9.5. Set of Real Numbers with Decimal Representation of Only 1s

Try these first

1. 1.

   Define Cardinality.

   Hint

   Think about how many elements a set has.

2. 2.

   What is an injective function?

   Hint

   Remember, no two outputs should be the same for different inputs.

3. 3.

   What does cardinality refer to?

   • Size of a set
   • Shape of a set
   • Color of elements

   Hint

   Think about how big the set is.

4. 4.

   True or False: An injective function maps two inputs to the same output.

   • True
   • False

   Hint

   Recall the definition of injective functions.

5. 5.

   Prove that for any set A, removing a finite number of elements will leave the cardinality either unchanged or smaller.

   Hint

   Consider the implications of finite vs infinite removals.

6. 6.

   Show that the union of two countably infinite sets is countably infinite.

   Hint

   Think about list ordering.