Power Set

Power Sets, also known as the “set of all subsets,” is one of the important concepts in Set Theory. Power Set is nothing but a collection of all the subsets of any set including the empty set, as the empty set is the subset of all possible sets. 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.

In this article, we will discuss all the topics related to Power Set in detail, including its definition, symbol, and examples. Other than that, we will also learn how to find power sets for any set and also see various solved examples for that. So, let us start our learning for the concept of Power Sets.

What is Power Set?

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.

Lets Say, 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 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 Set Definition

In order to define a power set, we can simply say that a power set is simply a set that contains all the subsets of the original set, including the null or empty set. Formally, 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.

Power Set Symbol

The power set of a set A is basically represented by P (A).

Power Set Example

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.

Power Set of the Empty Set

A power set is all possible subsets of the original set including the null or empty set. So a power set of the Empty Set is basically the empty set itself. We can prove this with simple steps.

Let’s find out the total number of elements of the power set.

No. of elements of empty set = 0 [From Definition of Empty Set]

No. of elements of power set = 2⁰ = 1

Therefore, the power set of empty set i.e., P(∅) = {∅}.

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)| = 2ⁿ

Where “n” is the number of elements of Set A.

Let’s consider an example for better understanding.

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.

Solved Examples on Power Set

Example 1: Find the total no. of elements of “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 = 2^n(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 :-

⇒ 2^x = 64

⇒ 2^x = 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.

FAQs on Power Set

1. Define Power Set.

A power set is a concept of set theory. A power set is a set of all possible subsets of a given set, including the null set or an empty set.

2. What is the Power Set of {1, 2, 3}?

P({1, 2, 3}) = {∅, {1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3}, {1, 2, 3}}

3. What is the relationship between the Power Set and the Original set?

A power set contains all possible subsets of the original set including the empty set.

4. How many Power Sets can a Set have?

A particular set has only one power set as the set containing all the subsets for any set have same elements.

5. How do you Represent the Power Set of Set A?

Power set of set A can be represented by P(A).

6. How many elements are there in a Power Set of an empty set?

There is only one element in the power set of an empty set, and that is the empty set itself.

7. Can a Power Set of a finite set be infinite?

No, power set of a finite set is always finite.

8. Can a Power Set be smaller than the original set (provided the set is non-empty)?

No, the power set of a non-empty set can never be smaller than the original set. Since a power set is a set of all possible subsets of a given set, including the null set or an empty set.