Introduction to Digital Logic and Computer Design

Chapter 4 homework:

1. Demonstrate by means of truth tables the validity of the following identities:
• A.  A + BC = (A + B)(A + C)
• B.  X Z' + X'Y' + YZ = Y'Z' + X'Z + XY
• C.  (XYZ)' = X' + Y' + Z'
2. Prove the identity of each of the following Boolean equations using algebraic manipulation:
• A.  A'B'CD + A'BC'D' + AB'C'D + ABCD' + BD = A'BC' + A'CD + ABC + AC'D
• B.  (X + Y')(X' + Z)(X' + Y) = (X + Y')(Y' + Z)(X' + Y)
• C.  (A + B + C)(A' + B + C')(A + B + C')(A' + B + C) = B
3. Using DeMorgan's theorem, express the function:
   • A'D + (BC)' + A(BC)'
   • A. with only OR and complement operations.
   • B. with only AND and complement operations.
4. Find the complement of the following expressions:
• A.  XY' + X'YZ + X'Y'Z'
• B.  (A' + B' + C)(A' + B)C'
• C.  (X' + Y)(Y' + Z)(X' + Z)
5. Simplify the following Boolean expressions to a minimum number of literals:
• A.  (Y + Z')(Y + Z)
• B.  ABC'D + C(A + B) + AC' + AC'D + B + CB'
• C.  (A + B' + C')(A' + C')
6. Draw the logic diagram for the following Boolean expressions. Show inverters when necessary. The diagram should correspond exactly to the equation.
• A.  (AB' + (CD)'(A + B))C
• B.  ((X + Y)'(Z'X + Y))' + X'YZ
• C.  (A + BC)(B' + AC)' + A'C'
7. Find all of the prime implicants for the following Boolean functions, and determine which are essential.
• A.  F(A,B,C,D) = SUMm(0,1,3,4,6,7,11,13,15)
• B.  F(A,B,C,D) = SUMm(3,5,11,13,14,15)
• C.  F(W,X,Y,Z) = PRODUCTM(2,4,5,6,7,8,10,11,15)
• D.  F(W,X,Y,Z) = PRODUCTM(0,1,4,5,6,11,12,13,15)
8. Simplify the following functions using a map:
• A.  F(A,B,C,D) = SUMm(0,4,6,7,9,11,12,13,14,15)
• B.  F(A,B,C,D) = PRODUCTM(2,4,5,6,7,8,10,12,14)
• C.  F(X,Y,Z) = SUMm(1,2,4,7)
• D.  F(X,Y,Z) = PRODUCTM(0,2,3,5,7)
9. Simplify the following Boolean expressions using a map:
• A.  WXY'Z' + W'XY' + W'X'Z + X'YZ' + WX'Y
• B.  A'BC' + A'CD + AC'D' + AB'D + ABC
10. Simplify the following expressions in (1) sum-of-products and (2) product of sums form:
• A.  C'D + (AD + A'B)C' + (AC + A'B')D
• B.  (A + B + D')' + (A' + B' + C')' + (A'C' + AC)D + A'CD'
• C.  X'Y + XZ + YZ
11. 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) = SUMm(2,4)
• B.  F(A,B,C) = SUMm(0,4), d(A,B,C) = SUMm(3,7)
• C.  F(W,X,Y,Z) = SUMm(4,5,6,9,12), d(W,X,Y,Z) = SUMm(0,2,3,7,8,11,15)
12. Implement the simplified functions from problem 11 with NAND gates.
13. Redraw the logic diagrams from problem 6A using (1) NAND gates and (2) NOR gates using inverters when necessary.
14. Write a VHDL description of a three-input-one-output circuit. The Boolean expression is:
• X = A'(Cin) + A'B + B(Cin)