6.1.7. Set of Real Numbers between 0 and 1

Try these first

1. 1.

   Define a countably infinite set.

   Hint

   Think about a set you can list or count.

2. 2.

   What is an uncountable set?

   Hint

   Consider whether you can list all elements.

3. 3.

   What does uncountable mean?

   • Able to be listed
   • Cannot be listed
   • Can be represented on a number line

   Hint

   Consider if you can list out all elements.

4. 4.

   True or False: All infinite sets are countable.

   • True
   • False

   Hint

   Think about the set of real numbers.

5. 5.

   Construct a set to show that the set of real numbers cannot be counted. Use Cantor's method to illustrate.

   Hint

   Think about how flipping each digit creates a new, unlisted number.

6. 6.

   Using a bijection, explain how the infinite binary strings are related to real numbers between 0 and 1.

   Hint

   Look at binary representations and their decimal equivalents in the specified interval.