Description
Problem 1 (15+1 points)
Let A be a countable set. Prove that the set of all finite subsets of A is countable.
Problem 2 (22+1 points)
Recall that the set of real numbers is uncountable. In this question, we will consider other uncountably infinite sets. First, let (a, b) be the open interval from a to b, i.e. (a, b) = {r ∈ R : a < r < b}. Similarly, [a, b] is the closed interval from a to b, i.e. [a, b] = {r ∈ R : a ≤ r ≤ b}. We can define a bijection from R to any finite interval. For example, we can create a bijection f : R → (0, 1) by letting f(x) = arctan(x) for x ∈ R. (a) (3 points) Let a, b ∈ R such that a < b. Prove that (a, b) bij [a, b]. (b) (7 points) Prove that there is a bijection between any two intervals of finite length. In other words, prove that [a, b] bij [c, d] for any a < b, c < d, where a, b, c, d ∈ R. (Hint: Try to define a bijection using a linear function.) (c) (12 points) Let C be the unit circle, C = {(x, y) ∈ R × R : x 2 + y 2 = 1}. Prove that R bij C. (Hint: Consider first defining a bijection with a finite interval, then use polar coordinates.)