6.1.3. Examples of Infinite Sets

Try these first

1. 1.

   What defines a countably infinite set?

   Hint

   Think about listing or counting.

2. 2.

   Give an example of a countably infinite set.

   Hint

   Consider sets that don't stop growing.

3. 3.

   Which of the following sets is countably infinite?

   • Set of all real numbers
   • Set of integers
   • Set of infinite binary strings

   Hint

   Think about how you'd list each member of the set.

4. 4.

   Cantor's diagonalization shows that which of the following sets is uncountable?

   • True
   • False

   Hint

   Consider if you can cover all possibilities.

5. 5.

   In Cantor's diagonal argument, demonstrate how one can create a new binary string from an infinite binary string list.

   Hint

   Utilize the diagonal method!

6. 6.

   Explain why the set of rational numbers is countable while the set of real numbers is uncountable.

   Hint

   Think about representation in decimal forms.