H4: Boolean algebra

Time Estimate1-2 hours   Grade Impact1%   DueSep 14 @ 12pm  

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Helpful Tip
WaveDrom is an easy place to generate timing diagrams.

A vector tool like Inkscape allows you to take the union and intersection of two shapes, which is what I used to generate Venn diagrams.

Notes

Show your work. You must show all steps of your work for full credit. While calculators can perform these calculations, demonstrating your process ensures understanding and mastery of the concepts.

Notation. To encourage flexibility between notations, the problems in this assignment use a variety of notations. We have discussed each style already; review the textbook or your notes if you need help.

Exercises

For exercises 1-5, use the theorems of Boolean algebra to prove whether the equations are equivalent. Recall that proofs require rigorous mathematical effort, and they should end with “QED” or a square.

1. $A + \overline{A}BC \overset{?}{=} A + BC$

2. $(A + B)(\overline{A} + \overline{B}C) \overset{?}{=} \overline{A}B + A\overline{B}C$

3. $\overline{A + BC} \overset{?}{=} \overline{A},\overline{B} + \overline{A},\overline{C}$

4. $(A + B)(\overline{A} + C)(B + C) \overset{?}{=} (A + B)(\overline{A} + C)$

5. $(A + B)(\overline{A} + C) \overset{?}{=} AC + \overline{A}B + BC$

Exercises 6-10 are based on L9.

6. Complete a Sum-of-Products expression representing F for the truth table below.

   A	B	C	F
   0	0	0	0
   0	0	1	1
   0	1	0	1
   0	1	1	0
   1	0	0	1
   1	0	1	0
   1	1	0	0
   1	1	1	0

7. Complete a Sum-of-Products expression representing F for the truth table below.

   A	B	C	F
   0	0	0	0
   0	0	1	0
   0	1	0	0
   0	1	1	1
   1	0	0	0
   1	0	1	1
   1	1	0	1
   1	1	1	0

8. Complete a Product-of-Sums expression representing F for the truth table below.

   A	B	C	F
   0	0	0	0
   0	0	1	1
   0	1	0	0
   0	1	1	1
   1	0	0	1
   1	0	1	1
   1	1	0	1
   1	1	1	1

9. Complete a Product-of-Sums expression representing F for the truth table below.

   A	B	C	F
   0	0	0	1
   0	0	1	0
   0	1	0	1
   0	1	1	1
   1	0	0	1
   1	0	1	1
   1	1	0	1
   1	1	1	0

10. Create a unique 3-variable truth table. Write PoS and SoP expressions for it, then use the theorems of Boolean algebra to simplify your expressions. Describe the complexity of the expressions.