Power Set (original) (raw)

Last Updated : 23 Feb, 2026

A power set is basically a set that contains all the possible subsets of the original given set, including the null or empty set. If we have a set A, then the power set of A contains all the subsets of A, including the empty set.

**Other example: Set A = {1, 2, 9}.

Then its power set will be {∅, {1}, {2}, {9}, {1, 2}, {1, 9}, {2, 9}, {1, 2, 9}}

Let's break down the above Power set:-

• Here ∅ represents the Null or Empty Set.
• {1}, {2}, {9} this represents all the subsets with one elements.
• {1, 2}, {1, 9}, {2, 9} this represents all the subsets with two elements.
• Lastly, {1, 2, 9}, which represents the set itself.

Power sets are used in various fields where a list of all possibilities from some finite number of elements is required, such as computer science, data analysis, and even artificial intelligence.

Mathematically, if S is a set, then the power set P(S) is defined as:

**P(S) = {T | T is a subset of S}

Where,

• **T represents a subset of the set S.
• ****"|"** denotes "such that."
• The curly braces i.e., ****{}** indicate a set.

Let see an example for a clear and better understanding, consider a set A = {a, e, i, o, u}, therefore power set of A is given by P(A), i.e.

**P(A) = {∅,

****{a}, {e}, {i}, {o}, {u},**

****{a, e}, {a, i}, {a, o}, {a, u}, {e, i}, {e, o}, {e, u}, {i, o}, {i, u}, {o, u},**

****{a, e, i}, {a, e, o}, {a, e, u}, {a, i, o}, {a, i, u}, {a, o, u}, {e, i, o}, {e, i, u}, {e, o, u}, {i, o, u},**

****{a, e, i, o}, {a, e, i, u}, {a, e, o, u}, {a, i, o, u}, {e, i, o, u},**

****{a, e, i, o, u}}**

Here ∅ represents a Null set or Empty set.

How to Find Power Set?

In order to find a power set, follow these steps:

• Start with a null or empty set.
• Then add all combinations of subsets with one element.
• Then add all combinations of subsets with two elements.
• Do this till you reach the subsets with N-1 elements (where N is the total number of elements in the original set).
• Then add the original set.

Cardinality of Power Set

**Cardinality (cardinality of a set means the number of elements of a set) of a power set denotes the number of elements present in the power set. It is denoted by |P(A)|. Thus, number of elements in the power set is given by:

****|P(A)| = 2** n

Where "n" is the number of elements of Set A.

**Example: Find the cardinality of the Power Set of A, where A = {1,2,9}.

**Answer:

As |A| = 3, thus number of elements in Power Set of A = 2|A|

Thus, |P(A)| = 23 = 8

Therefore, there are 8 elements in the power set of A.

Properties of Power Set

There are several properties of the power set, some of which are listed as follows:

• Total number of elements of a power set is 2n ( where n is the total number of elements of the original Set).
• Power Set always contains an empty set and the original set as its members.
• The elements of the power set are always greater than the elements of the original set (since it has 2n elements of the original set).
• The power set of an empty or null set is the set itself.
• An empty or null set's power set is the set itself. Following distributive rules, power sets can be utilized for set operations like union, intersection, and complement.
• Power set size is always 2n, where n is the size of the initial set.
• Each member of the original set's subsets makes is always a member of power set too.

**Also Check

• Set Symbols
• Universal Sets
• Supersets

Solved Examples on Power Set

**Example 1: Find the total no. of elements and power set for set A = {1, 2, 4, 9}

**Solution:

Number of elements of Set A i.e., n(A) = 4,

Total number of elements of Power set = 2n(A)= 24 = 16

**Example 2: Find the elements of the power set for Set A, where Set A = {9, 18, 5, 6}

**Solution:

Since power set contains all possible subset for the given set including the null or empty set.

Therefore Power set of A , P(A) = {∅, {9}, {18}, {5}, {6}, {9, 18}, {9, 5}, {9, 6}, {18, 5}, {18, 6}, {5, 6}, {9, 18, 5}, {9, 18, 6}, {9, 5, 6}, {18, 5, 6}, {9, 18, 5, 6} }

So, the power set of set A = {9, 18, 5, 6} contains 2^4 = 16 elements or subsets.

**Example 3: Find the number of elements of an empty set?

**Solution:

A = { }

Total number of elements of power set of A , P(A) = 20 = 1

P (A) = { }

Power set of an empty set is the set itself.

**Example 4: What is the size of the power set of a set A with 10 elements?

**Solution:

Applying the cardinality rule (|P(A)| = 2n) to calculate the elements of power set :-

No. of elements of power set of Set A or P(A) = 2n , where n is no. of elements of Set A.

Putting the value of n, we get :-

210 = 1024 elements for the power set.

**Example 5: How many elements are in set A if set A has a power set with 64 subsets?

**Solution:

Let the number of element of Set A be 'x' .

Applying the cardinality rule (|P(A)| = 2n) , we get :-

⇒ 2x = 64

⇒ 2x = 26

Comparing both sides , we get :

x = 6

Therefore no. of elements of set A = 6

Practice Questions on Power Set

**Q1: What will be the Power Set of the set A = {2x: -2 ≤ x ≤2}

**Q2: What will be the Power Set of set P = {x: x is a prime number and x ≤ 50}

**Q3: What will be the Cardinality of Power Set of set containing first five even natural numbers.

**Q4: What will be the cardinality of Power Set of the set containing first 7 multiples of 3.