3. Countable and Uncountable Sets

Try these first

1. 1.

   What is the cardinality of the set X = {Alice, Bob}?

   Hint

   Count the number of names in the set.

2. 2.

   Is the set Y = {1, 2, 3, 4} countable?

   Hint

   Check if it's possible to list elements.

3. 3.

   What is a countable set?

   • A set with finite elements
   • A set with the same cardinality as positive integers
   • Both A and B

   Hint

   Think about the definitions discussed regarding countable sets.

4. 4.

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

   • True
   • False

   Hint

   Consider Cantor’s diagonal argument!

5. 5.

   Show that the set of all even integers is countable by constructing a bijection with positive integers.

   Hint

   Use the formula relating each even integer to positive integers.

6. 6.

   Explain why the interval [0, 1] contains uncountably many numbers.

   Hint

   Reflect back on concepts involving real numbers and enumeration.