7.3. Construction of the Subset S

Try these first

1. 1.

   What is the power set of the set {a, b}?

   Hint

   List all possible subsets.

2. 2.

   Is the cardinality of {1, 2} greater than, less than, or equal to its power set?

   Hint

   Consider how many subsets there are.

3. 3.

   What does Cantor's theorem state about the cardinality of a set and its power set?

   • They are equal
   • Power set is greater
   • Power set is lesser

   Hint

   Reflect on the relationship between a set and all of its subsets.

4. 4.

   Is the following statement true or false? 'The power set of a finite set is also finite.'

   • True
   • False

   Hint

   Consider the formula for the size of the power set.

5. 5.

   Prove using Cantor's theorem that the set of real numbers cannot be put into one-to-one correspondence with the natural numbers.

   Hint

   Think about how you would apply the diagonal method to generate a new number.

6. 6.

   Consider a set B. If the cardinality of B is 5, determine the cardinality of P(B) and describe what this means in terms of Cantor's theorem.

   Hint

   Use the exponential relationship between set size and power sets.