Math proofs

1.   Suppose that x2y + xy2 + y3 = x3:

a) Using the method of proof by contrapositive, show that if x and y are not both zero, then y ≠ 0.

b) Using the method of proof by contradiction, show that if x and y are not both zero, then y ≠ 0.

2.  Prove the following two statements:

a) For every integer n, 72 | n iff 8 | n and 9 | n.

b) It is not true that for every integer n, 90 | n iff 6 | n and 15 | n.

3.   Let a, b, and c be real numbers with a ≠ 0. Prove that limx→c (ax + b) = ac + b.

4.   Suppose that {Ai | i ∈ I} is an indexed family of sets and B is a set. Prove that
(∩i∈I Ai) × B = ∩i∈I (Ai × B).

5.   Suppose R is a partial order on A and S is a partial order on B. Define a relation T on A × B such that (a1, b1) T (a2, b2) iff a1 R a2 and b1 S b2. Is T a partial order on A x B? Either provide a proof to show that this is true or provide a counterexample to show that this is false.