Prove that if a and b are countable sets then so is a b

Question

Problem 1 - Prove that if A and B are countable sets, then so is A ∪ B

Problem 2 - Prove that the set {0, 1}*, of finite binary strings, is countable.

Problem 3 - For each of the following sets indicate whether it is finite, countably infinite, or uncountable.

1. The set of solution to x³ - x = -0.1

Problem 4 - Describe which of the following sets have bijections between them.

Z                         R

C                         Q

P(Z)                     P(∅)

P(P(∅))                {0, 1}*

{0, 1}^ω                {T, F}

P({T, F})              P({0, 1}^ω)