6. Module No # 05

Try these first

1. 1.

   List three examples of countably infinite sets.

   Hint

   Think about sets that can be listed or counted.

2. 2.

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

   Hint

   Consider whether you can list all real numbers.

3. 3.

   Which of the following sets is uncountable?

   • Set of integers
   • Set of binary strings of infinite length
   • Set of rational numbers

   Hint

   Recall Cantor's diagonal argument.

4. 4.

   True or False: The rational numbers can be listed in a sequential manner.

   • True
   • False

   Hint

   Think about how you could express fractions as sequences.

5. 5.

   Considering Cantor's arguments, generate an enumeration of the real numbers between 0 and 1, and demonstrate why this enumeration is incomplete.

   Hint

   Pay close attention to how the new number differs from your enumeration.

6. 6.

   Use Cantor’s diagonalization argument to attempt to prove that the set of all finite binary strings is uncountable, and analyze where your reasoning fails.

   Hint

   Reflect on the nature of finite versus infinite sets.