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.
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