The document discusses showing that the set of rational numbers is countable by writing them in a 2D array that is then enumerated into a 1D list, with all rationals appearing in the list. It explains that rationals like 1/2, 2/4, and 3/6 would be duplicates in the list but removing duplicates would still result in a valid list that contains all rational numbers, demonstrating their countability.

1. Countability of Rationals Goal: show the rationals are a countable set! How? Write the rational numbers... In an ordered list Starting with a first element Idea: Write the rationals in a 2D array (easy) Enumerate the array to form a 1D list (tricky)

2. All rationals appear in this list! Denominator 1 Denominator 2 Denominator 3 Denominator 4 All rationals appear in this table! Enumerate them in a clever way....

3. Conclusion: The resulting list is Every rationalmust appear at least once! This list contains duplicates. For example, 1/2 = 2/4 = 3/6 = ... Not a problem! Just remove the duplicates, and the resulting list will still be valid.