Power Set

The set of all subsets of a given set

The power set of a set X, denoted 𝒫(X), is the set of all subsets of X, including X itself and the empty set.

Let us begin with the smallest of all sets, the empty set. If

then

A little more complicated case is a set with only one element. If

then

Now, let us examine the cases in which X has two, three, and four elements, respectively:

We have listed until now five sets, increasing by one the number of elements at each step. Counting the number of elements in each set and the relative power sets, we can notice the following relation:

We can derive a general rule from this short list: If n is the number of elements in a set, 2ⁿ is the number of elements in its power set. In other words, every time we increase by one the number of elements in a set, the number of subsets in its power set doubles.