H3: Equivalence

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

Additional Materials & Formats
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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

1. Use Venn diagrams to prove whether $(x_1 \lor x_2 \lor x_3) \land (x_1 \lor x_2 \lor \lnot x_3) = x_1 \lor x_2$

2. Use truth tables to prove whether $x = (x \lor y) \land (x \lor \lnot y)$

3. Use K-maps or Venn diagrams to prove whether $xy + yz + \overline{x}z = xy + \overline{x}z$

4. Use truth tables to prove the distributive rule (covered in the next lecture): $x + yz = (x+y)\cdot (x+z)$

5. Write 3 unique Boolean expressions that can be represented by the following Venn diagram:

6. Write 3 unique Boolean expressions that can be represented by the following truth table:

   x	y	z	Out
   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	1

7. Draw a K-map for the expression $xy + yz + xz$.

8. Draw a K-map for the expression $(x + y) \cdot (y + z) \cdot (x + z)$.

9. Draw 2 unique circuit diagrams for the following truth table:

   x	y	z	Out
   0	0	0	0
   0	0	1	1
   0	1	0	1
   0	1	1	0
   1	0	0	1
   1	0	1	1
   1	1	0	0
   1	1	1	1

10. Write two Boolean expressions with at least 3 variables each. Use your preferred method to prove whether they are equivalent. My favorite submission will be featured (anonymously) in our next lecture to demonstrate how much easier it can be to use algebraic manipulation.