7. Cantor's Theorem

Try these first

1. 1.

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

   Hint

   Remember, the power set includes all subsets.

2. 2.

   Define cardinality.

   Hint

   Think about how to measure the size of a set.

3. 3.

   What is the relationship between a set and its power set according to Cantor's Theorem?

   • The sizes are equal
   • The power set is larger
   • The power set is smaller

   Hint

   Think about how many subsets can be formed.

4. 4.

   True or False: A power set of a finite set always has a finite number of elements.

   • True
   • False

   Hint

   Consider the formula for power sets.

5. 5.

   Construct a set A with 4 elements and illustrate the power set P(A). Explain the relationship of sizes.

   Hint

   Count the number of subsets and relate it back to the power set definition.

6. 6.

   Explain how Cantor's diagonalization argument not only proves the non-existence of a surjective function but also illustrates the concept of uncountability.

   Hint

   Reflect on the steps in constructing the diagonal set.