 We'll introduce the following definition. The power set of a, written this way, is the set of all subsets of a. For example, let a be the set consisting of 0, 1, and the empty set. Find the power set of a, so we can find all the subsets. So it's important to note the difference between the empty set and the set containing the empty set. The empty set is the empty set. You can think of it as a bag that contains nothing. Meanwhile, the set that contains the empty set is a set that contains the empty set. So you can think about this as a bag that contains the bag that contains nothing. To find all the subsets, let's be systematic about it. First, there's the set with no elements. That's the empty set. Next, there are the sets with one element. Next, there are the sets with two elements. And finally, there's the set with three elements. Now, notice that one way we can form a subset is to go through each element of a and make a decision to include it or exclude it. And so there would be two to the cardinality of a elements of the power set, where the cardinality of a is the number of elements of a. So we could write the cardinality of the power set of a as two raised to the power of the cardinality. And this leads to an alternate notation for the power set. The power set of a is written two to the exponent a.