Solution Manual for Fundamentals of Logic Design 7th Edition by Roth ISBN
1133628478 9781133628477
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Unit 5 Problem Solutions
41 © 2014 Cengage Learning.All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part 0 1 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 1 0 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 1 0 1 1 0 0 1 0 1 1 1 1 0 0 1 1 0 1 * 0 1 1 * 0 1 * 1 1 0 1 1 0 1 1 * 1 * 1 1 * 1 1 1 * 1 1
Unit 5 Solutions
Solution
5.3 (a) a b c 0 1 00 01 11 5.3 (b) d e f 0 1 00 01 11 5.3 (c) r s t 0 1 00 01 11 5.3 (d) x y z 0 1 00 01 11 10 f1 = a'c' + a b'c + b c' 10 f2 = d'e' + d'f '+ e'f' 10 f3 = r' + t' 10 f4 = x'z + y + x z' 5.4 (a) A B C D 00 01 11 10 00 01 11 10 5.4 (b) A B C D 00 01 11 10 00 01 11 10 5.4 (c) A B C D 00 01 11 10 00 01 11 10 F = BD' + B'CD + ABC + ABC'D + B'D' F = D' + B'C + A B F = (A + B'+ D')(B + C + D') 5.5 (a) See FLD p. 735 for solution. 5.5 (b) X1 X2 C1 C2 00 01 11 10 00 01 11 10 Alt: Z = C1'X1'X2+ C1'X1X2 ' + C1C2X1'X2'+ C1X1X2+ C1'C2'X2 Z = C1'X1'X2+ C1'X1X2 ' + C1C2X1'X2'+ C1X1X2+ C1'C2'X1 Z = C1'X1'X2+ C1'X1X2 ' + C1C2X1'X2'+ C1X1X2+ C2'X1X2 5.6 (a) a b c d 00 01 11 10 00 01 11 10 5.6 (b) a b c d 00 01 11 10 00 01 11 10 Alt:
(*) Indicates a minterm that makes the corresponding prime implicant essential.
a'd →m5; a'b'c'→m0; b'cd→m11;abd'→m12
(*) Indicates a minterm that makes the corresponding prime implicant essential.
bd→m13 or m15; a'c→m3; b'd'→m8 or m10
Solutions 42 © 2014 Cengage Learning.All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part f = a'b'c' + a'd + b'cd + abd' + bcd' f = a'b'c' + a'd + b'cd + abd' + a'bc Alt: F = a ' c + b ' d ' + b d + a ' d ' F = a ' c + b ' d ' + b d + a ' b
Unit 5
(*) Indicates a minterm that makes the corresponding prime
(c'+ d')(b'+c')(a + b + c)(a'+ c + d)
= (a'+ c)(b'+ d')(b + d)(c'+ d)
= (a'+ c)(b'+ d')(b + d)(b'+ c')
a'b'd + bc'd' + cd
43 © 2014 Cengage Learning.All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part 1 1 1 * 1 X 0 0 X X 0 0 1 * X 1 * 0 1 * 1 1 1 1 1 1 1 0 0 X 0 X 1 1 X 1 1 X 1 0 0 1 1 1 0 1 1 0 1 1 0 0 1 0 1 0 1 1 1 c d a b 5. 00 01 11 10 7 (c) c d a b 5.7 00 01 11 10 (d) A B C D 00 01 11 00 00 00 01 01 01 11 11 11 10 10 10 f = a'b' + a'c'd' + abc f = b'c'd' + ab'c + a'bc + bc'd + ad' F = D + A C 0 1 0 0 0 1 1 1 X X X 0 1 0 X 1 0 1 0 0 0 1 1 1 X X X 0 1 0 X 1 0 1 X X 1 0 0 0 1 X X 1 X 0 X 0 0 1 X X 1 0 0 0 1 X X 1 X 0 X 0 5.6 (c) a b c d 00 01 11 10 00 01 11 10 5.7 (a) a b c d 00 01 11 10 00 01 11 10 F = a'd' + b' + c'd'
Unit 5 Solutions
c'd'→m12
Alt: f = a'c'd'
f = a'c'd' + a'cd + b'c'd' + abcd' + a'b'c 5.7 (b) 10 X 1 1 X 1 1 X 5.8 (a) a b c d 00 01 11 10 00 01 11 10 a b c d 00 01 11 10 00 01 11 10
f = a'bc'
5.8 (b) a b c d 00 01 11 10 00 01 11 10 a b c d 00 01 11 10 00 01 11 10 Alt: f
f
f
implicant essential.
; a'd'→m6; b'→m10 or m11
+ a'cd + b'c'd' + abcd' + a'b'd'
f =
+ ac'd + b'cd'
=
F = (A'+ B'+ E )(A'+ C'+ D + E )(C + D'+ E') (A + B + D'+ E )(A + B + C )(B'+ D + E')
Alt:
F = A'C D' + A'B E' + C D E + A B'C'D' + A B'D E' + B'C E
F = A'C D' + A'B E' + C D E + A B'C'E' + A B'C D + B'D'E
F = A'C D' + A'B E' + C D E + A B'C'D' + A B'D E' + B'D'E
F = A'C D' + A'B E' + C D E + A B'C'E' + A B'D E' + B'D'E
Essential prime implicants: c'd'e' (m ),
b'ce, c'd'e', a'cd'
Prime implicants: a'b'de, a'd'e', cd'e, a'ce', ace, a'ce' (m ), ace (m ), a'b'de (m )
5 Solutions 44 © 2014 Cengage Learning.All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part , m 14 3 16 24 31 5.9 (a) B C D E 00 01 11 10 B C D E 00 01 11 10 00 A 01 1 0 11 10 1 1 0 0 1 X X 0 0 1 0 0 1 0 0 1 X 0 0 1 X 1 1 X 1 0 1 0 1 1 0 1 00 A 01 1 0 11 10 1 1 0 0 1 X X 0 0 1 0 0 1 0 0 1 X 0 0 1 X 1 1 X 1 0 1 0 1 1 0 1 F = (A'+ B'+ C + E )(A'+ B + C'+ D')(A + B'+ C'+ E ) (B'+ D + E )(A + C'+ D)(A'+ C + D + E')(A'+ B'+ C'+ E') F = A'C'E + A'C'D + A'D E + A B'C D' + C'D E + A B C D E' + B'C'E' + A'B'D Alt: F = A'C'E + A'C'D + A'D E + A B'C D' + C'D E + A B C D E' + B'C'E' + A'B'E' 5.9 (b) B C D E 00 01 11 10 B C D E 00 01 11 10 00 A 01 1 0 11 10 1 0 0 0 0 1 X 1 1 1 0 0 X 1 X 0 0 1 1 0 0 1 1 0 1 1 0 0 X 0 1 1 00 A 01 1 0 11 10 1 0 0 0 0 1 X 1 1 1 0 0 X 1 X 0 0 1 1 0 0 1 1 0 1 1 0 0 X 0 1 1
Unit
5.10 (a) b c d e 00 a 01 1 0 11 10 00 01 11 10 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 5.10 (b) b c d e 00 a 01 1 0 11 10 00 01 11 10 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
a'b'c,
Unit 5 Solutions
F = a'b'c' + a'c'd + bcd +abc + a b' = (a'b'c' + ab') + a'c'd + bcd + (abc + a b')
= (a'c' + a)b' + (a'c'd + bcd) + a(bc + b')
= (c' + a)b' + (a'c'd + bcd +a'bd) + a(c + b')
= (b'c' + a'bd + a'c'd) + (bcd + a'bd + ac) + ab'
= (b'c' + ac + ab') + a'bd
= b'c' + ac + a'bd
PIs: C'D, C D', A'B, A B'C', A B'D' f = C'D + C D' + A'B
PIs: B'C D, A C', A D', A B', B C'D, B C D', A'C D, A'B D, A'B C f = B'C D + A C' + A D' C D 00 01 11 10 00 01 11 10
C D 00 01 11 10 00 01 11 10
PIs: A'B, B C D, A B'C', A B'D' f = A'B + B C D PIs: C D, A'B, A B' f = C D + A'B
5.22 (f) A B 5.22 (g) A B 5.23 (a) A B C D 00 01 11 10 00 01 11 10
C D 00 01 11 10 00 01 11 10
PIs: A'B, B C', B D', A C', A D', A B' f =
D 00 01 11 10 00 01 1 1 1 0
PIs: (B'+ C'+ D'), (A + C ), (A + D ), (B + D), (B + C), (A + B')
= (B'+ C'+ D') (A + D ) (B + C ) or
49 © 2014 Cengage Learning.All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part 1 1 1 1 1 1 1 1 1 1 1 1 X X 1 X X 1 X 1 X 1 X X 1 X 1 X 1 X 1 X 1 X X X X 1 X X 1 X X X 1 X 1 1 X X 1 X 1 X 1 X X 1 X 1 X X 1 X X X X X 1 X X X 1 X X X X X X X X X X 0 X 0 X X 0 X 0 0 X X 0 0 X 0 X X X 0 0 X X 5.21 a b 5.22 (a) A B 5.22 (b) A B c d 00 01 11 10 00 01 11 10 C D 00 01 11 10 00 01 11 10 C D 00 01 11 10 00 01 11 10
PIs: A B', B C', A D', B D', A C', A' B f = A B' + B D' + A C' or = A B' + B C' + B D' or = A B' + B C' + A D' 5.22 (c) A B 5.22 (d) A B 5.22 (e) A B C D 00 01 11 10 00 01 11 10
B C' f = A'B C D 00 01 11 10 00 01 11 10
B D' f =
PIs: B C D, A'C', A'D', A'B, B'D', B'C' f = B C D + A'B or f = B C D + A'D' or f = B C D + A'C'
5.23 (b) A B 5.23 (c) A B 5.23 (d) A B
PIs: (B'+ C'+ D'), (A + B ), (A + D'), (A + C') f = (B'+ C'+ D') (A + B )
C
f
Solutions 50 © 2014 Cengage Learning.All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part 0 0 X X X 0 X 0 0 X X 0 0 X 0 X 0 X 0 X 0 0 X 0 X = (B'+ C'+ D') (A + C ) (B + D ) or = (B'+ C'+ D') (A + C ) (A + D ) C D 00 01 11 10 00 01 11 10 PIs: (B + C + D ), (C'+ D'), (A'+ C + D ), (A' + B), (A + B' + D'), (A + B' + C') f = (B + C + D ) (C'+ D') (A'+ C + D ) C D 00 01 11 10 00 01 11 10 PIs: (B ), (A'+ C ), (A'+ D ), (C + D'), (C' + D), (A + D'), (B + D'), (A + C') f = (B ) (C + D') (A'+ D ) or = (B ) (A'+ C ) (C'+ D ) or = (B ) (A'+ C ) (A'+ D )
Unit 5
51 © 2014 Cengage Learning.All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part 0 0 X 0 X 0 X X 0 X 0 X 0 X X 0 X X X 0 X 0 X 0 X X X X 0 X X X 0 X 0 X 0 X X 0 X X X 1 1 1 1 1 X 1 1 X X 1 1 1 1 1 1 1 1 1 1 1 1 1 X 1 1 1 1 1 X X X 1 0 X 1 1 0 0 X 0 1 0 1 0 0 1 1 X 1 1 1 1 1 1 1 1 0 1 1 0 1 0 1 1 0 1 1 0 1 1 1 1 1 X 1 1 X 1 1 1 X 5.23 (e) A B 5.23 (f) A B 5.23 (g) A B C D 00 01 11 10 00 01 11 10 C D 00 01 11 10 00 01 11 10 C D 00 01 11 10 00 01 11 10 PIs: (C + D'), (A'+ D ), (B + D ), (A' + C), PIs: (B), (D'), (A'), (C') PIs: (B)(A'+ D)(A'+ C)(A + C') (B + C), (C' + D), (A + B' + D'), (A + B'+ C') f = (B ) (A') or (C'+ D)(C + D')(A + D') f = (A'+ C ) (B + C ) (C'+ D ) or = (B ) (C') or f = (B)(A'+ C)(C'+ D) or = (C + D') (A'+ D ) (B + D ) = (B ) (D') = (B)(A'+ D)(C+ D') or = (B)(A'+ D)(A'+ C) 5.24 (a) 5.24 (b) 5.24 (c) A B C D 00 01 11 10 00 01 11 10 A B C D 00 01 11 10 00 01 11 10 A B C D 00 01 11 10 00 01 11 10 Alt: F = A B C' + B'C D + A'C + A'B'D' + A'B D F = A B C' + B'C D + A'C + A'B'D' + B C'D Alt: F = A'C'D'+ B'C'D' + A'B'D' F = A'C'D'+ B'C'D' + A'B'C F = A'C'D+ A'B + B C'D 5.24 (d) w x 5.24 (e) A B 5.25 (a) a b y z 00 01 11 10 00 01 11 10 C D 00 01 11 10 00 01 11 10 c d 00 01 11 10 00 01 11 10 Alt: f = x'y' + w'z + y'z + wz' f = x'y' + wy' + w'z + wz' f = x'y' + wy' + w'z + wx' F = A'B'C D + BD' + AD' + AB f = a'd + a'bc' + c'd + bd 5.25 (b) a b 5.25 (c) a b 5.25 (d) a b c d 00 01 11 10 00 01 11 1 0 c d 00 01 11 10 00 01 11
Unit 5 Solutions
Unit 5 Solutions 52 © 2014 Cengage Learning.All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part 0 1 0 1 X X 0 0 X 0 1 1 0 0 1 1 10 c d 00 01 11 10 00 01 11 10 f= b'c'd + cd' + bd' + bc + ab f = b'd' + cd' + ac'd f = a'bc' + ab'd' + ac
f = (A'+ B'+ D)(A'+ C')(A + B + D)
f = C'D + AB'C' + A'B + A'D
f = x'y' + w'z + y'z + wz'
f = x'y' + wy' + w'z + wz'
f = x'y' + wy' + w'z + wx'
f = (w + x'+ z)(w + y'+ z)(w'+ y'+ z')
f = (w + x'+ z)(w + y'+ z)(w'+ x'+ y')
F = b'd' + a'd + c'd
Notice that abcd = 0101 and 1111 never occur, so minterms 5 and 15 are don’t cares.
F = A B'D' + A'B + A'C + C D
F' = ABD' + A'B'C' + AC'D
Solutions 53 © 2014 Cengage Learning.All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part 0 0 X 1 1 0 0 1 1 X 1 0 0 X 0 0 C D A B 00 01 11 10 5.26 (b) C D A B 00 01 11 10 00 00 01 01 11 11 10 10 0 1 0 1 1 X 1 1 1 X 0 0 0 1 0 0 0 1 0 1 1 X 1 1 1 X 0 0 0 1 0 0 1 0 1 1 X 1 1 1 1 1 0 X 0 X X 1 1 0 1 1 X 1 1 1 1 1 0 X 0 X X 1 1 1 1 X 1 1 1 1 X 1 1 0 1 0 1 0 1 0 0 1 1 1 1 1 1 0 1 1 0 1 0 1 0 1 1 0 0 0 0 0 0 1 0 5.26 (a) 0 0 0 0 X 0 0 0 X F = (C'+ D ) (A'+ B + D') (A + B'+ C ) (A + D ) F = (B'+ C ) (A'+ B + C') (A + D ) (C'+ D ) Alt: F = (B'+ C ) (A'+ B + C') (A + D ) (B'+ D ) 5.27 (a) A B C D 00 01 11 10 00 01 11 10 A B C D 00 01 11 10 00 01 11 10
Unit 5
5.27 (b) w x y z 00 01 11 10 00 01 11 10 w x y z 00 01 11 10 00 01 11 10
Alt:
Alt:
5.28 a b 5.29 (a) A B 5.29 (b) A B c d 00 01 11 10 00 01 11 10 C D 00 01 11 10 00 01 11 10 C D 00 01 11 10 00 01 11 10
F =
= (A
(A' +
D'
+ D) (A' +
+ C + D'
D)
∏ M(0, 1, 9, 12, 13, 14)
+ B + C + D)(A + B + C + D')
B + C +
)(A' + B' + C
B'
)(A' + B' + C' +
5.31 Prime implicants for f ': abc'e, ac'd', ab'e', a'ce, b'c'de', c'd'e, a'd'e
Prime implicants for f: a'd'e', ace, a'ce', bde', abc, bce', b'c'de, a'c'de, a'bc'd, ab'de
5.32 For F: b'c'de', a'ce, ab'e', ac'd', abc'e, c'd'e, a'd'e
For G: ab'ce, a'bcd, a'bde', cde, b'de, a'b'c'd, a'c'e'
5.33
5-variable mirror image map
5-variable diagonal map
(c + d'), (a'+ c), (b + c ), (a + b +d'), (a + b'+c'+d), (a'+b'+d'), (a'+b+d )
Solutions 54 © 2014 Cengage Learning.All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part 1 X 1 X 1 1 1 X 1 0 1 0 1 0 1 0 0 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 X 1 X X X X 1 X 1 1 X 1 X X 1 X X X X 1 X 1 1 X 1 X X X 0 0 X 0 X X X X X 5.29 (c) A B C D 00 01 11 10 00 01 11 10 5.30 A B C D 00 01 11 10 00 01 11 10 F = (A'+ B'+ D )(A + B + C )(A'+ C + D') F = D + AB C
Unit 5
abc de 000 001 011 010 110 111 101 100 00 01 11 10
f=
OtherPIs:
b c d e 00 01 11 10 1 1 1 00 1 1 1 1 a 01 1 1 1 0 1 11 1 1 1 10 1 5.34 (a) a b c d 00 01 11 10 00 01 11 10 5.34 (b) & (c) a b c d 00 01 11 10 00 01 11 10 PIs: bd', a'b, a'd', c, ab'd f = b d' + c or = a'b + c or = a'd' + c 5.34 (d) & (e) a b c d 00 01 11 10 00 01 11 10
f =
= (b + c
+
or (b + c )(a'+ c )
EssentialPIs: ab'c'e',a'bc'e,a'b'cd
a'b'cd+ a'bc'e+ ab'c'e'+ a'bcd'+ ab'ce+ ac'd'e+ acd'e'
ab'd',bcd'e',bc'd'e,a'bd'e,b'cde
PIs:
(c + d')(a'+ c ) or
)(c
d')
mirror image map
PIs: (A'+ B'), (A'+ C'), (A'+D + E'), (B'+D'), (B'+ C + E'), (C'+ E), (D'+E), (B + C'+ D), (A + C + D'), (A + B + E) F
(B'+D')(A + B + E) or
+ C + D')
5.36 (a), 5-variable mirror image map
5-variable diagonal map
PIs: A B, A D, A C'E, B C, B D'E, C D, D E', C E',
B'D, C'D'E,A'D'E, B'C'E, A'B' F
+
+ AB or
5.36 (d), (e) 5-variable mirror image map
5-variable diagonal map
55 © 2014 Cengage Learning.All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part X X 1 X X X 1 1 X 1 X X X X X X 1 X 1 X X X X X X X X X X 0 X X X X X X X X X X X 0 X X X X 0 X 0 X X X X X D E 000 001 011 010 110 111 101 100 D E 00 X X X X X X 01 X X 1 X 1 X X 11 X 1 X X X X 1 A 10 X X 1 X X X 1 X 0 X 0 X X X 0 X X X X X X 0 X X X 0 X X X X X X X X X 5.35
5-variable diagonal
(c) A B C D E 000 001 011 010 110 111 101 100 B C D E 00 01 11 10 00 11 10 PIs: A, C'D', B'E, C E, B D', D'E, B C'E', B'C D F= A + B'E + BD' or = A + C'D' + CE 00 A 01 1 0 11 10 1 X X X X X 1 X X X X 1 X 1 X 1 X X X X 1 X X X X X X X 5.35
5-variable
5-variable diagonal map A B C D E 000 001 011 010 110 111 101 100 B C D E 00 01 11 10 00 01 11 10
Unit 5 Solutions
(a), 5-variable mirror image map
map
(d), & (e)
=
= (C'+
00 A 01 1 0 11 10 X X X X 0 X X X X X X X X X X X X 0 X X X X 0 X 0 X
E)(A
(b) & (c) A B C B C 00 01 11 10 X X X 00
= A'C
= B'D
BC or = A'C
AB
AD 01 X X 1 X 1 X X X 11 X 1 X X 1 X X 10 X X 1 X
A'C,
B'D
+ AB +
+
+
A B C D E
B C D E 00 01 11 10 00 01 11 1 0 P I s : (A'+ B'+
D'),
B'+
(A'+
000 001 011 010 110 111 101 100
C'), (A'+B'+
(A'+
E),
C'+ E'),
56 © 2014 Cengage Learning.All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part (A'+C'+ D), (B'+ D'+ E'), (B'+ C+ D'), (D + E), (A + B + E), (A + C ), (B + D ), (C + E) F = (A + C )(B + D ) 00 A 01 1 0 11 10 0 X X X X X X 0 X 0 X X X X X X X X X 0 X X X X X X
Unit 5 Solutions
PIs: (B'+ C + E'), (C'+ E ), (D'+ E ), (A'+D +E'), (A'+ C'), (B'+D'), (A'+B'), (A + C+D'), (A + B + E)
F = (A'+ D + E')(C'+ E )(D'+ E )(B'+ D') or
= (A + B + E )(B'+ D')(A'+ B')(A'+ C') or
= (A + C + D')(C'+ E )(A'+ B')(A'+ C') or
= (B + C'+ D )(D'+ E )(B'+ D')(A'+ B') or
= (B'+ C + E')(C'+ E )(D'+ E )(A'+ C')
+ B'C'D' + B'D E or = A'C E + B'C'E +
C'D'E'
(*) Indicates a minterm that makes the corresponding prime implicant essential. a'b'd'→m
a'b'd', cd'e', bc'd'e, b'cd', ac'de', ab'ce', ab'de', a'c'd'e, a'bc'e, a'bc'd, bc'de', a'bde', a'bce'
57 © 2014 Cengage Learning.All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part 1 1 1 1 1* 1* 1 * 1 1 1 1 1 1 1 1 5.37 (a), (b) & D E B C 00 01 11 10 5.37 (d), & (e) D E B C 00 01 11 10 (c) 00 A 01 1 0 11 10 1 X X X X 1 X X 1 X 1 X 1 X X X 1 X X X X X 00 A 01 1 0 11 10 X 0 X X 0 X X 0 X 0 X X X 0 X X X 0 X 0 X 0 0 X 0 X
C D'E,
C'D E, A B'D'E',
B'C', A'B C'E',
E, A'B'E,
= A'B'E
A'B D'
or = A'C'D'
=
or = A'B D'
Unit 5 Solutions
PIs: A B C D E', B
A
A
A'B D', A'B'C D, A'D'E, A'C E, B'D
B'C'E, C'D'E', A'C'D', B'C'D' F
+
+ A B'C'
+ A'C E + A B'C' or
A'D'E + B'D E + C'D'E'
5.38 (a) b c d e 00 01 11 10 5.38 (b) b c d e 00 01 11 10 00 * 1 1 1 00 1 1 1 1 1 a 01 1 0 11 a 01 1 0 11 1 1 1 1 10
1;
1 1 1 10 1 1
cd'e'→m28; bc'd'e→m25; b'cd'→m21
5.39 (a) b c d e 00 01 11 10 5.39 (b) b c d e 00 01 11 10 1 00 1 a 01 1 1 0 1 11 1 1 10 1 1 1 1 1 1 X X 1 1 1 1 X 1 1 00 a 01 1 0 11 10 1 1 1 1 1 1 1 1 1 1 1 1 1
Unit 5 Solutions
f = a'b'c'e + a'bc'd' + bcde + ab'd'e' + abde + acd'e' +
+ acde
58 © 2014 Cengage Learning.All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part Alt: f = a'b'c + a'bc' + ab'c'd + c'd'e' + abd' + bcde + a'c'e f = a'b'c + a'bc' + ab'c'd + c'd'e' + abd' + bcde + a'b'e alt:
= a'b'c'e
acd'e' + a'b'd'e
f
+ a'bc'd' + bcde + ab'd'e' + abde +
+ ab'ce
f = a'b'c'e + a'bc'd' + bcde + ab'd'e' + abde + acd'e' + b'cd'e + ab'ce
b'cd'e
VZ + WX'YZ'+ VWX
F = (X + Y + Z) (V + Y' + Z') (V + X' + Z') (V + X' + Y') (V' + W + Z)
F = (c + d + e)(a'+ b)(a + b')(a + c'+ d'+ e')
F = (c + d + e)(a'+ b )(a + b')(b + c'+ d'+ e') Alt:
F = (c + d')(a + d'+ e')(b'+ c + e')(a'+ b'+ c'+ d) (a + c + e) (b + c'+ e)
F = (c + d')(a + d'+ e')(a + b'+ c)(b'+c + e') (a'+ b'+ c'+ d)(b + c'+ e)
Solutions 59 © 2014 Cengage Learning.All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part 1 5.40 B C D E 00 01 11 10 5.41 B C D E 00 01 11 10 0 0 1 0 1 X 00 00 1 0 1 0 1 1 A 01 1 1 1 1 1 1 1 1 0 A 01 1 1 1 1 X 0 0 0 0 1 11 0 1 0 1 0 1 0 1 10 0 1 0 0 0 11 1 X 1 1 1 10 1 X F = A'B'C'D' + BC'DE + BCD' + B'CDE' + ABC'D + A'B'CE + AD'E F = AB'CD'E' + BC'D' + ABDE' + A'B'C'E' + A'C'DE + A'CD'E 5.42 (a) W X Y Z 00 01 11 10 5.42 (b) Y Z W X 00 01 11 10 00 V 01 1 0 11 10 X 0 1 0 0 1 1 0 X 1 1 1 X 0 0 1 1 1 1 1 0 0 0 0 0 0 1 1 X 0 0 1 00 V 01 1 0 11 10 X 0 1 0 0 1 1 0 X 1 1 1 X 0 0 1 1 1 1 1 0 0 0 0 0 0 1 1 X 0 0 1 F = V'XY'Z'+ X'Y'Z+
Unit 5
5.43 (a) b c d e 00 01 11 10 5.43 (b) b c d e 00 01 11 10 00 a 01 1 0 11 10 0 0 1 0 0 1 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 0 0 0 0 0 1 1 1 1 0 0 00 a 01 1 0 11 10 1 0 0 1 X X 1 0 1 1 0 0 1 1 1 0 0 1 1 0 X 0 0 0 0 0 1 0 X 0 1 0
Alt:
F = (v'+ w'+ x'+ y + z')(w + y'+ z')(v + y')(w + x + y)
(v'+ x + y + z) (w'+ x + y')
F = (v'+ w'+ x'+ y + z')(w + y'+ z')(v + y')(w + x + y)
(v'+ w'+ x + z )(w'+ x + y')
F = (v'+ w'+ x'+ y + z')(w + y'+ z')(v + y')(w + x + y)
(v'+ w'+ x + z)(x + y'+ z')
F = (c + d + e) (a' + c' + d') (a' + b + c + d) (a + c' + d) (b + d' + e) (a + c + e)
F = ACD' + BC'D + B'C +
m4, m13, or m14 change the minimum sum of products, removing A'C', BC'D, or ACD', respectively
Changing m solution. to a don't care removes C'D from the
+ XYZ + VW'X'Y' + VW'Y Z' + W'XZ
Solutions 60 © 2014 Cengage Learning.All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 5.44 (a) w x y z 00 01 11 10 5.44 (b) b c d e 00 01 11 10 00 v 01 1 0 11 0 1 1 0 0 1 1 1 0 1 0 1 0 1 1 1 0 0 1 0 0 0 0 0 00 a 01 1 0 11 0 0 X 0 0 0 0 0 0 0 X X 1 1 1 0 10 0 0 0 0 0 0 X 10 0 0 0
Unit 5
Alt:
5.45 (a) A B C D 00 01 11 10 00 01 11 10 5.45 (b) A B C D 00 01 11 10 00 01 11 10
A'C'
F = C'D + BD + AB'
5.46 (a) W X Y Z 00 01 11 10 1 5.47 (a) f a b 00 1 X 1 1 1 V 01 X 1 X 1 c d 00 01 11 10 00 0 1 1 01 11 1 1 1 1 10 X 1 1
V'XY'
10 m4 m8 m31
F =
+ V'WZ'
Unit 5 Solutions
F = V'XY' + V'WZ' + XYZ + VW'X'Z' + VW'XY + W'Y'Z
F = V'XY' + V'WZ' + XYZ + VW'X'Y' + VW'Y Z' + W'Y'Z
5.46 (b)
F = V'XY' + V'WZ' + XYZ + VW'X'Z' + VW'Y Z' + W'Y'Z
V'WZ'→m8; XYZ→m31; V'XY'→m4
5.47 (b)
Essential: a′b′, ab, c′d′, cd,
Nonessential: a′c′, a′d, bc′, bd
5.47 (c)
f = a′b′ + ab + c′d′ + cd + a′d
f = a′b′ + ab + c′d′ + cd + bd
f = a′b′ + ab + c′d′ + cd + bc′
f = a′b′ + ab + c′d′ + cd + a′c′
© 2014 Cengage Learning.All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part
61
Another solution to 5.49
5.49 (a) f
a b
c d 00 01 11 10 00 01 11 10
f(a, b, c, d) = ∑m(0,
14, 15)
5.49 (b)
5.49 (c)
Essential: b', a'c', d', ac '
f = b' + a'c' + d' + ac
5.50 (a) f
+ a'd + ad'
a b
c d 00 01 11 10 00 01 11 10
f(a, b, c, d) = ∑m(0, 1, 4, 5, 12) + ∑d(3, 7, 8, 9, 10, 14)
5.50 (b)
5.50 (c)
Nonessential: a'd, c'd', a'c', b'c', ad'
f = a'd + c'd'
f = a'c' + c'd'
f = a'c' + ad'
Unit 5 Solutions 62 © 2014 Cengage Learning.All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 X 1 1 0 X X X 0 0 0 0 X X 1 1 1 1 1 1 1 1 1 1 1 1 1 1 5.48 (a) f a b 5.49 (a) f a b 5.48 (b) 5.48 (c) c d 00 01 11 10 00 01 11 10 f(a, b, c, d) = ∑m(0, 1, 2, 3, 4, 5, 7, 8, 9, 10, 12, 13, 14, 15) Essential: c' Nonessential: a'b', a'd, b'd', bd, ab, ad' f = c' + a'd + b'd' + ab f = c' + a'b' + ad' + bd 5.49 (b) 5.49 (c) c d 00 01 11 10 00 01 11 10 f(a, b, c, d) = ∑m(0, 1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 12, 13, 14) Essential: b', c', a'd, ad' f = b' + c'
2, 3, 4, 5, 6, 8, 9, 10, 11, 12,
1,
Unit 5 Solutions 63 © 2014 Cengage Learning.All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part 1 1 1 X 1 1 0 X X X 0 0 0 0 X X 5.50 (d) f a b c d 00 01 11 10 00 01 11 5.50 (e) 10 Essential: c', a' + d' Nonessential: a' + b ' f = (c')(a' + d')
