In this chapter we will understand the concept of power set with properties and examples.

To understand power sets, you should have good understanding about the subsets and its properties.

**Read about the subsets here**.

## What are Power sets ?

Power set is a **collection of all subset of a given set**.

If A is a set. Then collection of all subset of A is called Power set of A.

Power set is denoted by P(A)

### How to find Power set of given set ?

Follow the below steps;

(a) Find all the possible subsets of given set.

(b) Make new set of all the subsets

(c) This new set will be represented by P (set name)

Let us understand the above process with the help of examples.

#### Examples of Power set

**Example 01**

If set A = { 1, 2 }, then find power set P(A).

**Solution**

For the given set A, first find all the possible subsets.

Subsets of A are;

⟹ {𝜙}, {1}, {2}, {1, 2}

Form a new set with all above subsets;

P (A) = {{𝜙}, {1}, {2}, {1, 2}}

Hence, the collection of all subset is the Power set.

**Example 02**

If set X = { a, b, c }, then find the proper set P (X}

**Solution**

Find all the possible subset of X.

Subset of X are;

⟹ {𝜙}, {a}, {b}, {a, b}, {b, c}, {c, a}, {a, b, c}

Now form the set of above subsets

P(X) = {{𝜙}, {a}, {b}, {a, b}, {b, c}, {c, a}, {a, b, c}}

Hence the above set P(X) is the power set of set X.

**Example 03**

If A = { a, {b} }, then find the power set P(A)

**Solution**

Here the set A is given as;

A = { a, {b} }

Here inside the set A, we have another set in form of { b }.

Suppose that;

X = { b }

Then;

A = { a, X }

Now find all the subset of set A.

⟹ {𝜙}, {a}, {X}, {a, X}

Put the value of X in all subsets;

⟹ {𝜙}, {a}, {{b}}, {a, {b}}

Now form the set of all the subsets.

P (A) = { {𝜙}, {a}, {{b}}, {a, {b}} }

Hence, P(A) is the power set of given set A.

**Example 04**If A = { x : x is prime number less than 7 }.

Find he power set P (A)

**Solution**

First write the set A in Roster form.

A = { 2, 3, 5 }

Now write all the possible subsets of A

⟹ {𝜙}, {2}, {3}, {5}, {2, 3}, {3, 5}, {5, 2}, {2, 3, 5}

Now form the new set with all the above elements

P (A) = { {𝜙}, {2}, {3}, {5}, {2, 3}, {3, 5}, {5, 2}, {2, 3, 5} }

### Cardinality of Power set

The **number of unique element present** in given set is called cardinality.

If A is the given set, then;

Power set cardinality = Number of subsets of A

**Number of subset in set A is given by formula**;

Total subset = \mathtt{2^{n}}

Where;

n = number of unique element in set A

**Hence, the cardinality of power set is given as**;

Power set Cardinality = \mathtt{2^{n}}

Let us understand the concept with examples.**Example 01**

A = { 2, 4 }

Find cardinality of P (A)

**Solution**

Number of subsets of A = \mathtt{2^{2} =\ 4}

Listing all the 4 subsets

⟹ {𝜙}, {2}, {4}, {2, 4}

For power set, we will form the set of all subsets.

P (A) = { {𝜙}, {2}, {4}, {2, 4} }

There are 4 unique elements in set P(A).

Hence, cardinality of power set is 4.

**Note:**

To find the cardinality of power set just apply below formula.

Power set cardinality = \mathtt{2^{n}}

**Example 02**

Set A = { 9, 7, 10 }

Find the cardinality of power set A.

**Solution**

Cardinality of power set is given by following formula;

Power set Cardinality = \mathtt{2^{n}}

Power set cardinality = \mathtt{2^{3} =\ 8}

Hence, there are 8 unique elements in Power set.

## Properties of Power set

Given below are important properties of power set;

(a)** Power set is the set of subsets**.

(b) **Power set of empty set has one element**.

Empty set is the one which contains 0 elements.

Empty set is expressed as follows;

If A = { 𝜙 };

Subset of A will be { 𝜙 }

Then P (A) = { { 𝜙 } }

Hence, power set of empty set is not empty, it contain the null set A.

(c) **Cardinality of Power set is given by** \mathtt{2^{n}}

(d) Power set of finite set contain finite number of elements and is much larger than the original set.

(e) Power set of infinite set is also infinite.