Fundamentals of Logic Design 7th Edition Roth SOLUTIONS MANUAL
Unit 2 Problem Solutions

2.1

See FLD p. 731 for solution.

2.2 (a)
In both cases, if X = 0, the transmission is 0, and if  X = 1, the transmission is 1.
2.2 (b)
In both cases, if X = 0, the transmission is YZ, and if X = 1, the transmission is 1.

X X  X

X

2.3

X

Answer is in FLD p. 731

Y Z Y

2.4 (a)

2.5 (a)

2.6 (a)

F =
[(
A·
1)
+
(
A·
1)]
+ E + BCD = A + E + BCD
(
A + B
) (
C + B
) (
D' + B
) (
ACD' + E
)
=
(
AC + B
) (
D' + B
) (
ACD' + E
) By Dist. Law
=
(
ACD' + B
) (
ACD' + E
) By Dist. Law
= ACD' + BE
By Dist. Law
AB + C'D' =
(
AB + C'
) (
AB + D'
)
=
(
A + C'
) (
B + C'
) (
A + D'
) (
B + D'
)
2.4 (b)

2.5 (b)

2.6 (b)

Y =
(
AB' +
(
AB + B
))
B + A =
(
AB' + B
)
B + A

=
(
A + B
)
B + A = AB + B + A = A + B
(
A' + B + C'
) (
A' + C' + D
) (
B' + D'
)
=
(
A' + C' + BD
) (
B' + D'
) {By Distributive Law with
X = A' + C'
}
= A'B' + B'C' + B'BD + A'D' + C'D' + BDD'

= A'B' + A'D' + C'B' + C'D'

WX + WY'X + ZYX = X
(
W + WY' + ZY
)
= X
(
W + ZY
) {By Absorption}
= X
(
W +Z
) (
W + Y
)
2.6 (c)

2.6 (e)

2.7 (a)

A'BC + EF + DEF' = A'BC + E
(
F +DF'
)
= A'BC + E
(
F +D
)
=
(
A'BC + E
) (
A'BC + F + D
)
=
(
A' + E
) (
B + E
) (
C + E
) (
A' + F + D
) (
B + F + D
) (
C + F + D
)
ACD' + C'D' + A'C = D'
(
AC + C'
)
+ A'C

= D'
(
A + C'
)
+ A'C
By Elimination Theorem
=
(
D' + A'C
) (
A + C' + A'C
)
=
(
D' + A'
) (
D' + C
) (
A + C' + A'
) By Distributive Law and Elimination Theorem
=
(
A' + D'
) (
C + D'
) (
A + B + C + D
) (
A + B + C + E
) (
A + B + C + F
)
= A + B + C + DEF
Apply second Distributive Law twice
D

2.6 (d)

2.6 (f)

2.7 (b)

XYZ + W'Z + XQ'Z = Z
(
XY + W' + XQ'
)
= Z
[
W' + X
(
Y + Q'
)]
= Z
(
W' + X
) (
W' + Y + Q'
) By Distributive Law
A + BC + DE

=
(
A + BC + D
)(
A + BC + E
)
=
(
A + B + D
)(
A + C + D
)(
A + B + E
)(
A + C + E
)
WXYZ + VXYZ + UXYZ = XYZ
(
W + V + U
) By first Distributive Law
E

F  A B W

C  Z
2.8 (a)

2.8 (c)

2.9 (a)
[(
AB
)
' + C'D
]
' =  AB
(
C'D
)
' = AB
(
C + D'
)
= ABC + ABD'
((
A + B'
)
)
'
(
A + B
) (
C + A
)

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

= A'BC'

F =
[(
A + B
)
' +
(
A +
(
A +  B
)
)
] (
A +
(
A + B
)
)

=
(
A +
(
A +  B
)
)
By Elimination Theorem with
X=
(
A+
(
A+B
)
)
' = A'
(
A + B
)
= A'B

2.8 (b)

2.9 (b)
[
A + B
(
C' +  D
)]
' =  A'
(
B
(
C' +  D
))

=  A'
(
B' +
(
C' +  D
)
)
=  A'
(
B' + CD'
)
=  A'B' +  A'CD'
G = {[(R + S + T)' PT(R + S)']' T}'
=
(
R +  +
)
P
(
R +
)
+

= T' +
(
R'S 'T'
)
P
(
R'S'
)
T = T' + PR'S'T'T = T'

2.10 (a)
X

X

2.10 (b)
X

Y X

2.10 (c)

2.10 (e)

X Y'

X

X
'

X

X

Z Y
Z

X

Y'

2.10 (d)
A

A  B

A

C
'

B

B

2.10 (f)
X  X

Y Z Y Z
2.11 (a)

2.11 (c)

2.11 (e)
(
A' + B' + C
)(
A' + B' + C
)
'
= 0 By Complementarity Law
AB +
(
C' + D
)(
AB
)
' = AB + C' + D
By Elimination Theorem [
AB' +
(
C + D
)
' +E'F
](
C + D
) =
AB'
(
C + D
)
+ E'F
(
C + D
) Distributive Law
2.11 (b)

2.11 (d)

2.11 (f)

AB
(
C' + D
)
+ B
(
C' + D
)
= B
(
C' + D
) By Absorption (
A'BF + CD'
)(
A'BF + CEG
)
= A'BF + CD'EG
By Distributive Law
A'
(
B + C
)(
D'E + F
)
' +
(
D'E + F
)
= A'
(
B + C
)
+ D'E + F
By Elimination
2.12 (a)

2.12 (c)

2.12 (e)

2.13 (a)

2.13 (c)

2.14 (a)

2.15 (a)

2.16 (a)

2.17 (a)

2.17 (c)
(
X + Y'Z
)
+
(
X + Y'Z
)
'
= 1 By Complementarity Law (
V'W + UX
)
'
(
UX + Y + Z + V'W
)
=
(
V'W + UX
)
(
Y + Z
) By Elimination Theorem (
W' + X
)(
Y + Z'
)
+
(
W' + X
)
(
Y + Z'
)
=
(
Y + Z'
) By Uniting Theorem
F
1
= A'A + B +
(
B + B
)
=
0
+ B + B = B

F
3
=
[(
AB + C
)
'D
][(
AB + C
)
+ D
]
=
(
AB + C
)
'D
(
AB + C
)
+
(
AB + C
)
' D

=
(
AB + C
)
' D
By Absorption
ACF
(
B + E + D
)
f ' =
{[
A +
(
BCD
)
][(
)
' + B
(
C' + A
)]}

=
[
A +
(
BCD
)
]
' +
[(
)
' + B
(
C' + A
)]

= A'
(
BCD
)
'' +
(
)
''
[
B
(
C' + A
)]

[
B' +
(
C' + A
)
]
[
B' + C''A'
]
[
B' + CA'
]
f
D
=
[
A +
(
BCD
)
][(
)
' + B
(
C' + A
)]
D

=
[
A
(
B + C + D
)
]
+
[(
A + D
)
(
B + C'A
)]
f =
[(
A' + B
)
]
+
[
A
(
B + C'
)]
= A'C + B'C + AB + AC'

= A'C + B'C + AB + AC' + BC

= A'C + C + AB + AC' = C + AB + A = C + A

f =
(
A' + B' + A
)(
A + C
)(
A' + B' + C' + B
) (
B + C + C'
)
=
(
A + C
)
2.12 (b)

2.12 (d)

2.12 (f)

2.13 (b)

2.13 (d)

2.14 (b)

2.15(b)

2.16 (b)

2.17 (b)

2.18 (a)
[
W + X'
(
Y +Z
)][
W' + X'
(
Y + Z
)]
= X'
(
Y + Z
) By Uniting Theorem (
UV' + W'X
)(
UV' + W'X + Y'Z
)
= UV' + W'X
By Absorption Theorem (
V' + U + W
)[(
W + X
)
+ Y + UZ'
]
+
[(
W + X
)
+ UZ' + Y
]
=
(
W + X
)
+ UZ' + Y
By Absorption
F
2
= A'A' + AB' = A' + AB' = A' + B'

Z =
[(
A + B
)
]
' +
(
A + B
)
CD =
[(
A + B
)
]
' + D
By Elimination with
X =
[(
A + B
)
]

= A'B' + C' + D' W + Y + Z + VUX

f ' =
[
AB'C +
(
A' + B + D
)(
ABD' + B'
)]

=
(
AB'C
)
[(
A' + B + D
)(
ABD' + B'
]

=
(
A' + B'' + C'
)[(
A' + B + D
)
' +
(
ABD'
)
'B''
]
=
(
A' + B + C'
)[
A''B'D' +
(
A' + B' + D''
)
B
]
=
(
A' + B + C'
)[
AB'D' +
(
A' + B' + D
)
B
]
f
D
=
[
AB'C +
(
A' + B + D
)(
ABD' + B'
)]
D
=
(
A + B' + C
)[
A'BD +
(
A + B + D'
)
B'
)
f = A'C + B'C + AB + AC' = A + C
product term, sum-of-products, product-of-sums)

= D
(
A' + B' + AC'
)(
C + AC'
) 0 0 0 0 0 0
= D
(
A' + B' + C'
)(
C + A
) 0 0 1 1 1 x  0 1 0 1 0 1
D  B'
1 0 0 0 0 0
C'
1 1 0 0 0 0  1 1 1 1 1 x
2.18 (b)

2.18 (d)
sum-of-products sum term, sum-of-products, product-of-sums
2.18 (c)

2.18 (e)
none apply  product-of-sums
2.19

Z
+

+

X

Z

X Y
+

2.20 (a)

2.20 (b)

F = D
[(
A' + B'
)
C + AC'
]
F = D
[(
A' + B'
)
C + AC'
]
= A' CD + B' CD +AC' D

A ' C D  B ' C D

A C' D

2.20 (c)

F = D
[(
A' + B'
)
C + AC'
]
2.21
A B C H F G

A'

2.22 (a)

2.22 (b)

2.22 (c)

2.23 (a)

A'B' + A'CD + A'DE'

= A'
(
B' + CD + DE'
)
= A'
[
B' + D
(
C + E'
)]
= A'
(
B' + D
)(
B' + C + E'
)
H'I' + JK

=
(
H'I' + J
)(
H'I' + K
)
=
(
H' + J
)(
I' + J
)(
H' + K
)(
I' + K
)
A'BC + AB'C + CD'

= C
(
A'B + AB' + D'
)
= C
[(
A + B
)(
A' + B'
)
+ D'
]
= C
(
A + B + D'
)(
A' + B' + D'
)
W + U'YV =
(
W + U'
)(
W + Y
)(
W + V
)
2.22 (d)

2.22 (e)

2.22 (f)

2.23 (b)

A'B' +
(
CD' + E
)
= A'B' +
(
C + E
)(
D' + E
)
=
(
A'B' + C + E
)(
A'B' + D' + E
)
=
(
A' + C + E
)(
B' + C + E
) (
A' + D' + E
)(
B' + D' + E
)
A'B'C + B'CD' + EF' = A'B'C + B'CD' + EF'

= B'C
(
A' + D'
)
+ EF'

=
(
B'C + EF'
)(
A' + D' + EF'
)
=
(
B' + E
)(
B' + F'
)(
C + E
)(
C + F'
) (
A' + D' + E
)(
A' + D' + F'
)
WX'Y + W'X' + W'Y' = X'
(
WY + W'
)
+ W'Y'

= X'
(
W' + Y
)
+ W'Y'

=
(
X' + W'
)(
X' + Y'
)(
W' + Y + W'
)(
W' + Y + Y'
)
=
(
X' + W'
)(
X' + Y'
)(
W' + Y
)
TW + UY' + V

=
(
T+U+Z
)(
T+Y'+V
)(
W+U+V
)(
W+Y'+V
)

2.23 (c)

A'B'C + B'CD' + B'E' = B'
(
A'C + CD' + E'
)
= B'
[
E' + C
(
A' + D'
)]
= B'
(
E' + C
)(
E' + A' + D'
)
2.23 (d)

ABC + ADE' + ABF' = A
(
BC + DE' + BF'
)
= A
[
DE' + B
(
C + F'
)]
= A
(
DE' + B
)(
DE' + C + F'
)
= A
(
B + D
)(
B + E'
)(
C + F' + D
)(
C + F' + E'
)