9.1. Tutorial 5

Try these first

1. 1.

   What is cardinality?

   Hint

   Think about how we count items.

2. 2.

   Give an example of a countably infinite set.

   Hint

   Can you list numbers sequentially starting from one?

3. 3.

   What does the Schröder-Bernstein theorem establish?

   • That every set has the same cardinality
   • That two sets have the same cardinality if both have injective mappings
   • That uncountable sets cannot be defined

   Hint

   Recall if injective mapping implies equality in size of sets.

4. 4.

   Is the intersection of two uncountable sets always uncountable?

   • True
   • False

   Hint

   Consider the examples of different types of sets.

5. 5.

   Prove that the union of a countably infinite number of countable sets remains countable.

   Hint

   Focus on how indices can dictate ordering.

6. 6.

   Given two uncountable sets, describe how their intersection can ever be countable.

   Hint

   Assess the boundaries set by each uniqueness.