What Is the Power Set?

One question in set theory is whether a set is a subset of another set. A subset of A is a set which is formed by using some of the elements from the set A. In order for B to be a subset of A, every element of B must also be an element of A.

Every set has several subsets. Sometimes it is desirable to know all of the subsets that are possible. A construction known as the power set helps in this endeavor. The power set of the set A is a set with elements that are also sets. This power set formed by including all of the subsets of a given set A.

Example 1

We will consider two examples of power sets. For the first, if we begin with the set A = {1, 2, 3}, then what is the power set? We continue by listing all of the subsets of A.

  • The empty set is a subset of A. Indeed the empty set is a subset of every set. This is the only subset with no elements of A.
  • The sets {1}, {2}, {3} are the only subsets of A with one element.
  • The sets {1, 2}, {1, 3}, {2, 3} are the only subsets of A with two elements.
  • Every set is a subset of itself. Thus A = {1, 2, 3} is a subset of A. This is the only subset with three elements.
A
A
A

Example 2

For the second example, we will consider the power set of B ={1, 2, 3, 4}. Much of what we said above is similar, if not identical now:

  • The empty set and B are both subsets.
  • Since there are four elements of B, there are four subsets with one element: {1}, {2}, {3}, {4}.
  • Since every subset of three elements can be formed by eliminating one element from B and there are four elements, there are four such subsets: {1, 2, 3}, {1, 2, 4}, {1, 3, 4}, {2, 3, 4}.
  • It remains to determine the subsets with two elements. We are forming a subset of two elements chosen from a set of 4. This is a combination and there are C (4, 2 ) =6 of these combinations. The subsets are: {1, 2}, {1, 3}, {1, 4}, {2, 3}, {2, 4}, {3, 4}.
B
B

Notation

There are two ways that the power set of a set A is denoted. One way to denote this is use the symbol P( A), where sometimes this letter P is written with a stylized script. Another notation for the power set of A is 2A. This notation is used to connect the power set to the number of elements in the power set.

Size of the Power Set

We will examine this notation further. If A is a finite set with n elements, then its power set P( A ) will have 2n elements. If we are working with an infinite set, then it is not helpful to think of 2n elements. However, a theorem of Cantor tells us that the cardinality of a set and its power set cannot be the same.

It was an open question in mathematics whether the cardinality of the power set of a countably infinite set matches the cardinality of the reals. The resolution of this question is quite technical, but says that we may choose to make this identification of cardinalities or not. Both lead to a consistent mathematical theory.

Power Sets in Probability

The subject of probability is based upon set theory. Instead of referring to universal sets and subsets, we instead talk about sample spaces and events. Sometimes when working with a sample space, we wish to determine the events of that sample space. The power set of the sample space that we have will give us all possible events.