= + g= +

Problem 4.12

9

x 1 x 2 x 3 x 4

1

00 01 11 10

1 0 0

0 0 0 1

0 1 0 d

1 1 0 d

00

01

11

10

(a) f

x 1 x 2 x 3 x 4

d

00 01 11 10

1 0 0

0 0 d 1

0 0 1 0

1 0 d 1

00

01

11

10

(a) g

f= + g= +

Cost = 5AND+2OR+16ins-AND+8ins-or = 31 < (15+21)=36

Problem 4.12

10

x 1 x 2 x 3 x 4

1

00 01 11 10

1 0 0

0 0 0 1

0 1 0 d

1 1 0 d

00

01

11

10

(a) f

x 1 x 2 x 3 x 4

d

00 01 11 10

1 0 0

0 0 d 1

0 0 1 0

1 0 d 1

00

01

11

10

(a) g

f= Cost = 3AND+1OR+8ins-AND+3ins-or

=15

g= + Cost = 4AND+1OR+12ins-AND+4ins-or

=21

Problem 1 and 3

11

x 1 x 2 x 3 x 4

0

00 01 11 10

1 1 1

0 1 1 0

0 0 0 0

0 1 0 0

00

01

11

10

(a) f

g= F = ()()

x1x2x3

0 0

1 0

d d

1 1

00 01 11 10

0

1

Problem 5

12

0

00 01 11 10

0 1 1

0 1 1 0

1 1 d d

0 0 d d

00

01

11

10

(a) f

f= + f = ()()

wxyz

0

00 01 11 10

0 1 1

0 1 1 0

1 1 d d

0 0 d d

00

01

11

10

wxyz

Problem 5

13

0

00 01 11 10

1 1 0

1 0 0 1

0 0 0 0

0 1 1 0

00

01

11

10

g

g= D g= () ()(B)

ABCD

0

00 01 11 10

1 1 0

1 0 0 1

0 0 0 0

0 1 1 0

00

01

11

10

ABCD

14

x 1 x 2 x 3 x 4

0

00 01 11 10

1 d 0

0 d 1 0

0 d d 0

d 1 1 d

00

01

11

10

(a)

x 1 x 2 x 3 x 4

0

00 01 11 10

0 0 0

1 0 0 1

1 0 0 1

1 1 1 1

00

01

11

10

(b)

f= +

15

x 1 x 2 x 3 x 4

0

00 01 11 10

1 d 0

0 d 1 0

0 d d 0

d 1 1 d

00

01

11

10

(a)

x 1 x 2 x 3 x 4

0

00 01 11 10

0 0 0

1 0 0 1

1 0 0 1

1 1 1 1

00

01

11

10

(b)

f = () () ()