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 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.

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.

Also Check,

• Set Symbols
• Universal Sets
• Supersets

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.