Exercise 1) Let and let . How many elements does A have? How many elements does B have? Does A=B?
Solution: A has 5 elements, and B has 3! The result for A should be clear, but the result for B follows because the elements in B are , where are single elements—they’re sets in their own right, but when viewed in B they are single elements! Accordingly, A and B are not equal as sets. We can see this many ways. First of all, a set with 3 elements can never be equal to a set with 5 elements. More concretely, we see that the element 1 is not in B, nor is the element 2, or 3, or 4. Conversely, the element is not in A, nor is the element . When viewed in this way, it becomes clear that these sets are not equal.
Exercise 2) Knowing that a set with elements has a power set with elements, how many elements does the power set of the power set have? In the notation of letting denote the power set of , the question is to find the number of elements in . (hint: what if where M is some other number?)
Solution: Suppose A has N elements. Then from this lesson we know that has elements. Now we ask how many elements the power set of has. Well, we simply apply this logic again to the set . Namely, we can let , and then we’re just asking what the power set of a set with elements is. We already know this answer: it’s . But, recalling how we defined M, we see that has elements. For example, if A has 2 elements, then has elements. If A has 3 elements, then has elements. Clearly, the size of grows extremely fast as the size of A grows!