Computer Architecture

Chapter 6 Solutions:

• 1. Rework problem 4 from the chapter 4 homework. Optimize the design and draw the circuit.
  See the chapter 4 solution.
• 2. Rework problem 7 from the chapter 4 homework. Optimize the design and draw the circuit.
  See the chapter 4 solution.
• 3. Vahid 6.3
  ab + bc + ac
• 4. Vahid 6.4
  a + c
• 5. Vahid 6.5
  bc' + abd
• 6. Vahid 6.6
  ab + a'b'd'
• 7. Vahid 6.10 part A
  Both prime implicants are essential: c, a'b
• 8. Vahid 6.11 part B
  Essential prime implicants: ab'c', a'bc
  Non-essential prime implicants: bcd, abd, ac'd
• 9. Find all the prime implicants for the following Boolean functions and determine what are essential:
  • A. F(W,X,Y,Z) = SUMm(0,1,3,4,6,7,11,13,15)
    Essential prime implicants: CD, ABD
    Non-essential prime implicants: A'BC, A'BD', A'C'D', A'B'C', A'B'D
  • B. F(A,B,C,D) = SUMm(3,5,11,13,14,15)
    Essential prime implicants: BC'D, ABC, B'CD
    Non-essential prime implicants: ABD, ACD
  • C. F(W,X,Y,Z) = PRODUCTM(2,4,5,6,7,8,10,11,15)
    Essential prime implicants: WXZ', W'X'Y', W'X'Z
    Non-essential prime implicants: WXY', WY'Z, X'Y'Z
  • D. F(W,X,Y,Z) = PRODUCTM(0,1,4,5,6,11,12,13,15)
    Essential prime implicants: WX'Y', W'YZ, WYZ'
    Non-essential prime implicants: W'X'Y, X'YZ', WX'Z'
• 10. Simplify the following functions using a map:
  • A. F(A,B,C,D) = SUMm(0,4,6,7,9,11,12,13,14,15)
    Possible solutions:
    F = A'C'D' + AB + AD + BC
    F = A'C'D' + BD' + AD + BC
    F = (A' + B + D)(A + C + D')(A + B + C')
  • B. F(A,B,C,D) = PRODUCTM(2,4,5,6,7,8,10,12,14)
    Possible solutions:
    F = (A + B')(C' + D)(A' + D)
    F = A'B'C' + B'D + AD
  • C. F(X,Y,Z) = SUMm(1,2,4,7)
    Possible solutions:
    F = X'Y'Z + X'YZ' + XY'Z' + XYZ
    F = (X + Y + Z)(X + Y' + Z')(X' + Y + Z')(X' + Y' + Z)
  • D. F(X,Y,Z) = PRODUCTM(0,2,3,5,7)
    Possible solutions:
    F = (X + Z)(X' + Z')(X + Y')
    F = (X + Z)(X' + Z')(Y' + Z')
    F = X'Y'Z + XZ'
• 11. Simplify the following Boolean expressions using a map:
  • A. WXY'Z' + W'XY' + W'X'Z + X'YZ' + WX'Y
    Possible solutions:
    XY'Z' + W'Y'Z + X'Y
    (X + Y + Z)(X' + Y')(W' + Y + Z')
  • B. A'BC' + A'CD + AC'D' + AB'D + ABC
    Possible solutions:
    CD + AD + A'BC' + ABC
    (B + D)(A + B + C)(A' + C + D)(A + C' + D)
• 12. Simplify the following expressions in sum-of-products and product of sums form:
  • A. C'D + (AD + A'B)C' + (AC + A'B')D
    Possible SOP solutions:
    B'D + AD + A'BC'
    possible POS solutions:
    (A + B' + C')(B + D)(A' + D)
  • B. (A + B + D')' + (A' + B' + C')' + (A'C' + AC)D + A'CD'
    Possible SOP solutions:
    A'C'D + A'B'C + BCD' + ACD
    A'C'D + B'CD + ABC + A'CD'
    Possible POS solutions:
    (C + D)(A' + C)(A + B' + C' + D')(A' + B + D)
  • C. X'Y + XZ + YZ
    Possible SOP solutions:
    X'Y + XZ
    Possible POS solutions:
    (X + Y)(X' + Z)
• 13. Simplify the following functions F together with the don't care conditions d:
  • A. F(A,B,C) = SUMm(0,1,5,6), d(A,B,C) = m(2,4)
    Possible solutions:
    B' + C'
  • B. F(A,B,C) = SUMm(0,4), d(A,B,C) = m(3,7)
    Possible solutions:
    B'C'
  • C. F(W,X,Y,Z) = SUMm(4,5,6,9,12), d(WXYZ) = m(0,2,3,7,8,11,15)
    Possible solutions:
    Y'Z' + W'X + WX'Y'
    Y'Z' + W'X + WX'Z
    (W + X)(W' + Y')(W' + X' + Z')