Power Set – Definition, Formula, Cardinality, Properties, Examples

What Is a Power Set in Math?

The power set in set theory is a set of all subsets of a given set. Thus, the empty set and the set itself are always included in the power set. We denote the power set of set A as P(A) or ℘(A).

To define the power set of a set, we write the set of all subsets of the given set.

Example: 

Power Set: Definition

The set of all the subsets of a given set is called the power set in math. The original set and an empty set are always the members of a power set. 

Cardinality of a Power Set

The number of subsets of a set with cardinality n can be determined by the formula $2^{n}$. Thus, the cardinality of the power set of a set having “n” elements is $2^{n}$.

Example: If a set has 2 elements, its power set has $2^{2} = 4$ elements.

$A = \left\{1,\; 2\right\}$ 

$n(A) = | A | = 2$

$P(A) = \left\{ \left\{1\right\}, \left\{2\right\}, \left\{1, 2\right\}, ∅ \right\}$

$n( P(A) ) = | P(A) | = 4$

How to Find the Power Set

Power set symbol is P. The power set of a set A is denoted by P(A). To define the power set, start listing down the subsets of the given set. 

Example: $S = \left\{x, y, z\right\}$

Let P(S) be the powerset of S. It contains all the subsets of S.

Cardinality of $S = 3$

The power set will have $2^{n} = 2^{3} =  8$ elements.

Let’s list down the subsets of S to define P(S). The subsets that can be formed using the elements x, y, z are listed below.

1. $\left\{\right\}$
2. $\left\{x\right\}$
3. $\left\{y\right\}$
4. $\left\{z\right\}$
5. $\left\{x, y\right\}$
6. $\left\{y, z\right\}$
7. $\left\{x, z\right\}$
8. $\left\{x, y, z\right\}$

Therefore, the power set of set S will be written as

$P(S) = \left\{ ∅, \left\{x\right\}, \left\{y\right\}, \left\{z\right\}, \left\{x, y\right\}, \left\{y, z\right\}, \left\{x, z\right\}, \left\{x, y, z\right\}\right\}$

Power Set of an Empty Set

The only subset of an empty set is the empty set itself. Thus, the power set of an empty set has only one element.

Empty set has no elements (zero number of elements). 

So, the power set of the empty set has $2^{0} = 1$ element.

$P(∅) = \left\{ ∅ \right\}$

Power Set Properties

• Power sets are larger sets compared to the original set.
• The power set has $2^{n}$ elements, where n is the number of members in a set.
• The power set of a countable finite set is countable.
• ​​The power set of an empty set has only one element.
• Empty set and the set itself are the fixed elements of any power set.

Examples of Power Set

Set	Power Set
$A = \left\{1\right\}$	$P(A) = \left\{ ∅, \left\{1\right\} \right\}$
$B = \left\{ X, Y \right\}$	$P(B) = \left\{ ∅, \left\{X\right\}, \left\{Y\right\}, \left\{X, Y\right\} \right\}$
$C = \left\{0\right\}$	$P(C) = \left\{ ∅, \left\{0\right\} \right\}$
$D = \left\{1, 2, 3. 4, 5\right\}$	$P(D) = \left\{∅, \{1\right\}, \left\{2\right\}, \left\{3\right\}, \left\{4\right\}, \left\{5\right\}, \left\{1, 2\right\}, \left\{1, 3\right\}, \left\{2, 3\right\}, \left\{1, 4\right\}, \left\{2, 4\right\}, \left\{3, 4\right\}, \left\{1, 5\right\}, \left\{2, 5\right\}, \left\{3, 5\right\}, \left\{4, 5\right\}, \left\{1, 2, 3\right\}, \left\{1, 2, 4\right\}, \left\{1, 3, 4\right\}, \left\{2, 3, 4\right\}, \left\{1, 2, 3, 4\right\}, \left\{1, 2, 5\right\}, \left\{1, 3, 5\right\}, \left\{2, 3, 5\right\}, \left\{3, 4, 5\right\}, \left\{1, 2, 3, 5\right\}, \left\{1, 4, 5\right\}, \left\{2, 4, 5\right\}, \left\{1, 2, 4, 5\right\}, \left\{1, 3, 4, 5\right\}, \left\{2, 3, 4, 5\right\}, \left\{1, 2, 3, 4, 5\right\}$

Facts about Power Set

Conclusion

In this article, we learned about the power set, how to find the power set, its cardinality, properties, and examples. Let’s solve a few more examples and practice problems for better understanding.

Solved Examples on Power Set

1. Determine the power set of set $X = \left\{\text{june, july, august}\right\}$.

Solution: 

$X = \left\{\text{june, july, august}\right\}$

There are three elements in set X. 

$N(X) = 3$

The power set of set X will have $2^{n} = 2^{3} = 222 = 8$ elements.

We need to list 8 subsets of the set X.

Subset of $X = \left\{\right\}, \left\{\text{june}\right\}, \left\{\text{july}\right\}$,

$\left\{\text{august}\right\}, \left\{\text{june, july}\right\}, \left\{\text{june, august}\right\}$,

$\left\{\text{july, august}\right\}, \left\{\text{june, july, august}\right\}$

The power set of subset X of a set will be written as

$P(X) = \left\{\{\right\}, \left\{\text{june}\right\}, \left\{\text{july}\right\}$,

$\left\{\text{august}\right\}, \left\{\text{june, july}\right\}, \left\{\text{june, august}\right\}$,

$\left\{\text{july, august}\right\}, \left\{\text{june, july, august}\right\}$

2. Find the power set of set $A = \left\{0, 1, 2, 3\right\}$.

Solution: 

$A= \left\{0, 1, 2, 3\right\}$

Here $n (A) = 4$,

P(A) has $2^{4} = 2 \times 2 \times 2 \times 2 = 16$

Thus, it is clear that we will have 16 subsets of set A.

Subsets of $A = \left\{\right\}, \left\{0\right\}, \left\{1\right\}, \left\{2\right\}$,

$\left\{3\right\}, \left\{0, 1\right\}, \left\{0, 2\right\}, \left\{0, 3\right\}, \left\{1, 2\right\}$,

$\left\{1, 3\right\}, \left\{2, 3\right\}, \left\{0, 1, 2\right], \left\{0, 1, 3\right\}$,

$\left\{0, 2, 3\right\}, \left\{1, 2, 3\right\}, \left\{0, 1, 2, 3\right\}$

Now, the power set subset A will be written as

$P(A) = \left\{\{\right\}, \left\{0\right\}, \left\{1\right\}, \left\{2\right\}, \left\{3\right\}$,

$\left\{0, 1\right\}, \left\{0, 2\right\}, \left\{0, 3\right\}, \left\{1, 2\right\}, \left\{1, 3\right\}$,

$\left\{2, 3\right\}, \left\{0, 1, 2\right], \left\{0, 1, 3\right\}, \left\{0, 2, 3\right\}, \left\{1, 2, 3\right\}$,

$\left\{0, 1, 2, 3\right\}\}$

3. Find the power set of $M = \left\{\text{Black, White}\right\}$. Determine the total number of elements.

Solution: 

Set $M = \left\{\text{Black, White}\right\}$

Here $n(M) = 2$

Therefore, applying the power set formula, we can calculate the following:

$P(M) = 2^{2} = 4$

Thus, we will have 4 subsets of set M, i.e.,

Subsets of $M = \left\{\right\}, \left\{\text{Black}\right\}, \left\{\text{White}\right\}, \left\{\text{Black, White}\right\}$

Now, the power set of subsets in set A will be determined as

$P(M) = \left\{ \left\{\right\}, \left\{\text{Black}\right\}, \left\{\text{White}\right\}, \left\{\text{Black, White}\right\}\right\}$

4. How many subsets does a set with 6 elements have?

Solution:

A set has 6 elements.

Thus, its power set will have $2^{6} = 64$ elements.

It means that a set with 6 elements has 64 subsets.

Practice Problems on Power Set

Frequently Asked Questions about Power Set