1

2-2.

a) X Y + XY + XY =X + Y

= (XY+ X Y ) + (X Y + XY)

= X(Y + Y) + Y(X + X) +

= X + Y

Verification of DeMorgan’s Theorem

X YZ XYZ XYZ X+Y+Z

0 0 0 0 1 1

001 0 1 1

0 1 0 0 1 1

0 1 1 0 1 1

100 0 1 1

101 0 1 1

110 0 1 1

111 1 0 0

The Second Distributive Law

X Y Z YZ X+YZ X+Y X+Z (X+Y)(X+Z)

0 000000 0

001 0 0 0 1 0

0 100010 0

0 111111 1

100 0 1 1 1 1

101 0 1 1 1 1

110 0 1 1 1 1

111 1 1 1 1 1

X YZX

YYZXZXY+YZ+XZ XY YZ XZXY+YZ+XZ

0 00000 0 000 0

001010 1 001 1

0 10100 1 010 1

0 11100 1 001 1

100001 1 100 1

101010 1 100 1

110001 1 010 1

111000 0 000 0

XYZ X Y Z++=

XYZ+XY+()XZ+()⋅=

XY YZ XZ++ XY YZ XZ++=

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2

Problem Solutions – Chapter 2

b) A B+ B C + AB + B C = 1

= (A B+ AB) + (B C + B C)

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

= B + B

= 1

c) Y + X Z + X Y = X + Y + Z

= Y + X Y + X Z

= (Y + X)(Y + Y) + X Z

= Y + X + X Z

= Y + (X + X)(X + Z)

= X + Y + Z

d) X Y + Y Z + XZ + XY + Y Z = X Y + XZ + Y Z

= X Y + Y Z(X + X) + XZ + XY + Y Z

= X Y + X Y Z + X Y Z + XZ + XY + Y Z

= X Y (1 + Z) + X Y Z +XZ + XY + Y Z

= X Y + XZ(1 + Y) + XY + Y Z

= X Y + XZ + XY (Z + Z)+ Y Z

= X Y + XZ + XY Z +Y Z (1 + X)

= X Y + XZ(1 + Y) + Y Z

= X Y + XZ + Y Z

2-7.

a) X Y + XYZ + XY = X + XYZ = (X + XY)(X + Z)

= (X + X)(X + Y)(X + Z) = (X + Y)(X + Z) = X + YZ

b) X + Y(Z + X Z) = X + YZ + X Y Z = X + (YZ + X)(YZ + YZ) = X + Y(X + YZ)

= X + XY + YZ = (X + X)(X + Y) + YZ = X + Y + YZ = X + Y

c) WX(Z + YZ) + X(W + W YZ) = WXZ + WXYZ + WX + WXYZ

= WX + WXZ + WXZ = WX + WX = X

d)

=

=

= A + C + A(BCD)

= A + C + BCD

= A + C + C(BD)

= A + C + BD

2-9.

a)

b)

c)

d)

AB AB+()CD CD+()AC+

ABCD ABCD ABCD ABCD A C+++++

ACABCD++

FAB+()AB+()=

FVW+()XY+()Z=

FWX+YZ+()YZ+()+[]WX+YZ YZ++[]=

FABCAB+()CABC+()++=

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3

Problem Solutions – Chapter 2

2-10.

a) Sum of Minterms: XYZ + XYZ + XYZ + XYZ

Product of Maxterms: (X + Y + Z)(X + Y + Z)(X + Y + Z)(X + Y + Z)

b) Sum of Minterms: A B C + A B C + A B C + A B C

Product of Maxterms: (A + B + C)(A + B + C)(A + B + C)(A + B + C)

c) Sum of Minterms: W X Y Z + W X Y Z + W X Y Z + W X Y Z + W X Y Z + W X Y Z

+ W X Y Z

Product of Maxterms: (W + X + Y + Z)(W + X + Y + Z)(W + X + Y + Z)

(W + X + Y + Z)(W + X + Y + Z)(W + X + Y + Z)

(W + X + Y + Z)(W + X + Y + Z)(W + X + Y + Z)

2-12.

a) (AB + C)(B + CD) = AB + BC + ABCD = AB + BC s.o.p.

= B(A + C) p.o.s.

b) X + X ((X + Y)(Y + Z)) = (X + X)(X + (X + Y)(Y + Z))

= (X + X + Y)(X + Y + Z) = X + Y + Z s.o.p. and p.o.s.

c) (A + BC + CD)(B + EF) = (A + B + C)(A + B + D)(A + C + D)(B + E)(B + F) p.o.s.

(A + BC + CD)(B + EF) = A(B + EF) + BC(B + EF) + CD(B + EF)

=AB + AEF + BCEF + BCD + CDEF s.o.p.

2-15.

Truth Tables a, b, c

XYZaABCbWX Y Z c

0000 0001 00000

0010 0011 00010

0100 0100 00101

0111 0111 00110

1000 1000 01000

1011 1010 01010

1101 1100 01101

1111 1111 01110

10000

10010

10101

10110

11001

11011

11101

11111

X

Y

Z

A

B

C

a) b) c)

X Z + XY A + CB B + C

A

B

C

1

11

11

1

111 1

1

1

1

1

1

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4

Problem Solutions – Chapter 2

2-18.

2-19.

Using K-maps:

a) Prime = XZ, WX, X Z, W Z b) Prime = CD, AC, B D, ABD, B C c) Prime = AB, AC, AD, BC, BD, CD

Essential = XZ, X Z Essential = AC, B D, ABD Essential = AC, BC, BD

2-22.

Using K-maps:

a) s.o.p. CD + AC + B D b) s.o.p.A C + B D + A D c) s.o.p.B D + ABD + (ABC or ACD)

p.o.s.(C + D)(A + D)(A + B + C) p.o.s.(C + D)(A + D)(A + B + C)p.o.s.(A + B)(B + D)(B + C + D)

2-25.

2-28.

W

X

Y

Z

A

B

C

D

a) b) c)

X

Y

Z

Σm3567,,,() Σm34579131415,,,,, , ,()

Σm0 2 6 7 8 10 13 15,,,,, , ,()

11

1

1

11

1

1

11

1

111

11

1

1

1

1

Primes = AB, AC, BC, A B C

Essential = AB, AC, BC

F = AB + AC + BC

Primes = X Z, XZ, WXY, WXY, W Y Z, WYZ

Essential = X Z

F = X Z + WXY + WXY

Primes = AB, C, AD, BD

Essential = C, AD

F = C + AD + (BD or AB)

W

X

Y

Z

A

B

C

D

a) b) c)

A

B

C

1

1

1

1

11

1

1

1

11

111

1

1

X

XX

X

XX

XX

X

XX

A

B

A

B

C

D

C

D

A

B

CD

A

BDC

A

B

A

B

CD

C

D

A

B

C

D

A

B

D

C

4-input NAND

from 2-input NANDs

and NOTs

a)

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annapurna

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University:
Punjab Engineering College

Subject:
Digital Logic Design

Upload date:
20/07/2012