# Countability study problems

### Problem 1

Let's define sets A and B as follows:

A = {0, 2, 4, 6, 8, 10, 12, ...}, i.e. the even numbers starting with 0.

B = {1, 4, 9, 16, 25, 36, 49, ...}, i.e. perfect squares starting with 1.

Show that $$|A| = |B|$$.

### Problem 2

A pretty wheel'' graph is a wheel graph whose vertices are colored with colors from the set {red, green, blue, violet, yellow}. Two pretty wheels are distinct if they have a different number of nodes or if they have the same number of nodes but a different color pattern. Is the set of distinct pretty wheels countable or uncountable? Briefly justify your answer.