3.4. Countably Finite Sets and Countably Infinite Sets

Try these first

1. 1.

   Define a countably finite set.

   Hint

   Think about a set you can count entirely—like a box of candies.

2. 2.

   Is the set of all even numbers countable?

   Hint

   Consider how you can list them.

3. 3.

   Which of the following is a countably finite set?

   • Set of all positive integers
   • Set of continents
   • Set of all real numbers

   Hint

   Think of how many continents exist.

4. 4.

   True or False: The set of all prime numbers is uncountable.

   • True
   • False

   Hint

   Recall that primes can be listed as 2, 3, 5, 7, ...

5. 5.

   Prove that the set of all even integers is countably infinite.

   Hint

   Show that every positive integer has a unique even counterpart.

6. 6.

   Explain Cantor’s diagonal argument and how it demonstrates that the reals are uncountable.

   Hint

   Focus on the construction of a new number that differs from any given entry in the list.