need help with the following questions:

Problem 1)

Let A = {x, y, z} and B = {p, r}. How many functions are there from A to B?

Problem 2)

Let f : (R R +) R be a function given as f(x, y) = |2x| 3y (where R + is the set of positive real numbers). Answer the following questions, and explain your answers. (a) Is f one-to-one? (b) Is f onto?

Problem 3)

Let A and B be any two subsets of the same universal set U. Is it always true that P(A) P(B) = P(A B)? If your answer is "yes", then prove it; otherwise, show a counterexample.

Problem 4)

Let A, B, C be subsets of the same universal set U. Which one of the following sets is equal to (A B) C for all A, B, C? [No justification needed.] a) A B C b) (A C) (B C) c) C A B d) (A C) (B C) e) C A B.

Problem 5)

Prove that if n 5 + 3 is even, then n is odd.

Problem 6)

Prove that if a + b 100, where a and b are integers, then a 75 or b 25.

Problem 7)

Let D = E = R. Write negations for each of the following statements and determine which is true, the given statement or its negation. Explain your answer. (i) x D, y E such that xy + 2 = x. (ii) x D such that y E, x > y.

Problem 8)

Suppose that {T, S, R, Q} is an inconsistent set of propositions. Is it always true that (T S R Q) W is a tautology for any proposition W? Explain your answer.

Problem 9)

Construct a truth table for the proposition (p q) (p q). Is this proposition a tautology, a contradiction, or a contingency?

Problem 10)

Translate the given statement into propositional logic using the propositions provided. "You can graduate only if you have completed the requirements of your major, you have finished all the paperwork and you do not have an overdue library book." Express your answer in terms of g: "You can graduate", p: "You have finished all the paperwork", r: "You have completed the requirements of your major", and b: "You have an overdue library book".