1. Determine whether these propositions are equivalent: and

. Justify your answer.

2. On the island of knights and knaves, you encounter two people, *A* and *B*. Person *A* says “*B* is a knave”. Person *B* says “At least one of us is a knight”. Determine whether each person is a knight or a knave.

Assuming that knights always tell the truth and knaves always lie:

Case 1: Assume *A* is a knight, *B *is a knight:

Case 2: Assume *A* is a knight, *B *is a knave:

Case 3: Assume *A* is a knave, *B *is a knight:

Case 4: Assume *A* is a knave, *B *is a knave:

3. If *P*(*x*, *y*) means *x* + 2*y* = *xy*, where *x* and *y* are integers, determine the truth value of

.

4. Show that the hypotheses “I left my notes in the library or I finished the rough draft of the paper” and “I did not leave my notes in the library or I revised the bibliography” imply that “I finished the rough draft of the paper or I revised the bibliography.”

5. Prove: If *x* and* y* are odd integers, then *x* + *y* is even.

6. Give a proof by cases that *x* ≤ |*x*| for all real numbers *x*.

Case *x* ≥ 0:

Case *x* < 0:

7. Prove or disprove: If *A*, *B*, and *C* are sets, then .

Part 1:

Part B:

8. Find

9. Let *A* = {*a*, *b*, *c*}. True or false: .

10. Prove that between every two rational numbers there is a rational number.

11. Give an example of a function *f*: *Z* → *N* that is one-to-one and not onto *N*.

Purchase the answer to view it