9.5.3. Union of Sets S

Try these first

1. 1.

   What is the cardinality of the set of integers?

   Hint

   Think about the natural numbers.

2. 2.

   Is the set of all real numbers between 0 and 1 countably infinite?

   Hint

   Consider Cantor's diagonal argument.

3. 3.

   What is the definition of a countable set?

   • A set that has a finite number of elements.
   • A set that can be listed in correspondence with natural numbers.
   • Any set.
   • An infinite set.

   Hint

   Think about how we can list elements of a set.

4. 4.

   True or False: The intersection of two uncountable sets is always uncountable.

   • True
   • False

   Hint

   Consider overlapping intervals.

5. 5.

   Given two infinite sets, A and B, with A included in B, prove A's cardinality is less than or equal to B.

   Hint

   Start by demonstrating elements of A can map to B.

6. 6.

   Construct a sequence in which the union of subsets remains countable. Provide an example.

   Hint

   List down individual subsets and combine interestingly.