20 May Logic And Set Theory
Part I
Demonstrate DeMorgan’s Laws using a Venn diagram.
- Draw a Venn diagram showing the elements of sets A, B, and the universe for all 4 regions.
- Draw a second diagram showing only the elements of the complement of set A.
- Draw a third diagram showing only the elements of the complement of set B.
- Draw a fourth diagram showing the union of steps 2 and 3 for the first DeMorgan’s law or the intersection of steps 2 and 3 for the second DeMorgan’s law.
- Draw a fifth diagram showing the elements of the intersection of sets A and B for the first DeMorgan’s law or the union of sets A and B for the second DeMorgan’s law.
- Finally, draw the last diagram showing the complement of step 5. Compare the results from step 4 against those in step 6 to prove both DeMorgan’s laws. In the absence of data elements, you can use 4 different colors to clearly indicate the regions of the universe and follow all of the steps.
Part II
Define two propositions (simple statements that can be either true or false). Give a real-world example of 2 propositions that r and s can represent. Call them r and s. Create a truth table that shows all values of the following:
| Proposition | Definition | Your example explained |
| r | ||
| s | ||
| ¬ r | ||
| ¬s | ||
| r ∧ s | ||
| ¬ r v s | ||
| r ∧ ¬s |
Interpret the columns of the truth table for those proposition examples. Interpret the operations on the propositions and the values in the table based on the operations and the values of r and s.
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