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.

$P(A) = \{ \phi, \{1\}, \{2\}, \{1, 2\}\}$

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.

Cardinality ( $n$ )	Total Subsets ( $2^n$ )
0	1
1	2
2	4
3	8
4	16
5	32
6	64
7	128
8	256
9	512
10	1024

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.

Set	Power Set
$A = \phi$	$\{ \phi \}$
$A = \{1\}$	$\{ \phi,\ \{1\}\ \}$
$A = \{1, 2\}$	$\{ \phi,\ \{1\},\ \{2\},\ \{1,2\}\ \}$
$A = \{1, 2, 3\}$	$\{ \phi,\ \{1\},\ \{2\},\ \{3\},\ \{1,2\},\ \{1,3\},\ \{2,3\},\ \{1,2,3\}\ \}$
$A = \{1, 2, 3, 4\}$	$\{ \phi,\ \{1\},\ \{2\},\ \{3\},\ \{4\},\ \{1,2\},\ \{1,3\},\ \{1,4\},\ \{2,3\},\ \{2,4\},\ \{3,4\},\ \{1,2,3\},\ \{1,2,4\},\ \{1,3,4\},\ \{2,3,4\},\ \{1,2,3,4\}\ \}$
$A = \{1, 2, 3, 4, 5\}$	$\{ \phi,\ \{1\},\ \{2\},\ \{3\},\ \{4\},\ \{5\},\ \{1,2\},\ \{1,3\},\ \{1,4\},\ \{1,5\},\ \{2,3\},\ \{2,4\},\ \{2,5\},\ \{3,4\},\ \{3,5\},\ \{4,5\},\ \{1,2,3\},\ \{1,2,4\},\ \{1,2,5\},\ \{1,3,4\},\ \{1,3,5\},\ \{1,4,5\},\ \{2,3,4\},\ \{2,3,5\},\ \{2,4,5\},\ \{3,4,5\},\ \{1,2,3,4\},\ \{1,2,3,5\},\ \{1,2,4,5\},\ \{1,3,4,5\},\ \{2,3,4,5\},\ \{1,2,3,4,5\}\ \}$

Write Powers Sets - Interactive Exercise

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

Power Set Exercise - Create the Power Set of the given set.

A = \{a\}

Write one subset at a time without outer curly brackets and press enter.

\{

\}

P(A) = \{

\}

Power Set Exercise - Create the Power Set of the given set.

A = \{a, b, c\}

Write one subset at a time without outer curly brackets and press enter.

\{

\}

P(A) = \{

\}

Power Set Exercise - Create the Power Set of the given set.

A = \{0, \{0, 1\}\}

Write one subset at a time without outer curly brackets and press enter.

\{

\}

P(A) = \{

\}

Power Set Exercise - Create the Power Set of the given set.

A = \{1, 2, 3, 4\}

Write one subset at a time without outer curly brackets and press enter.

\{

\}

P(A) = \{

\}

True or False Activity!!!

Read each statement carefully and decide if it is true or false.

If a set has $n$ elements, its power set has $2^n$ elements.

The power set of a finite set is always finite.

The power set of an infinite set is infinite.

$\emptyset$ is always an element of the power set of any set.

If $A$ has 3 elements, $P(A)$ has 6 elements.

End of Lesson