(Sheet 2) Pudding - Countability

Here are three countability questions for you to ponder.

1. Sheet 2 Q7 asks you to show that any open interval \( (a,b) \) in the real numbers contains a rational number and contains an irrational number.  Can you go further, and show that this interval \( (a,b) \) in fact contains uncountably many irrational numbers?
2. Let \( A \) be a collection of pairwise-disjoint discs in the plane. (Pairwise disjoint means that the intersection of any two discs is empty.) Can \( A \) be uncountable?  What if \( A \) is instead a collection of pairwise-disjoint circles in the plane?
3. Is the set of finite subsets of \( N \) countable or uncountable?  What about the set of all subsets of \( N \)?  (The set of all subsets of \( N \) is called the power set of the natural numbers.)