1. Expand each of the following functions into a canonical sum-of-products expression.
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1 CHAPTER 4 PROLEMS 1. Expand each of the following functions into a canonical sum-of-products expression. (a) F(x, y, z) = xy + y z + x (b) F(w, x, y, z) = x y + wxy + w yz (c) F(A,,C,D) = AC + CD + C D (d) F(A,,C,D) = A + CD + D 2. Expand each of the following functions into a canonical product-of-sums expression. (a) F(x, y, z) = (x + y )(x + z ) (b) F(w, x, y, z) = (x + y + z )(w + z) 3. Find the minterm and maxterm list forms for each of the following functions. (a) F(A,,C,D) = A + C D + A' D (b) F(A,,C,D) = (A + + C') (A + C' + D') (A' + D) (c) F(A,,C,D) = ( A + )(C + + D)(C + D) (d) F(A,,C,D) = ( + C )(A + + C )D (e) F(A,,C,D,E) = (A + )( + C + D + E)(C + E ) (f) F(A,,C,D,E) = (A )D + C DE + AD 4. Find the minterm and maxterm list forms for the complement of each of the functions in Problem Given f(w,x,y,z) = m(0, 1, 2, 4, 6, 8, 10, 12, 14), find (a) the canonical sum-of-products expression for f. (b) the canonical product-of-sums expression for f. (c) the simplest sum-of-products expression for f. (d) the simplest product-of-sums expression for f. 6. Find the minterm list form for F(A,D,,C) if F(A,,C,D) = m(0, 1, 3, 5, 7, 9, 10, 13, 14) 7. Find the minterm and maxterm list forms for f(a,b,c,d) if a b = 11 (both a and b are 1) never occur in input states. f(a,b,c,d) = a' b + a c d + a' d' + c' d 8. Given f(w,x,y,z) = m (1,2,4,5,7,11,12,15) and g(w,x,y,z) = m (0,1,2,7,8,9,12,14,15), 1
2 find the minterm list forms for (a) f (b) g (c) f g (d) f + g, (e) f g (f) f g (g) f + g (h) f g 9. Simplify each of the following expressions to a sum-of-products expression. (a) (b) (c) a' b' c a (a b + c') (a' b 1) (a b )+ (c c' 0) (a + d e) (a + b') 10. Find the minterm list form for each of the following functions. (a) (b) (c) f(a,b,c,d) = a'b b'cd cd f(a,b,c,d) = (a' + b) (c + d) (a + c + d ) f(a,b,c,d) = 1 bcd acd 11. Given F(A,,C,D) = (A C D) + D + A CD find all the sub-function of F with A and D as expansion variables. Express each sub-function as a simplest sum-of-products expression. 12. Find the simplest sum-of-products expression for F(A,,C,D,E) using the following sub-functions. F D = 00 = A + E F D = 01 = AE F D = 10 = 0 F D = 11 = A(C + E ) 13. Find the simplest sum-of-products expression for the sub-function F AC = 00 of a 5- variable function F(A,,C,D,E) if F ACD = 000 = E F ACD = 001 = + E 14. Find the minterm list form for F(A,,C,D,E) using the following sub-functions. F A = 00 = DE + C D F A = 01 = (C + D )E F A = 10 = C + D E F AD = 101 = C F AD = 111 = C + E 15. Given in Figure P4.1 is the timing diagrams of a oolean function F(A,,C), find the simplest product-of-sums expression for F. 2
3 A C F Figure P Find the simplest sum-of-products expressions for X, Y, and Z in each of the logic circuits in Figure P4.2. Figure P4.2 Circuit (a) a b c X Y Z Circuit (b) a b c d X Y Z Circuit (c) a b c d X Y Z 3
4 17. Find the simplest sum-of-products and product-of-sums expressions for the logic circuits in Figure P4.3 without eliminating the internal and output inversions. Figure P4.3 A C D D C F 18. Minimize the number of internal and output inversions for the circuit in Figure P.4.3 and then determine the simplest sum-of-products expression for F. 19. A switching circuit has four inputs a, b, c, d and an output /V. The input combination abcd is a (6, 3, 1, 1) weighted-code given in the rightmost column of Table 2.3. /V is an active-low output which is asserted if and only if the input combination is a valid (6, 3, 1, 1) weighted code. Construct a truth table for /V and find the simplest sum-of-products and simplest product-of-sums expressions for /V. 20. A switching circuit has four inputs a, b, c, d and an output V. The input combination abcd is the reflected code given in Table 2.4. V is an active-high output which is asserted if and only if the input combination is a valid reflected code. Find the minterm and maxterm list forms for V. 21. A switching circuit has four inputs a, b, c, d and four outputs W, X, Y, Z. The input combination abcd is an excess-3 code and the output combination WXYZ is the reflected code given in Table 2.4. Find the minterm and maxterm list forms for W, X, Y, and Z. 4
5 ANSWERS TO CHAPTER 4 PROLEMS 1. (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) ACD + ACD + A CD + A CD + A C D + A C D + A C D + AC D (d) A C D + A C D + A CD + A CD + A CD + A CD + A C D + A CD + A C D + A CD + AC D + ACD 2. (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 (3,5,7,11,12,13,14,15) = M (0,1,2,4,6,8,9,10 ) (b) m (0,1,4,5,6,13,15) = M (2,3,7,8,9,10,11,12,14) (c) M (0,1,2,3,4,6,10,12,14) = m(5,7,8,9,11,13,15) (d) M (1,2,3,5,6,7,9,11,13,14,15) = m(0,4,8,10,12) (e) M(5,7,8,13,15,16-24,29,31) = m(0-4,6,9-12,14,25-28,30) (f) m(8,9,11,12.13,16,17,20.21,24,25,27,28,29) = M(0-7,10,14,15,18,19,22,23,26,30,31) 4. (a) m (0,1,2,4,6,8,9,10) = M(3,5,7,11,12,13,14,15) (b) m (2,3,7,8,9,10,11,12,14) = M(0,1,4,5,6,13,15) (c) m (0,1,2,3,4,6,10,12,14) = M(5,7,8,9,11,13,15) (d) m (1,2,3,5,6,7,9,11,13,14,15) = M(0,4,8,10,12) (e) m (5,7,8,13,15,16-24,29,31) = M(0-4,6,9-12,14,25-28,30) (f) M(8,9,11,12.13,16,17,20.21,24,25,27,28,29) = m (0-7,10,14,15,18,19,22,23,26,30,31) 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,,C) = m(0,4,5,6,7,9,11,12,14) 7. m (0,1,2,4,5,6,7,9,11) + d(12,13,14,15) = M(3,8,10) D(12,13,14,15) 8. (a) f g = m(1,2,7,12,15) (b) f + g = m(0,1,2,4,5,7,8,9,11,12,14,15) (c) f g = m (0,8,9,14) (d) f + g = (f g) = m (0,3,4,5,6,8,9,10,11,13,14) 5
6 (e) f g = f g + f g = m(0,8,9,14) + m(4,5,11) = m(0,4,5,8,9,11,14) 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 10. (a) m(2,3,4,5,7,10,11,14) (b) m(1,3,5,6,8,13,14,15) (c) m(0,1,2,3,4,5,6,8,9,11,12,13) 11. F AD=00 = C, F AD=01 = + C, F AD=10 =, F AD=11 = + C 12. F = D (A + E ) + D(AE ) + D (0) + DA(C + E ) = A D + D E + ADE + ACD 13. F AC=00 = E + D 14. The correct given sub-functions are as follows: (corrections in red) F A = 00 = DE + C D F A = 01 = (C + D )E F A = 10 = C + D E F AD = 110 = C F AD = 111 = C + E F = A DE + A C D + A CE + A D E + A C + A D E + ACD + ACD + ADE = A DE + A C D + A CE + A D E + A C + A D E + AC + ADE = A DE + A C D + A CE + A D E + AC + A D E + AC + ADE = A E ( D + C + D ) + A C D + AC + A D E + AC + ADE = A E ( D + C + CD+ D ) + A C D + AC + A D E + AC + ADE (Consensus Th.) = A E ( D + CD + D ) + A C D + AC + A D E + AC + ADE (Consensus Th.) = A DE + A CDE + A D E + A C D + AC + A D E + AC + ADE = A DE + CDE + A D E + A C D + AC + A D E + AC + ADE 15. f = M(0,3,4,6) = ( A + + C )( + C ) ( A + C ) 16. Circuit (a) x = a b + ab y = a bc z = b c + bc Circuit (b) x = a b y = a b + c z = 1 Circuit (c) x = ab y = ab + c + d z = c + d 17. F = (A + + C ) (A + D) ( + D) (C + D) = A D + D + C D + A C 6
7 D C A F (c) (a) (b) F(x,y) y Figure x 4.4 Logic symbol for NAND 18 F = A D + D + C D + A C A C D D C F 19. Let the inputs of the (6,3,1,1) weighted code be A,,C,D. Simplest SOP /V = C D +AC Simplest POS /V = (A + C )( + C )(C + D) 20. V(a,b,c,d) = m (0,1,2,3,6,8,9,10,11,14) = M (4,5,7,12,13,15) 21. W(a,b,c,d) = m (8,9,10,11,12) + d(0,1,2,13,14,15) = M (3,4,5,6,7) D(0,1,2,13,14,15) X(a,b,c,d) = m (7,8) + d(0,1,2,13,14,15) = M (3,4,5,6,9,10,11,12) D(0,1,2,13,14,15) Y(a,b,c,d) = m (5,6,7,8,9,10) + d(0,1,2,13,14,15) = M (3,4,11,12) D(0,1,2,13,14,15) Z(a,b,c,d) = m (4,5,10,11) + d(0,1,2,13,14,15) = M (3,6,7,8,9,12) D(0,1,2,13,14,15) 7
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