Power Set

The power set of $A$ is denoted as $P(A)$ where $P(A)$ is the set containing all the possible subsets of $A$.

Consider the set $A = \{1, 2 \}$. How many subsets does it have? It has four subsets given below.

• $\phi$ or the empty set which is a subset of all sets be default.
• $\{ 1 \}$ is a subset of $A$
• $\{ 2 \}$ is a subset of $A$
• $\{ 1, 2 \}$ is also subset of $A$ because each set is also a subset of itself.

Putting these four subsets of $A$ in one set makes the power set of $A$ which has four elements, each of them being a set.

How many subsets does a given set have?

The number of subsets of a set can be found by the relation $2^n$ where $n$ is the cardinality of the set. This table shows the relation between the number of sets and cardinality more clearly.

Note that the empty set also has one subset which is also an empty set. And a singleton set has two subsets, itself and the empty set.

This table displays the power sets of sets of cardinality from 0 to 5 so that you can see how to construct them intuitively.

Write Powers Sets - Interactive Exercise

In these next exercises, press enter with input field empty to enter the null set.