Math questions help

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 + 2y = 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.