9.4.2. Listing of Set S

Try these first

1. 1.

   What is the definition of cardinality?

   Hint

   Think about how we can count elements in a group.

2. 2.

   Give an example of an injective mapping.

   Hint

   Consider functions that don’t pair different elements together.

3. 3.

   What theorem states that two sets have the same cardinality when there are injective mappings both ways?

   • Cantor's Theorem
   • Schroder-Bernstein Theorem
   • Euler's Theorem

   Hint

   Recall the theorem focused on injective mappings.

4. 4.

   True or False: There exists a countably infinite subset in every infinite set.

   • True
   • False

   Hint

   Think about the properties of sets with infinite particles.

5. 5.

   Using the mapping concept, prove that both (0, 1) and (0, 1] have the same cardinality systematically.

   Hint

   Consider visual aids for the mappings.

6. 6.

   Construct two uncountable sets with finite intersection. Explore their properties.

   Hint

   Use boundaries of sets for example constructions.