Solution Manual For Digital Design, 5th Edition

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2
CHAPTER 1
1.1 Base-10: 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32
Octal: 20 21 22 23 24 25 26 27 30 31 32 33 34 35 36 37 40
Hex: 10 11 12 13 14 15 16 17 18 19 1A 1B 1C 1D 1E 1F 20
Base-12 14 15 16 17 18 19 1A 1B 20 21 22 23 24 25 26 27 28
1.2 (a) 32,768 (b) 67,108,864 (c) 6,871,947,674
1.3 (4310)5 = 4 * 53 + 3 * 52 + 1 * 51 = 58010
(198)12 = 1 * 122 + 9 * 121 + 8 * 120 = 26010
(435)8 = 4 * 82 + 3 * 81 + 5 * 80 = 28510
(345)6 = 3 * 62 + 4 * 61 + 5 * 60 = 13710
1.4 16-bit binary: 1111_1111_1111_1111
Decimal equivalent: 216 -1 = 65,53510
Hexadecimal equivalent: FFFF16
1.5 Let b = base
(a) 14/2 = (b + 4)/2 = 5, so b = 6
(b) 54/4 = (5*b + 4)/4 = b + 3, so 5 * b = 52 – 4, and b = 8
(c) (2 *b + 4) + (b + 7) = 4b, so b = 11
1.6 (x – 3)(x – 6) = x2 –(6 + 3)x + 6*3 = x2 -11x + 22
Therefore: 6 + 3 = b + 1m, so b = 8
Also, 6*3 = (18)10 = (22)8
1.7 64CD16 = 0110_0100_1100_11012 = 110_010_011_001 _101 = (62315 )8
1.8 (a) Results of repeated division by 2 (quotients are followed by remainders):
43110 = 215(1); 107(1); 53(1); 26(1); 13(0); 6(1) 3(0) 1(1)
Answer: 1111_10102 = FA16
(b) Results of repeated division by 16:
43110 = 26(15); 1(10) (Faster)
Answer: FA = 1111_1010
1.9 (a) 10110.01012 = 16 + 4 + 2 + .25 + .0625 = 22.3125
(b) 16.516 = 16 + 6 + 5*(.0615) = 22.3125
(c) 26.248 = 2 * 8 + 6 + 2/8 + 4/64 = 22.3125
(d) DADA.B16 = 14*163 + 10*162 + 14*16 + 10 + 11/16 = 60,138.6875

3
(e) 1010.11012 = 8 + 2 + .5 + .25 + .0625 = 10.8125
1.10 (a) 1.100102 = 0001.10012 = 1.916 = 1 + 9/16 = 1.56310
(b) 110.0102 = 0110.01002 = 6.416 = 6 + 4/16 = 6.2510
Reason: 110.0102 is the same as 1.100102 shifted to the left by two places.
1011.11
1.11 101 | 111011.0000
101
01001
101
1001
101
1000
101
0110
The quotient is carried to two decimal places, giving 1011.11
Checking: 1110112 / 1012 = 5910 / 510 ≅ 1011.112 = 58.7510
1.12 (a) 10000 and 110111
1011 1011
+101 x101
10000 = 1610 1011
1011
110111 = 5510
(b) 62h and 958h
2Eh 0010_1110 2Eh
+34 h 0011_0100 x34h
62h 0110_0010 = 9810 B38
82A
9 5 8h = 239210
1.13 (a) Convert 27.315 to binary:
Integer Remainder Coefficient
Quotient
27/2 = 13 + ½ a0 = 1
13/2 6 + ½ a1 = 1
6/2 3 + 0 a2 = 0
3/2 1 + ½ a3 = 1
½ 0 + ½ a4 = 1

4
2710 = 110112
Integer Fraction Coefficient
.315 x 2 = 0 + .630 a-1 = 0
.630 x 2 = 1 + .26 a-2 = 1
.26 x 2 = 0 + .52 a-3 = 0
.52 x 2 = 1 + .04 a-4 = 1
.31510 ≅ .01012 = .25 + .0625 = .3125
27.315 ≅ 11011.01012
(b) 2/3 ≅ .6666666667
Integer Fraction Coefficient
.6666_6666_67 x 2 = 1 + .3333_3333_34 a-1 = 1
.3333333334 x 2 = 0 + .6666666668 a-2 = 0
.6666666668 x 2 = 1 + .3333333336 a-3 = 1
.3333333336 x 2 = 0 + .6666666672 a-4 = 0
.6666666672 x 2 = 1 + .3333333344 a-5 = 1
.3333333344 x 2 = 0 + .6666666688 a-6 = 0
.6666666688 x 2 = 1 + .3333333376 a-7 = 1
.3333333376 x 2 = 0 + .6666666752 a-8 = 0
.666666666710 ≅ .101010102 = .5 + .125 + .0313 + ..0078 = .664110
.101010102 = .1010_10102 = .AA16 = 10/16 + 10/256 = .664110 (Same as (b)).
1.14 (a) 0001_0000 (b) 0000_0000 (c) 1101_1010
1s comp: 1110_1111 1s comp: 1111_1111 1s comp: 0010_0101
2s comp: 1111_0000 2s comp: 0000_0000 2s comp: 0010_0110
(d) 1010_1010 (e) 1000_0101 (f) 1111_1111
1s comp: 0101_0101 1s comp: 0111_1010 1s comp: 0000_0000
2s comp: 0101_0110 2s comp: 0111_1011 2s comp: 0000_0001
`
1.15 (a) 25,478,036 (b) 63,325,600
9s comp: 74,521,963 9s comp: 36,674,399
10s comp: 74,521,964 10s comp: 36,674,400
(c) 25,000,000 (d) 00000000
9s comp: 74,999,999 9s comp: 99999999
10s comp: 75,000,000 10s comp: 100000000
1.16 C3DF C3DF: 1100_0011_1101_1111
15s comp: 3C20 1s comp: 0011_1100_0010_0000
16s comp: 3C21 2s comp: 0011_1100_0010_0001 = 3C21
1.17 (a) 2,579 → 02,579 →97,420 (9s comp) → 97,421 (10s comp)
4637 – 2,579 = 2,579 + 97,421 = 205810
(b) 1800 → 01800 → 98199 (9s comp) → 98200 (10 comp)
125 – 1800 = 00125 + 98200 = 98325 (negative)
Magnitude: 1675
Result: 125 – 1800 = 1675

5
(c) 4,361 → 04361 → 95638 (9s comp) → 95639 (10s comp)
2043 – 4361 = 02043 + 95639 = 97682 (Negative)
Magnitude: 2318
Result: 2043 – 6152 = -2318
(d) 745 → 00745 → 99254 (9s comp) → 99255 (10s comp)
1631 -745 = 01631 + 99255 = 0886 (Positive)
Result: 1631 – 745 = 886
1.18 Note: Consider sign extension with 2s complement arithmetic.
(a) 0_10010 (b) 0_100110
1s comp: 1_01101 1s comp: 1_011001 with sign extension
2s comp: 1_01110 2s comp: 1_011010
0_10011 0_100010
Diff: 0_00001 (Positive) 1_111100 sign bit indicates that the result is negative
Check:19-18 = +1 0_000011 1s complement
0_000100 2s complement
000100 magnitude
Result: -4
Check: 34 -38 = -4
(c) 0_110101 (d) 0_010101
1s comp: 1_001010 1s comp: 1_101010 with sign extension
2s comp: 1_001011 2s comp: 1_101011
0_001001 0_101000
Diff: 1_010100 (negative) 0_010011 sign bit indicates that the result is positive
0_101011 (1s comp) Result: 1910
0_101100 (2s complement) Check: 40 – 21 = 1910
101100 (magnitude)
-4410 (result)
1.19 +9286 → 009286; +801 → 000801; -9286 → 990714; -801 → 999199
(a) (+9286) + (_801) = 009286 + 000801 = 010087
(b) (+9286) + (-801) = 009286 + 999199 = 008485
(c) (-9286) + (+801) = 990714 + 000801 = 991515
(d) (-9286) + (-801) = 990714 + 999199 = 989913
1.20 +49 → 0_110001 (Needs leading zero extension to indicate + value);
+29 → 0_011101 (Leading 0 indicates + value)
-49 → 1_001110 + 0_000001→ 1_001111
-29 → 1_100011 (sign extension indicates negative value)
(a) (+29) + (-49) = 0_011101 + 1_001111 = 1_101100 (1 indicates negative value.)
Magnitude = 0_010011 + 0_000001 = 0_010100 = 20; Result (+29) + (-49) = -20
(b) (-29) + (+49) = 1_100011 + 0_110001 = 0_010100 (0 indicates positive value)
(-29) + (+49) = +20

6
(c) Must increase word size by 1 (sign extension) to accomodate overflow of values:
(-29) + (-49) = 11_100011 + 11_001111 = 10_110010 (1 indicates negative result)
Magnitude: 01_001110 = 7810
Result: (-29) + (-49) = -7810
1.21 +9742 → 009742 → 990257 (9's comp) → 990258 (10s) comp
+641 → 000641 → 999358 (9's comp) → 999359 (10s) comp
(a) (+9742) + (+641) → 010383
(b) (+9742) + (-641) →009742 + 999359 = 009102
Result: (+9742) + (-641) = 9102
(c) -9742) + (+641) = 990258 + 000641 = 990899 (negative)
Magnitude: 009101
Result: (-9742) + (641) = -9101
(d) (-9742) + (-641) = 990258 + 999359 = 989617 (Negative)
Magnitude: 10383
Result: (-9742) + (-641) = -10383
1.22 6,514
BCD: 0110_0101_0001_0100
ASCII: 0_011_0110_0_011_0101_1_011_0001_1_011_0100
ASCII: 0011_0110_0011_0101_1011_0001_1011_0100
1.23
0111 1001 0001 ( 791)
0110 0101 1000 (+658)
1101 1110 1001
0110 0110
0001 0011 0100
0001 0001
0001 0100 0100 1001 (1,449)
1.24 (a) (b)
6 3 1 1 Decimal
0 0 0 0 0
0 0 0 1 1
0 0 1 0 2
0 1 0 0 3
0 1 1 0 4 (or 0101)
0 1 1 1 5
1 0 0 0 6
1 0 1 0 7 (or 1001)
1 0 1 1 8
1 1 0 0 9
6 4 2 1 Decimal
0 0 0 0 0
0 0 0 1 1
0 0 1 0 2
0 0 1 1 3
0 1 0 0 4
0 1 0 1 5
1 0 0 0 6 (or 0110)
1 0 0 1 7
1 0 1 0 8
1 0 1 1 9
1.25 (a) 6,24810 BCD: 0110_0010_0100_1000
(b) Excess-3: 1001_0101_0111_1011
(c) 2421: 0110_0010_0100_1110
(d) 6311: 1000_0010_0110_1011

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8
1.27 For a deck with 52 cards, we need 6 bits (25 = 32 < 52 < 64 = 26). Let the msb's select the suit (e.g.,
diamonds, hearts, clubs, spades are encoded respectively as 00, 01, 10, and 11. The remaining four bits
select the "number" of the card. Example: 0001 (ace) through 1011 (9), plus 101 through 1100 (jack,
queen, king). This a jack of spades might be coded as 11_1010. (Note: only 52 out of 64 patterns are
used.)
1.28 G (dot) (space) B o o l e
11000111_11101111_01101000_01101110_00100000_11000100_11101111_11100101
1.29 Steve Jobs
1.30 73 F4 E5 76 E5 4A EF 62 73
73: 0_111_0011 s
F4: 1_111_0100 t
E5: 1_110_0101 e
76: 0_111_0110 v
E5: 1_110_0101 e
4A: 0_100_1010 j
EF: 1_110_1111 o
62: 0_110_0010 b
73: 0_111_0011 s
1.31 62 + 32 = 94 printing characters
1.32 bit 6 from the right
1.33 (a) 897 (b) 564 (c) 871 (d) 2,199
1.34 ASCII for decimal digits with even parity:
(0): 00110000 (1): 10110001 (2): 10110010 (3): 00110011
(4): 10110100 (5): 00110101 (6): 00110110 (7): 10110111
(8): 10111000 (9): 00111001
1.35 (a)
a b c
f
g
a
b
c
f
g
1.36
a b
f
g
a
b
f
g

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9
CHAPTER 2
2.1 (a)
x y z
0 0 0
0 0 1
0 1 0
0 1 1
1 0 0
1 0 1
1 1 0
1 1 1
x + y + z
0
1
1
1
1
1
1
1
(x + y + z)'
1
0
0
0
0
0
0
0
x'
1
1
1
1
0
0
0
0
y'
1
1
0
0
1
1
0
0
z'
1
0
1
0
1
0
1
0
x' y' z'
1
0
0
0
0
0
0
0
x y z
0 0 0
0 0 1
0 1 0
0 1 1
1 0 0
1 0 1
1 1 0
1 1 1
(xyz)
0
0
0
0
0
0
0
1
(xyz)'
1
1
1
1
1
1
1
0
x'
1
1
1
1
0
0
0
0
y'
1
1
0
0
1
1
0
0
z'
1
0
1
0
1
0
1
0
x' + y' + z'
1
1
1
1
1
1
1
0
(b) (c)
x y z
0 0 0
0 0 1
0 1 0
0 1 1
1 0 0
1 0 1
1 1 0
1 1 1
x + yz
0
0
0
1
1
1
1
1
(x + y)
0
0
1
1
1
1
1
1
(x + z)
0
1
0
1
1
1
1
1
(x + y)(x + z)
0
0
0
1
1
1
1
1
x y z
0 0 0
0 0 1
0 1 0
0 1 1
1 0 0
1 0 1
1 1 0
1 1 1
x(y + z)
0
0
0
0
0
1
1
1
xy
0
0
0
0
0
0
1
1
xz
0
0
0
0
0
1
0
1
xy + xz
0
0
0
0
0
1
1
1
(c) (d)
x y z
0 0 0
0 0 1
0 1 0
0 1 1
1 0 0
1 0 1
1 1 0
1 1 1
x
0
0
0
0
1
1
1
1
y + z
0
1
1
1
0
1
1
1
x + (y + z)
0
1
1
1
1
1
1
1
(x + y)
0
0
1
1
1
1
1
1
x y z
0 0 0
0 0 1
0 1 0
0 1 1
1 0 0
1 0 1
1 1 0
1 1 1
yz
0
0
0
1
0
0
0
1
x(yz)
0
0
0
0
0
0
0
1
xy
0
0
0
0
0
0
1
1
(xy)z
0
0
0
0
0
0
0
1
(x + y) + z
0
1
1
1
1
1
1
1
2.2 (a) xy + xy' = x(y + y') = x
(b) (x + y)(x + y') = x + yy' = x(x +y') + y(x + y') = xx + xy' + xy + yy' = x
(c) xyz + x'y + xyz' = xy(z + z') + x'y = xy + x'y = y
(d) (A + B)'(A' + B')' = (A'B')(A B) = (A'B')(BA) = A'(B'B)A = 0
(e) (a + b + c')(a'b' + c) = aa'b' + ac + ba'b' + bc + c'a'b' + c'c = ac + bc +a'b'c'
(f) a'bc + abc' + abc + a'bc' = a'b(c + c') + ab(c + c') = a'b + ab = (a' + a)b = b
2.3 (a) ABC + A'B + ABC' = AB + A'B = B

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10
(b) x'yz + xz = (x'y + x)z = z(x + x')(x + y) = z(x + y)
(c) (x + y)'(x' + y') = x'y'(x' + y') = x'y'
(d) xy + x(wz + wz') = x(y +wz + wz') = x(w + y)
(e) (BC' + A'D)(AB' + CD') = BC'AB' + BC'CD' + A'DAB' + A'DCD' = 0
(f) (a' + c')(a + b' + c') = a'a + a'b' + a'c' + c'a + c'b' + c'c' = a'b' + a'c' + ac' + b'c' = c' + b'(a' + c')
= c' + b'c' + a'b' = c' + a'b'
2.4 (a) A'C' + ABC + AC' = C' + ABC = (C + C')(C' + AB) = AB + C'
(b) (x'y' + z)' + z + xy + wz = (x'y')'z' + z + xy + wz =[ (x + y)z' + z] + xy + wz =
= (z + z')(z + x + y) + xy + wz = z + wz + x + xy + y = z(1 + w) + x(1 + y) + y = x + y + z
(c) A'B(D' + C'D) + B(A + A'CD) = B(A'D' + A'C'D + A + A'CD)
= B(A'D' + A + A'D(C + C') = B(A + A'(D' + D)) = B(A + A') = B
(d) (A' + C)(A' + C')(A + B + C'D) = (A' + CC')(A + B + C'D) = A'(A + B + C'D)
= AA' + A'B + A'C'D = A'(B + C'D)
(e) ABC'D + A'BD + ABCD = AB(C + C')D + A'BD = ABD + A'BD = BD
2.5 (a)
x y
F
F simplified
(b)
x y
F
F simplified
(c)

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15
(e)
A B C D
F
F simplified
2.8 F' = (wx + yz)' = (wx)'(yz)' = (w' + x')(y' + z')
FF' = wx(w' + x')(y' + z') + yz(w' + x')(y' + z') = 0
F + F' = wx + yz + (wx + yz)' = A + A' = 1 with A = wx + yz
2.9 (a) F' = (xy' + x'y)' = (xy')'(x'y)' = (x' + y)(x + y') = xy + x'y'
(b) F' = [(a + c) (a + b')(a' + b + c')]' = (a + c)' + (a + b')' + (a' + b + c')'
=a'c' + a'b + ab'c
(c) F' = [z + z'(v'w + xy)]' = z'[z'(v'w + xy)]' = z'[z'v'w + xyz']'
= z'[(z'v'w)'(xyz')'] = z'[(z + v + w') +( x' + y' + z)]
= z'z + z'v + z'w' + z'x' + z'y' +z' z = z'(v + w' + x' + y')
2.10 (a) F1 + F2 = Σ m1i + Σm2i = Σ (m1i + m2i)
(b) F1 F2 = Σ mi Σmj where mi mj = 0 if i
≠ j and mi mj = 1 if i = j
2.11 (a) F(x, y, z) = Σ(1, 4, 5, 6, 7)
(b) F(a, b, c) = Σ(0, 2, 3, 7)
x y z
0 0 0
0 0 1
0 1 0
0 1 1
1 0 0
1 0 1
1 1 0
1 1 1
F
0
1
0
0
1
1
1
1
F = xy + xy' + y'z
a b c
0 0 0
0 0 1
0 1 0
0 1 1
1 0 0
1 0 1
1 1 0
1 1 1
F
1
0
1
1
0
0
0
1
F = bc + a'c'
2.12 A = 1011_0001
B = 1010_1100

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Solution Manual For Digital Design, 5th Edition

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