Solution Manual For Digital Design, 5th Edition

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2CHAPTER 11.1Base-10:1617181920212223242526272829303132Octal:2021222324252627303132333435363740Hex:101112131415161718191A1B1C1D1E1F20Base-1214 151617 18191A1B2021222324252627281.2(a)32,768(b)67,108,864(c)6,871,947,6741.3(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= 137101.416-bit binary:1111_1111_1111_1111Decimalequivalent:216-1 =65,53510Hexadecimalequivalent:FFFF161.5Let 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 = 111.6(x–3)(x–6) = x2–(6 + 3)x + 6*3 = x2-11x + 22Therefore:6 + 3 = b + 1m,so b = 8Also, 6*3 = (18)10= (22)81.764CD16= 0110_0100_1100_11012=110_010_011_001_101=(62315)81.8(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_10101.9(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

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3(e)1010.11012=8 + 2 + .5 + .25+ .0625 = 10.81251.10(a)1.100102=0001.10012= 1.916= 1 + 9/16 = 1.56310(b)110.0102= 0110.01002= 6.416= 6 + 4/16 = 6.2510Reason:110.0102is the same as 1.100102shifted to the left by two places.1011.111.11101| 111011.000010101001101100110110001010110The quotient is carried to two decimal places, giving 1011.11Checking: 1110112/ 1012= 5910/ 510≅1011.112= 58.75101.12(a)10000 and 11011110111011+101x10110000= 161010111011110111= 5510(b)62hand 958h2Eh0010_11102Eh+34h0011_0100x34h62h0110_0010= 9810B3882A9 5 8h= 2392101.13(a)Convert 27.315 to binary:IntegerRemainderCoefficientQuotient27/2 =13+½a0= 113/26+½a1= 16/23+0a2= 03/21+½a3= 1½0+½a4= 1

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42710= 110112IntegerFractionCoefficient.315 x 2=0+.630a-1= 0.630 x 2=1+.26a-2= 1.26 x 2=0+.52a-3= 0.52 x 2=1+.04a-4= 1.31510≅.01012= .25 + .0625 = .312527.315≅11011.01012(b)2/3≅.6666666667IntegerFractionCoefficient.6666_6666_67x 2=1+.3333_3333_34a-1=1.3333333334x 2=0+.6666666668a-2=0.6666666668x 2=1+.3333333336a-3=1.3333333336x 2=0+.6666666672a-4= 0.6666666672 x 2=1+.3333333344a-5= 1.3333333344 x 2=0+.6666666688a-6= 0.6666666688 x 2=1+.3333333376a-7= 1.3333333376 x 2=0+.6666666752a-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_10101s comp:1110_11111s comp:1111_11111s comp:0010_01012s comp:1111_00002s comp:0000_00002s comp:0010_0110(d)1010_1010(e)1000_0101(f)1111_11111s comp:0101_01011s comp:0111_10101s comp:0000_00002s comp:0101_01102s comp:0111_10112s comp:0000_0001`1.15(a)25,478,036(b)63,325,6009s comp:74,521,9639s comp:36,674,39910s comp:74,521,96410s comp:36,674,400(c)25,000,000(d)000000009s comp:74,999,9999s comp:9999999910s comp:75,000,00010s comp:1000000001.16C3DFC3DF:1100_0011_1101_111115s comp:3C201s comp:0011_1100_0010_000016s comp:3C212s comp:0011_1100_0010_0001=3C211.17(a)2,579→02,579→97,420(9s comp)→97,421(10scomp)4637–2,579=2,579 + 97,421 = 205810(b)1800→01800→98199 (9s comp)→98200 (10 comp)125–1800 = 00125 + 98200 = 98325 (negative)Magnitude: 1675Result: 125–1800 = 1675

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5(c)4,361→04361→95638(9s comp)→95639(10s comp)2043–4361= 02043 + 95639=97682(Negative)Magnitude: 2318Result: 2043–6152 =-2318(d) 745→00745→99254 (9s comp)→99255 (10s comp)1631-745 = 01631 + 99255 = 0886 (Positive)Result: 1631–745 = 8861.18Note: Consider sign extension with 2s complement arithmetic.(a)0_10010(b)0_1001101s comp:1_011011s comp:1_011001with sign extension2s comp:1_011102s comp:1_0110100_100110_100010Diff:0_00001(Positive)1_111100sign bit indicates that the result is negativeCheck:19-18 = +10_0000111s complement0_000100 2s complement000100 magnitudeResult:-4Check:34-38 =-4(c)0_110101(d)0_0101011s comp:1_0010101s comp:1_101010with sign extension2s comp:1_0010112s comp:1_1010110_0010010_101000Diff:1_010100(negative)0_010011sign bit indicates that the result ispositive0_101011(1s comp)Result: 19100_101100(2s complement)Check: 40–21 =1910101100(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 =9899131.20+49→0_110001(Needs leading zeroextension toindicate + 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

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6(c)Must increase word size by 1 (sign extension) to accomodateoverflow of values:(-29) + (-49) =11_100011+11_001111 =10_110010 (1 indicates negative result)Magnitude:01_001110= 7810Result: (-29) + (-49) =-78101.21+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= 009102Result: (+9742) + (-641) = 9102(c)-9742) + (+641) = 990258 +000641 = 990899(negative)Magnitude: 009101Result: (-9742) + (641) =-9101(d) (-9742) + (-641) = 990258 + 999359 = 989617 (Negative)Magnitude: 10383Result: (-9742) + (-641) =-103831.226,514BCD:0110_0101_0001_0100ASCII:0_011_0110_0_011_0101_1_011_0001_1_011_0100ASCII:0011_0110_0011_0101_1011_0001_1011_01001.23011110010001 ( 791)011001011000(+658)110111101001011001100001 00110100000100010001010001001001 (1,449)1.24(a)(b)6 3 1 1Decimal0 0 0 000 0 0 110 0 1 020 1 0 030 1 1 04 (or0101)0 1 1 151 0 0 061 0 1 07 (or1001)1 0 1 181 1 0 096 4 2 1Decimal0 0 0 000 0 0 110 0 1 020 0 1 130 1 0 040 1 0 151 0 0 06 (or0110)1 0 0 171 0 1 081 0 1 191.25(a)6,24810BCD: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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71.266,2489s Comp:3,7512421 code:0011_0111_0101_00011s compc:1001_1101_1011_0001 (2421 code alternative #1)6,24824210110_0010_0100_1110(2421 code alternative #2)1s comp c1001_1101_1011_0001Match

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81.27For 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 bitsselect the "number" of the card. Example: 0001 (ace) through 1011 (9), plus 101 through 1100 (jack,queen, king). This a jackof spades might be coded as 11_1010. (Note: only 52 out of 64 patterns areused.)1.28G(dot)(space)Boole11000111_11101111_01101000_01101110_00100000_11000100_11101111_111001011.29Steve Jobs1.3073 F4 E5 76 E5 4A EF 62 7373:0_111_0011sF4:1_111_0100tE5:1_110_0101e76:0_111_0110vE5:1_110_0101e4A:0_100_1010jEF:1_110_1111o62:0_110_0010b73:0_111_0011s1.3162 + 32 = 94printing characters1.32bit 6 from the right1.33(a)897(b)564(c)871(d)2,1991.34ASCIIfor decimal digits withevenparity:(0):00110000(1):10110001(2):10110010(3):00110011(4):10110100(5):00110101(6):00110110(7):10110111(8):10111000(9):001110011.35(a)abcfgabcfg1.36abfgabfg

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9CHAPTER 22.1(a)x y z0 0 00 0 10 1 00 1 11 0 01 0 11 1 01 1 1x + y + z01111111(x + y + z)'10000000x'11110000y'11001100z'10101010x' y' z'10000000x y z0 0 00 0 10 1 00 1 11 0 01 0 11 1 01 1 1(xyz)00000001(xyz)'11111110x'11110000y'11001100z'10101010x' + y' + z'11111110(b)(c)x y z0 0 00 0 10 1 00 1 11 0 01 0 11 1 01 1 1x + yz00011111(x + y)00111111(x + z)01011111(x + y)(x + z)00011111x y z0 0 00 0 10 1 00 1 11 0 01 0 11 1 01 1 1x(y + z)00000111xy00000011xz00000101xy + xz00000111(c)(d)x y z0 0 00 0 10 1 00 1 11 0 01 0 11 1 01 1 1x00001111y + z01110111x + (y + z)01111111(x + y)00111111x y z0 0 00 0 10 1 00 1 11 0 01 0 11 1 01 1 1yz00010001x(yz)00000001xy00000011(xy)z00000001(x + y) + z011111112.2(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')(AB) =(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 = b2.3(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 = BD2.5(a)xyFFsimplified(b)xyFFsimplified(c)

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11xyzFsimplifiedF(d)BFFsimplifiedA0(e)xyzFsimplifiedF(f)

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12xyzFFsimplified2.6(a)ABCFFsimplified(b)xyzFFsimplified(c)xyFFsimplified

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13(d)wxyzFsimplifiedF(e)ABCDFsimplified= 0F(f)wxyzFsimplifiedF2.7(a)ABCDFsimplifiedF

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14(b)wxyzFsimplifiedF(c)ABCDFsimplifiedF(d)ABCDFsimplifiedF

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15(e)ABCDFFsimplified2.8F'=(wx + yz)' = (wx)'(yz)' = (w' + x')(y' + z')FF' =wx(w' + x')(y' + z') + yz(w' + x')(y' + z') = 0F + F' = wx + yz + (wx + yz)' = A + A' = 1with A = wx + yz2.9(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=Σm1i+Σm2i=Σ(m1i+ m2i)(b)F1 F2 =ΣmiΣmjwheremimj= 0ifi≠jandmimj= 1ifi = j2.11(a)F(x, y, z) =Σ(1, 4, 5, 6, 7)(b)F(a, b, c) =Σ(0, 2, 3, 7)x y z0 0 00 0 10 1 00 1 11 0 01 0 11 1 01 1 1F01001111F = xy + xy' + y'za b c0 0 00 0 10 1 00 1 11 0 01 0 11 1 01 1 1F10110001F = bc + a'c'2.12A = 1011_0001B = 1010_1100

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16(a)A AND B = 1010_0000(b)A OR B = 1011_1101(c)A XOR B = 0001_1101(d)NOT A = 0100_1110(e)NOT B = 0101_00112.13(a)uxyY = [(u + x')(y' + z)]z(u + x')(y' + z)(b)uxyY = (uxor y)' + x(uxor y)'x(c)uxyY = (u'+ x')(y + z')z(u'+ x')(y + z')(d)uxyY = u(xxor z) + y'zu(xxor z)y'(e)uxyY = u + yz +uxyzuyzuxy(f)

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