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**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* *Y Z XYZ XYZ X*+*Y*+*Z
*

0 0 0 0 1 1

0 0 1 0 1 1

0 1 0 0 1 1

0 1 1 0 1 1

1 0 0 0 1 1

1 0 1 0 1 1

1 1 0 0 1 1

1 1 1 1 0 0

The Second Distributive Law

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

0 0 0 0 0 0 0 0

0 0 1 0 0 0 1 0

0 1 0 0 0 1 0 0

0 1 1 1 1 1 1 1

1 0 0 0 1 1 1 1

1 0 1 0 1 1 1 1

1 1 0 0 1 1 1 1

1 1 1 1 1 1 1 1

*X* *Y Z XY YZ XZ XY+YZ+XZ XY YZ XZ XY+YZ+XZ
*

0 0 0 0 0 0 0 0 0 0 0

0 0 1 0 1 0 1 0 0 1 1

0 1 0 1 0 0 1 0 1 0 1

0 1 1 1 0 0 1 0 0 1 1

1 0 0 0 0 1 1 1 0 0 1

1 0 1 0 1 0 1 1 0 0 1

1 1 0 0 0 1 1 0 1 0 1

1 1 1 0 0 0 0 0 0 0 0

*XYZ X Y Z*+ +=

*X YZ*+ *X Y*+( ) *X Z*+( )⋅=

*XY YZ XZ*+ + *XY YZ XZ*+ +=

**1
**

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** 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*+ + + + +

*A C ABCD*+ +

*F A B*+( ) *A B*+( )=
*F V W*+( )*X Y*+( )*Z*=
*F W X*+ *Y Z*+( ) *Y Z*+( )+[ ] *W X*+ *YZ YZ*+ +[ ]=
*F ABC A B*+( )*C A B C*+( )+ +=

** 2
**

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** 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

*X Y Z *a *A B C *b *W X Y Z *c

0 0 0 0 0 0 0 1 0 0 0 0 0

0 0 1 0 0 0 1 1 0 0 0 1 0

0 1 0 0 0 1 0 0 0 0 1 0 1

0 1 1 1 0 1 1 1 0 0 1 1 0

1 0 0 0 1 0 0 0 0 1 0 0 0

1 0 1 1 1 0 1 0 0 1 0 1 0

1 1 0 1 1 1 0 0 0 1 1 0 1

1 1 1 1 1 1 1 1 0 1 1 1 0

1 0 0 0 0

1 0 0 1 0

1 0 1 0 1

1 0 1 1 0

1 1 0 0 1

1 1 0 1 1

1 1 1 0 1

1 1 1 1 1

X

Y

Z

A

B

C

**a) b) c)
**

*X Z + XY A + CB B + C
*

A

B

C

1

11

1 1

1

1 1 1 1

1

1

1

1

1

** 3
**

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** 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

Σ*m *3 5 6 7, , ,( ) Σ*m *3 4 5 7 9 13 14 15, , , , , , ,( ) Σ*m *0 2 6 7 8 10 13 15, , , , , , ,( )

11

1

1

11

1

1

1 1

1

1 11

1 1

1

11

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

1 1

1

1

1

1 1

1 1 1

1

1

X

X X

X

X X

X X

X

X X

AB

AB

C

DC

D AB

C D AB DC

A B

A B

C D C

D

A B

C D

A B

D C

4-input NAND from 2-input NANDs and NOTs

**a)**

** 4
**

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** Problem Solutions – Chapter 2**

**2-30.
**

**2-34.
**

**2-37.
**

**2-39.
**

4 × 0.5 = 2 ns

**2-44.
**

P-Logic N-Logic

X Y NAND NOR X Y NAND NOR X Y NAND NOR

L L H H 0 0 1 1 1 1 0 0

L H H L 0 1 1 0 1 0 0 1

H L H L 1 0 1 0 0 1 0 1

H H L L 1 1 0 0 0 0 1 1

A B

C

D

A B

A B

C D

C D

**b)
**

A

B

C

D

**a) b)
**

W

X

Y

Z
*F A B C*+ +( ) *A C*+( ) *A D*+( )= *F W X*+( ) *W X*+( ) *Y Z*+( ) *Y Z*+( )=

1 1

1 1 1 1

1

11

A C A D

1

1

B A

C 1 Y Z Y Z A

W X W X

*X Y*⊕ *XY XY*+=

Dual (X *Y *)⊕ Dual *XY XY*+( )=

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

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

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

16 inputs 16 inputs

6 inputs

** 5
**

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