3.5. Theorem on Countable Sets

Try these first

1. 1.

   Define a countable set.

   Hint

   Think about the types of numbers you can count.

2. 2.

   What is a bijection?

   Hint

   Consider how each element from one set relates to the other.

3. 3.

   What is a countable set?

   • A set that is finite and can’t be listed
   • A set with more than one element only
   • A set that can be put into one-to-one correspondence with positive integers

   Hint

   Focus on how we understand counting beyond finite numbers.

4. 4.

   True or False: The set of real numbers is countable.

   • True
   • False

   Hint

   Recall the characteristics of uncountable sets.

5. 5.

   Prove that the set of natural numbers is countable by establishing a bijection with the integers.

   Hint

   Consider how you can start pairing from 0.

6. 6.

   Show why the set of rational numbers is countable.

   Hint

   Think about how to organize and sequence pairs.