Answers to Problems For Fundamentals of Logic Design second edition (corrected version) 3
Answers to hapter Problems. (a) 5.875 (b) 38.5 (c) 53.5 (d) 574.75 (e) 4.4535 (f) 474.5. (a) = 33 9 (b) = A 6 (c) 78 = 3 5 (d) 3467 = 35 7 3. (a).36 8 (b).78 6 (c).3 5 (d). 4. (a) 6 = 496 = 3544 6 (b) 3 = 8 = 3 6 (c) 4 5 = 59 = 35 6 (d) 3467 8 = 847 = 35 6 5. R = 8 6. (a) = 64 6 = ( ) = 44 8 (b) 7 = D9 6 = ( ) = 33 8 (c) 47.65 = D8.A 6 = (. ) = 73.5 8 (d) 56.35 =.8 6 = (. ) = 4. 8 7. (a) 47 = ( ) = ( ) = (b) (c) (d) = ( ) 8. (a) 5 (b) 5 (c) 76 (d) 947 9. (a) (b) (c) (d). (a) 3. 4 = 9 E. 6 6 (b). 3 4 = (. ) (c) 3. 4 = 3.5 (d) 3 4 = 36 7. (a) ( ) = 3 4 (b) () 6 = ( ) 4 (c) ( ) 8 = ( ) = ( ) 4 (d) 745. 5 = 3. 4. (a) ( 58 ) = ( ) D (b) ( ) = ( 3 ) = ( ) D (c) ( ) D = ( 9357 ) (d) ( ) D = ( ) = ( ) 4
3. (a) ( 78 ) = ( ) Excess-3 (b) ( ) Excess-3 = ( 68 ) = ( ) D (c) ( ) = ( 8 ) = ( ) Excess-3 (d) ( F ) 6 = ( 485 ) = ( ) D 4. (a) (,,,,,,,,, ) or (,,,,,,,,, ) (b) (,,,,,,,,, ) or (,,,,,,,,, ) 5. (a) ecause it is impossible to encode decimal digit. (b) Yes. For example : (6, 4, 3, ) weighted code. How about (6, 4, 3, ), (6, 4, 3, ), and ( 6, 4, 3, )? 5
Answers to hapter 3 Problems. AD f(a,,, D) (a) (b) (c). (a) (b) (c) (d) A 3. (a) A + (b) A + D (c) A + + (d) + D 4. (a) x + y (b) x + y (c) (d) x + y 5. (a) a (b + c + d) (b) a b (c + d) (c) b d (d) a b + ab 6. (a) y + w xz (b) b (a + c) (c) ad + c(b + d) 7. (a) ab + ad + cd (b) ab + b d + c d + ad (c) a bd + bcd (d) ac + a b + a c d 8. (c) LHS = A + A + = A + A + + + A + A = + A + A = RHS 9. (a) ac + abd + a bc (b) ac d + abd (c) a d + bd. (a) (a + c)(a + b)(a + d ) (b) d( a + c )(b + c )(a + b) (c) a(b + c + d ) 6
. (a) (a + b + c)[(e + d'c') a'] (b) (a + b + d)[c' + (a' + b') d'] (c) [(w' + x) + (y + z) u] (u' + w' + xy). (a) (a + b + c') (b + c + d) (a' + c' + d') (b) (a + b'd) (a' + c' + be) (c) (d' + a'c + b'c') (a + d) 3. (a) c + a bd + ab d (b) ab' + ad' + b'c' (c) y' + u' x' w z + w' x z 4. (a) (a' +b')(a' + d)(a + c') (b) (b' + d')(a + b + d)(b + c + d) (c) (a + b + d)(b + c +d)(a + b + c)(b + c +d ) 7
Answers to hapter 4 Problems. (a) xy z +xy z + x y z + x y z + x yz + x yz (b) w x y z + w x y z + wx y z + wx y z + wxy z + wxy z + w x yz + w xyz (c) AD + AD + A D + A D + A D + A D + A D + A D (d) A D + A D + A D + A D + A D + A D + A D + A D + A D + A D + A D + AD. (a) (x + y + z )(x + y + z)(x +y + z )(x +y + z ) (b) (w +x+y +z )(w+x+y +z )(w+x +y +z)(w+x +y+z)(w+x+y +z)(w+x+y+z) 3. (a) m (3,5,7,,,3,4,5) = M (,,,4,6,8,9, ) (b) m (,,4,5,6,3,5) = M (,3,7,8,9,,,,4) (c) M (,,,3,4,6,,,4) = m(5,7,8,9,,3,5) (d) M (,,3,5,6,7,9,,3,4,5) = m(,4,8,,) (e) M(5,7,8,3,5,6-4,9,3) = m(-4,6,9-,4,5-8,3) (f) m(8,9,,.3,6,7,.,4,5,7,8,9) = M(-7,,4,5,8,9,,3,6,3,3) 4. (a) m (,,,4,6,8,9,) = M(3,5,7,,,3,4,5) (b) m (,3,7,8,9,,,,4) = M(,,4,5,6,3,5) (c) m (,,,3,4,6,,,4) = M(5,7,8,9,,3,5) (d) m (,,3,5,6,7,9,,3,4,5) = M(,4,8,,) (e) m (5,7,8,3,5,6-4,9,3) = M(-4,6,9-,4,5-8,3) (f) M(8,9,,.3,6,7,.,4,5,7,8,9) = m (-7,,4,5,8,9,,3,6,3,3) 5. (a) w x y z +w x y z+w x yz +w xy z +w xyz +wx y z +wx yz +wxy z +wxyz (b) (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 ) (c) w x y + z (d) (w + z ) (x + z ) (y + z ) 6. F(A,D,,) = m(,4,5,6,7,9,,,4) 7. m (,,,4,5,6,7,9,) + d(,3,4,5) = M(3,8,) D(,3,4,5) 8. (a) f = m(,3,6,8,9,,3,4) (b) g = m(3,4,5,6,,,3) (c) f g = m(,,7,,5) (d) f + g = m(,,,4,5,7,8,9,,,4,5) (e) f g = m (,8,9,4) (f) f g = m(4,5,) 8
(g) f + g = (f g) = m (,3,4,5,6,8,9,,,3,4) (h) f g = f g + f g = m(,8,9,4) + m(4,5,) = m(,4,5,8,9,,4) 9. (a) a' b c' + a b' c' + a' b' c + a b c (b) a' c' + a b' c (c) a + b + d e. (a) m(,3,4,5,7,,,4) (b) m(,3,5,6,8,3,4,5) (c) m(,,,3,4,5,6,8,9,,,3). F AD= =, F AD= = +, F AD= =, F AD= = +. F = D (A + E ) + D(AE ) + D () + DA( + E ) = A D + D E + ADE + AD 3. F A= = E + D 4. The correct given sub-functions are as follows: (corrections in red) F A = = DE + D F A = = ( + D )E F A = = + D E F AD = = F AD = = + E F = A DE + A D + A E + A D E + A + A D E + AD + AD + ADE = A DE + A D + A E + A D E + A + A D E + A + ADE = A DE + A D + A E + A D E + A + A D E + A + ADE = A E ( D + + D ) + A D + A + A D E + A + ADE = A E ( D + + D+ D ) + A D + A + A D E + A + ADE (onsensus Th.) = A E ( D + D + D ) + A D + A + A D E + A + ADE (onsensus Th.) = A DE + A DE + A D E + A D + A + A D E + A + ADE = A DE + DE + A D E + A D + A + A D E + A + ADE 5. f = M(,3,4,6) = ( A + + )( + ) ( A + ) 6. ircuit (a) x = a b + ab y = a bc z = b c + bc ircuit (b) x = a b y = a b + c z = ircuit (c) x = ab y = ab + c + d z = c + d 7. F = (A + + ) (A + D) ( + D) ( + D) = A D + D + D + A 8. F = A D + D + D + A 9
A D D F 9. Let the inputs of the (6,3,,) weighted code be A,,,D. Simplest SOP /V = D +A Simplest POS /V = (A + )( + )( + D). V(a,b,c,d) = m (,,,3,6,8,9,,,4) = M (4,5,7,,3,5). W(a,b,c,d) = m (8,9,,,) + d(,,,3,4,5) = M (3,4,5,6,7) D(,,,3,4,5) X(a,b,c,d) = m (7,8) + d(,,,3,4,5) = M (3,4,5,6,9,,,) D(,,,3,4,5) Y(a,b,c,d) = m (5,6,7,8,9,) + d(,,,3,4,5) = M (3,4,,) D(,,,3,4,5) Z(a,b,c,d) = m (4,5,,) + d(,,,3,4,5) = M (3,6,7,8,9,) D(,,,3,4,5) 3
Answers to hapter 5 Problems. (a) m (,,4,5,6) (b) m (-7,,,3,4) (c) m (-5,7-) (d) m (,,3,6,7,,4,6-,,4,6,8,3). (a) M (,4,5,7,8,-3,5) (b) M (,3,6-5,7,9,,3,4,5,7,8,9) 3. (a) Distinguished -cell: 3,5,8,4 Prime implicant: A D (3,7), A D (8,), D (5,7,3,5), A (,3,4,5) (b) Distinguished -cell: 4,6,,3 Prime implicant: (,3,5,7), (4,5,6,7), (,5,9,3), (,3,9,) (c) Distinguished -cell:,9,,5 Prime implicant: (,,8,), (,,8,9), (,5), (5,7), (7,5) (d) Distinguished -cell: 5,8,9,4 Prime implicant: (5,7), (,6,,4), (8,9,,), (,3,6,7), (,3,,) 4. (a) Distinguished -cell: 4,6 Prime implicate: (,,4,6), (,), (,9), (9,), (,), (,) (b) Distinguished -cell:,,,5 Prime implicate: (,,8,), (8,,,4), (4,5) (c) Distinguished -cell: 3,4,6,,3,4 Prime implicate: (3,), (4,6,,4), (,3) (d) Distinguished -cell:,5 Prime implicate: (,), (3,5), (,4), (4,), (,3) 5. (a) A D + A D + D + A (b) A + D + D (c) D + + D + A D or D + + D + A D (d) A D + D + A + or A D + D + A + A 6. (a) (A+D)(++D )(A ++ ) (b) (+D)(A +D)(A + + ) (c) (+ +D )( +D)(A + +) (d) (A++)(A + +D )( ++D) 7. (a) Distinguished -cell: 3,5,8,4,,3,6 Prime implicant: (,3,5,7,7,9,,3), (6,7), (6,4), (8,9,4,5), (,9,7,5), (7,9,5,7), (4,5,6,7) (b) Distingusished -cell:,4,,7 Prime implicant: (,5,9,3), (4,5), (5,7,3,5), (8,,6,3), (,3,3,3), (5,7), (6,7,3,3), (,9,7,5), (,,8,6), (7,5,3,3) 3
8. (a) E + D + A DE + A (b) A D + DE + D E + A E + AD + AD 9. (a) (++E)( +D+E)( + +E )(A + +E)(A+ ++D ) (b) (A++D +E )(A+ +D +E)(++D +E )(+D+E)(A + +D)( +D+E). f = A + AD + A f = A D + A + AD + A f 3 = A D + A + AD + A. (a) D + D + AD or D + D + A (b) + A (c) + A D E + D E + ADE. (a) f(a,,,d) = m (,,3,6,7,,4) (b) f(a,,,d) = m (,,3,4,5,7,,3) 3. (a) f = f D = m (9,,3,5) m (7,5) = m (7,9,,3) (b) f = f A D = m (,3,6,7) m (5,7) = m (,3,5,6) (c) f = f D = m (8,9,,,,3,4,5) m (7,5) = m (7,8,9,,,,3,4) (d) f = f D = m (,,4,5,8,9,,3) m (5,7,3,5) = m (,,4,7,8,9,,5) 4. (a) A + A D + A D = A D (b) A + A D + A D = A A D (c) A + A + AD + A D = A D (d) A + AD +A D = A D 5. A F D = F D = F D = F D = 3 4 6 5 7 4 5 8 9 3 6. (a) F = = A, F = a = A + D, F = = A, F = = AD (b) F = = A, F = a =, F = = A, F = = AD 3
7. (a) F ADE = =, F ADE = = +, F ADE = =, F ADE = =, F ADE = =, F ADE = =, F ADE = =, F ADE = = (b) F ADE = =, F ADE = =, F ADE = =, F ADE = =, F ADE = =, F ADE = =, F ADE = =, F ADE = = 33
Answers to hapter 6 Problems. (a), (b) f = A + + A D = [ (A ) ( ) (A D) ] (c) f = (A + + ) ( + ) (A + ) ( + + D) or (A + + ) ( + ) (A + ) (A + + D) (d) f = [ (A + + ) + ( + ) + (A + ) + ( + + D)] or [ (A + + ) + ( + ) + (A + ) + (A + + D)] (e) f = ( A + + A + D ) or ( A + + A + A D ). (a), (b) f = + A D + AE + D E = [ ( ) (A D) (AE) (D E) ] (c) f = (A + + ) ( + + D) (A + + + D ) (A + + E) ( + D + E) (d) f = [(A + + ) +( + + D) +(A + + + D ) +(A + + E) +( + D + E)] (e) f = ( A + D + A D + AE + D E ) 3. (b) F = m (7,,3,4,5) (c) F = M (-6,8,9,,) (d) F = A'D + A'D + A'D + AD' + AD = D + AD + AD + A + AD = D + AD + AD + A (e) F = (A + ) (A + ) (A + D) ( + ) ( + D) ( + D) (f) F' = [ (A + ) (A + ) (A + D) ( + ) ( + D) ( + D) ]' = (A + )' + (A + )' + (A + D)' + ( + )' + ( + D)' + ( + D)' = A'' + A'' + A'D' + '' + 'D' + 'D' (g) F' = [ D + AD + AD + A ]' = (D)' (AD)' (AD)' (A)' = ( A' + ' + ') ( A' + ' + D') ( A' + ' + D' ) ( ' + ' + D' ) (h) From (d), F = [ (A)' (AD)' (AD)' (D)' ]' (i) From (e), F = [(A + ) + (A + ) + (A + D) + ( + )' + ( + D)' + ( + D)' ]' (j) From (f), F' = [ (A'')' (A'')' (A'D')' ('')' ('D')' ('D')' ]' (k) From (g), F' = [(A' + ' + ')' +(A' + ' + D')' + (A' + ' + D')' + (' + ' + D')']' (l) From (f), F = ( A'' + A'' + A'D' + '' + 'D' + 'D' )' (m) From (d), F' = ( A + AD + AD + D )' 4. (a) f = ( A D + + D + A D ) (b) f = ( A D ) ( ) ( D ) ( A D ) (c) f = ( + D ) (A + + ) (A + + D) f = [ ( + D ) (A + + ) (A + + D) ] (d) f = ( + D ) + (A + + ) + (A + + D) 5. (a) f = + A D + DE + ADE (b) f = [ (A D ) (DE) (ADE ) ] (c) f = (A + + D + E) (A + + D) ( + + E ) (A + + ) (d) f = [ (A + + D + E) + (A + + D) + ( + + E ) + (A + + ) ] (e) f = ( A DE + AD + E + A ) 34
(f) f = ( A DE ) (AD ) ( E) (A ) (g) f = [ (A + + D) ( + D + E ) (A + D + E) ] (h) f = + (A + + D) + ( + D + E ) + (A + D + E) 6. (a) f(a,,,d) = A (D + ) + A ( + D) D ' D' A ' A' f (b) f(a,,,d) = A + A + A D = A ( + ) + A D A D ' f ' ' A' 7. (a) f(a,,,d) = (A++)(A+ +D)(A + +D)(A + +D ) = [A + ( + )( + D)] [A + ( + D)( + D )] ' D ' D ' D' A A' f (b) f(a,,,d,e) = (A++)(A +D)(A + )= (A++)(A + D) D' A A' f 35
8. (a) f(a,,,d) = A D + A + A + A D = D(A +A ) + (A + A ) = D (A ) + (A ) (b) f(a,,,d) = A + A + D + D = (A ) + D ( ) D A f A f D (a) (b) 9. EQ = (a b ) (a b ) GT = a b + (a b ) a b LT = a b + (a b ) a b. EQ = (a b ) (a b ) (a b ) GT = a b + (a b ) a b + (a b ) (a b ) a b LT = a b + (a b ) a b + (a b ) (a b ) a b 36
Answers to hapter 7 Problems. D D 7 : Output of 3-to-8 decoder D D 5 : Output of 4-to-6 decoder Logic Logic W Logic /E 3 4 5 6 7 X Y Z X Y Z /E /E D D 7 (3-to-8) / D 8 D 5 8 D D 7 (3-to-8) / D D 7 8. A 3 4 5 6 7 D D D D 3 D 4 D 5 D 6 D 7 A I /E I /E 3 3 D D 3 D 5 D 7 D D D 4 D 6 3. /d (b 3,b,b,b ) = M D(-5) = b 3 + b + b + b /d (b 3,b,b,b ) = M D(-5) = b 3 + b + b + b /d (b 3,b,b,b ) = M D(-5) = b + b + b /d 3 (b 3,b,b,b ) = M 3 D(-5) = b + b + b /d 4 (b 3,b,b,b ) = M 4 D(-5) = b + b + b /d 5 (b 3,b,b,b ) = M 5 D(-5) = b + b + b /d 6 (b 3,b,b,b ) = M 6 D(-5) = b + b + b /d 7 (b 3,b,b,b ) = M 7 D(-5) = b + b + b /d 8 (b 3,b,b,b ) = M 8 D(-5) = b 3 + b /d 9 (b 3,b,b,b ) = M 9 D(-5) = b 3 + b 4. F(A,,) = (A + )(A + ) = = m (3, 7) 5. (a) F(A,,) = m (3, 4, 6, 7) = (D + D + D + D 5 ) (b) F(A,,) = m (,, 5, 7) = (D + D 3 + D 4 + D 6 ) 6. (a) f(a,,,d) = m (,,3,4,5,6,8,9,,,4) = D + A + A + A 37
A D 3 4 5 6 7 f 7. A 3 4 5 6 7 F F 8. y = a 3 + a y = (a 3 + a )(a 3 + a ) /Idle = a 3 + a + a + a a 3 a y y a /Idle a 9. F = W X + W Y + WXZ + VW + VZ. A = w x = (wx) = w + x = (y+z)(y +z) = z D = y +z F = u w x + u vw + u vx + uv z + uvy + uvz 38
. I I I I 3 I 4 I 5 I 6 I 7 A Y. f = A D (I ) + A D(I ) + AD (I ) + AD(I 3 ) = A D ( + E) + A D() + AD ( + E) + AD( + E) = A D + A D E + A D + A D + A D E + AD + A DE = A + A D E + A D + AD + A E = A + A D E + D + D + A E = A + A D E + D + D + A E + A D E = A + D E + D + D + A E or f = A D + A D E + A D + A D + A D E + AD + A DE = m (,4-7,-5,7,9,,,5,7,3,3) = A + D E + D + D + A E 3. I = A +, I = A ( + E), I =, I 3 = A + A E 4. I = A, I = (A + )(A + ), I = A +, I 3 = (Three -input gates.) 39
Answers to hapter 8 Problems. 4-bit D inputs: A,,, D 4-bit excess-3 outputs: W, X, Y, Z W = m (5,6,7,8,9) + d(-5) X = m (,,3,4,9) + d(-5) Y = m (,3,4,7,8) + d(-5) Z = m (,,4,6,8) + d(-5) X X X X X 3 X X 4 X X X A 3 5 X 6 X X 7 X X 8 X X X 9 X X D 3 4 5 W X Y Z. W = m (,4,6,8,,,4,5) X = m (,,4,5,6,7,8,9,,,4,5) Y = m (,,4,5,,5) Z = m (,,3,4,5,6,7,8,9,,,4,5) X X X X X X X 3 X 4 X X X X A 3 5 X X X 6 X X X 7 X X 8 X X X 9 X X D X X X X X X X 3 4 X X X 5 X X X X W X Y Z 4
3. A D f = A D + D + AD f = A + A D + AD X X X X X X X X X X X X X X f 3 = D + A D X X X X X X X X X X X f 4 = A + D X X X f f f 3 f 4 4. A D X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X W X Y Z 5. f = A D + D + AD f = A + A D + AD f 3 = D + f 4 = A + D 6. V = A + + (Use two -input OR) W = D + D + AD (Use one 4-input OR) X = D + A + + A D (Use one 4-input OR) Y = A + AD (Use one -input OR) Z = X + A D (Use one -input OR) 4
Answers to hapter 9 Problems. Q = D = Q = Q = T = Q = (c), Q = JK =, Q =,,. haracteristic equation Q + = M Q + N Q, MN =, Q =, Q =, 3. haracteristic table for xy latch. Present inputs x y Present state Q Next state Q + haracteristic equation Q + = x + yq Necessary condition x + y = d d, xy = Q = Q =, 4
4. clock /preset /clear D Q 5. clock /preset /clear J K Q 43
Answers to hapter Problems.. 3. Q Q Present Next state, z State x = x = A A,, A, D, A,, D A,, Q Q Present Next state, z State x = x = A A, D,,, D, A, D,, Q Q Present Next state state x = x = Z A D A D A A D D 4. D = Q 3 D = Q D = Q D 3 = Q 5. D 3 = x Q Q Q + x Q Q D = x Q Q + x Q Q + x Q Q + x Q D = x Q z = Q Q + x Q Q + x Q 6. (a) D = x D = Q z = Q Q (b) J = x, K = x J = y K = y z = y y 7. (a) D = x Q D = (x + Q + Q )(x + Q + Q )(x + Q + Q ) z = x Q Q (b) T = x Q Q + xq + Q Q T = x Q + xq Q z = x Q Q (c) J = x Q K = x + Q J = x Q K = x Q z = x Q Q 44
8. J = x K = x J = xq Q K = Q J = x Q + Q Q K = x + Q z = Q Q + xq + x Q Q Q + xq Q 9. A/ / E/ D/ /. y y Y Y x = x = Z A D D = xy + y y D = x Z = y y. y y A Unuse d Y Y, Z x = x =,,,,,, d d, d d d, d D = xy D = x Z = x y. Presen Next state y y y t state x = x = Z A A A D D E E E F F E A T = x y y y + xy y T = x(y y ) + x y y T = y ( x y ) Z = y 45
Answers to hapter Problems. (a) (b) (c) (d) (e) (f). (a) 3 (b) +96 (c) (d) 6 (e) 8 (f) +7 3. (a) (b) (c) (d) (e) (f) 4. (a) A = = +85 = = + A = = 85 = = A + = + = 85 + = +95 A = + = () 85 = +75 (it in parentheses discarded) A + = + = 85 + = 75 A = + = () 85 = 95 (b) A = = +7 = = +4 A = = 7 = = 4 A + = + = 7 + 4 = +49 The greatest positive number for 8-bit signed number is +7. The result of A+ in 's complement arithmetic is a negative number, which is an overflow. A = + = () 7 4 = +65 A + = + = 7 + 4 = 65 A = + = () 7 4 = 49 The smallest negative number for 8-bit signed number is 8. The result of A in 's complement arithmetic is a positive number, which is an overflow. (c) A = = - = = +47 A = = + = = 47 A + = + = () + 47 = +5 A = + = () 47 = 69 A + = + = + 47 = +69 A = + = 47 = 5 (d) A = = 8 = = +7 A = = (?) = = 7 A + = + = 8 + 7 = A = + = () 8 7 = 55 ( can't do A + & A because A = +8 can't be represented by 8 bits.) 46
5. 6. RESET S A Z = X? X? Z = Z = Z = S X? Z = Z = X? Z = X? S4 D Z = X? X? Z = Z = E Z = S5 X? Z = X? Z = S3 X? Z = Z = S X? Z = Z = 47
7. y 3 = T + T + T = T 3 y = X T + X T + X T 3 = X ( T +T + T 3 ) = X T y = X T + T 3 y = X T + X T + X T + T 3 = X ( T +T + T ) + T 3 = X T 3 + T 3 = X + T 3 + + 8. D = Q = x (Q + Q 5 ) D = Q = x Q + D = Q = Q + + D 3 = Q 3 = x (Q + Q 4 ) D 4 = Q 4 = Q 3 + D 5 = Q 5 = x (Q + Q 4 + Q 5 ) (Refer to Figure. for RESET.) 9.,, A, A, A +, A, A, A +. (a) (b) (c) (d) (e) (f) (g) (h) 48