Exercises 2 (22 pts)

1. (2 pts) Using the postulates of Boolean algebra and the theorems given in the text's Appendix prove:

a. a'b' + ab + a'b = a' + b
b. a' + a(a'b + b'c)' = a'+b+c'

2. (2 pts) Prove the following equivalence for switching expressions by evaluating them for all possible assignments.

        xyz + yw + x'z' + xy' = y'z' + yw + xz + x'y'z
The proof is done by completing the following table. The xyz+yw+x'z'+xy' column should be identical to the y'z'+yw+xz+x'yz' column.
w x   y z |xyz|yw |x'z'|xy'|xyz+yw+x'z'+xy' |y'z'|yw |xz |x'yz'|y'z'+yw+xz+x'yz'
0 0 0 0 | 0 | 0 | 1 | 0 |            1              | 1 | 0 | 0 |   0 |      1
0 0 0 1 |     |     |     |    |                            |    |      |    |       |____
0 0 1 0 |     |     |     |    |                            |    |      |    |       |____
0 0 1 1 |     |     |     |    |                            |    |      |    |       |____
0 1 0 0 |     |     |     |    |                            |    |      |    |       |____
0 1 0 1 |     |     |     |    |                            |    |      |    |       |____
0 1 1 0 |     |     |     |    |                            |    |      |    |       |____
0 1 1 1 |     |     |     |    |                            |    |      |    |       |____
1 0 0 0 |     |     |     |    |                            |    |      |    |       |____
1 0 0 1 |     |     |     |    |                            |    |      |    |       |____
1 0 1 0 |     |     |     |    |                            |    |      |    |       |____
1 0 1 1 |     |     |     |    |                            |    |      |    |       |____
1 1 0 0 |     |     |     |    |                            |    |      |    |       |____
1 1 0 1 |     |     |     |    |                            |    |      |    |       |____
1 1 1 0 |     |     |     |    |                            |    |      |    |       |____
1 1 1 1 |     |     |     |    |                            |    |      |    |       |____

3. (2 pt) Find the sum of minterms and product of maxterm expressions for the following switching function:

a b c | f
0 0 0 | 0
0 0 1 | 1
0 1 0 | 1
0 1 1 | 1
1 0 0 | 0
1 0 1 | 1
1 1 0 | 0
1 1 1 | 0

4. (2 pts) Obtain the equivalent sum of minterms and product of maxterms for the following expressions. Use m- and M- notation.

a. a'b + ac + bc
b. (ab + c)(d'e + f)

5. (2 pts) Convert the following canonical expressions of (w, x, y, z) into the other canonical forms (e.g. convert PM to Sm, etc.).

a. Sm(0,1,3,4,11,12,14,15)
b. PM(0,1,3,5,6,9,10,13)

6. (8 pts) Recall the 7-segment display is used to display the digits from 0-9 on calculators and many other displays. The 7-segment display appears as:

where each segment is ON or OFF dependent upon the digit to be displayed. For example, to display the digit 0 would require segments a, b, c, d, e, and f to be ON with segment g OFF. To display the digit 2 would require segments a, b, g, e, and d to be ON and f and c segments OFF as in the above.

a. Obtain the sum of products and product of sums for the switching functions f(x₃,x₂,x₁,x₀) and g(x₃,x₂,x₁,x₀), that is the function that determines when f or g segment is on or off.
b. Obtain the sum of minterms and product of maxterms for the switching functions f(x₃,x₂,x₁,x₀) and g(x₃,x₂,x₁,x₀).

7. (1 pt) Determine the switching function (truth table) represented by the following switching expression.

E(x,y,z) = xyz+x'y+xyz'

8. (3 pts) A stairwell has four floors and one switch per floor to control the lights. If all switches are DOWN, the lights are OFF; any single change in the position of any switch changes the state of the lights (if lights state is ON, then a switch change in position changes the lights state to OFF). Describe the combinational system required to control the lights. Give: