Is the empty set a part of every power set?

Yes. The empty set (ϕ) is a subset of every set, by definition. Therefore, it is always an element of the power set of any set.

Is the empty set included in the power set?

Yes, the empty set (ϕ) is always included in the power set.

What is the difference between subset and power set?

A subset is any set derived from choosing some or all elements of A. A power set is the collection of all such subsets.

Can a set have two different power sets?

No. The power set of a given set is unique. It's the set of all possible subsets, and for any given set, the collection of its subsets is fixed and well-defined.

Home

JEE Maths

Power Set

Frequently Asked Questions

Join ALLEN!

(Session 2026 - 27)

Name

Mobile Number

Class

Choose class

Goal

Choose your goal

Preferred Programs

Preferred Mode

State

Choose State

I agree to

I authorise ALLEN Career Institute Pvt Ltd to send me regular updates via Phone calls, Whatsapp, SMS, Robocalls (Automated Calls), Emails, or on postal address.

Power Set

1.0What is a Power Set (or) Definition of Power Set

A Power Set of a set A is the set of all possible subsets of A, including the empty set and the set A itself.

If A={1,2}, then the Power Set of A, denoted by P(A), is: P(A)={ ϕ,{1},{2},{1,2} }

Thus, the power set represents the complete collection of subsets.

Notation of Power Set

The power set of A is denoted by: P(A)
If A has n elements, then $∣ P (A) ∣ = 2^{n}$

2.0Formula for Power Set

Number of Subsets

For a finite set A with n elements: $∣ P (A) ∣ = 2^{n}$

This is because each element of A can either be included or excluded in a subset, giving 2 choices per element.

General Expression

If $A = {a_{1}, a_{2}, \dots, a_{n}}$

Then, P(A)={ S:S⊆A }

3.0How to Write a Power Set?

Step-by-Step Method

Start with the empty set (ϕ).
List all 1-element subsets.
List all 2-element subsets.
Continue until you list the n-element subset, which is the set A itself.

Example 1: Power Set of {1, 2}: P({1,2})={ ϕ,{1},{2},{1,2} }

Example 2: Power Set of {a, b, c}: P({a,b,c})={ ϕ,{a},{b},{c},{a,b},{a,c},{b,c},{a,b,c} }

4.0Properties of Power Set

If a set A has n elements, then P(A) has $2^{n}$ elements.
The empty set (ϕ) is always a member of P(A).
The set A itself is always a member of P(A).
P(A) is always larger than A, unless A is the empty set.
P(ϕ)={ϕ}.

5.0Examples of Power Set

Example 1: Find the power set of {1, 2}

Sol: P({1,2})={ ϕ,{1},{2},{1,2} }

Example 2: Find the power set of {a, b, c}

Sol: P({a,b,c})={ ϕ,{a},{b},{c},{a,b},{a,c},{b,c},{a,b,c} }

Example 3: If A has 5 elements, find ∣P(A)∣.

Sol: $∣ P (A) ∣ = 2^{5} = 32$

6.0Applications of Power Set

Relations and Functions

A relation from set A to B is a subset of A×B.
Understanding power sets helps in visualizing all possible relations.

Probability

Sample space in probability can be considered as a power set of outcomes.
Example: Tossing a coin twice → Sample space = {HH, HT, TH, TT}. Its power set covers all possible events.

Logic and Boolean Algebra: Power sets are closely related to truth tables and logical structures.
Combinatorics: Many combinatorial problems in JEE rely on counting subsets, which directly use the power set formula.

7.0Practice Questions on Power Set

Find the power set of {x, y}.
A set A has 6 elements. Find the number of subsets in its power set.
If ∣P(A)∣=64, find ∣A∣.
List all subsets of {1, 2, 3, 4}.
How many subsets of a set with 10 elements contain at least 3 elements?

Join ALLEN!

(Session 2026 - 27)

Name

Mobile Number

Class

Choose class

Goal

Choose your goal

Preferred Programs

Preferred Mode

State

Choose State

I agree to

I authorise ALLEN Career Institute Pvt Ltd to send me regular updates via Phone calls, Whatsapp, SMS, Robocalls (Automated Calls), Emails, or on postal address.

Power Set

1.0What is a Power Set (or) Definition of Power Set

A Power Set of a set A is the set of all possible subsets of A, including the empty set and the set A itself.

If A={1,2}, then the Power Set of A, denoted by P(A), is: P(A)={ ϕ,{1},{2},{1,2} }

Thus, the power set represents the complete collection of subsets.

Notation of Power Set

The power set of A is denoted by: P(A)
If A has n elements, then $∣ P (A) ∣ = 2^{n}$

2.0Formula for Power Set

Number of Subsets

For a finite set A with n elements: $∣ P (A) ∣ = 2^{n}$

This is because each element of A can either be included or excluded in a subset, giving 2 choices per element.

General Expression

If $A = {a_{1}, a_{2}, \dots, a_{n}}$

Then, P(A)={ S:S⊆A }

3.0How to Write a Power Set?

Step-by-Step Method

Start with the empty set (ϕ).
List all 1-element subsets.
List all 2-element subsets.
Continue until you list the n-element subset, which is the set A itself.

Example 1: Power Set of {1, 2}: P({1,2})={ ϕ,{1},{2},{1,2} }

Example 2: Power Set of {a, b, c}: P({a,b,c})={ ϕ,{a},{b},{c},{a,b},{a,c},{b,c},{a,b,c} }

4.0Properties of Power Set

If a set A has n elements, then P(A) has $2^{n}$ elements.
The empty set (ϕ) is always a member of P(A).
The set A itself is always a member of P(A).
P(A) is always larger than A, unless A is the empty set.
P(ϕ)={ϕ}.

5.0Examples of Power Set

Example 1: Find the power set of {1, 2}

Sol: P({1,2})={ ϕ,{1},{2},{1,2} }

Example 2: Find the power set of {a, b, c}

Sol: P({a,b,c})={ ϕ,{a},{b},{c},{a,b},{a,c},{b,c},{a,b,c} }

Example 3: If A has 5 elements, find ∣P(A)∣.

Sol: $∣ P (A) ∣ = 2^{5} = 32$

6.0Applications of Power Set

Relations and Functions

A relation from set A to B is a subset of A×B.
Understanding power sets helps in visualizing all possible relations.

Probability

Sample space in probability can be considered as a power set of outcomes.
Example: Tossing a coin twice → Sample space = {HH, HT, TH, TT}. Its power set covers all possible events.

Logic and Boolean Algebra: Power sets are closely related to truth tables and logical structures.
Combinatorics: Many combinatorial problems in JEE rely on counting subsets, which directly use the power set formula.

7.0Practice Questions on Power Set

Find the power set of {x, y}.
A set A has 6 elements. Find the number of subsets in its power set.
If ∣P(A)∣=64, find ∣A∣.
List all subsets of {1, 2, 3, 4}.
How many subsets of a set with 10 elements contain at least 3 elements?

Frequently Asked Questions

Is the empty set a part of every power set?

Is the empty set included in the power set?

What is the difference between subset and power set?

Can a set have two different power sets?

Frequently Asked Questions

Is the empty set a part of every power set?

Is the empty set included in the power set?

What is the difference between subset and power set?

Can a set have two different power sets?

Join ALLEN!

Power Set

1.0What is a Power Set (or) Definition of Power Set

Notation of Power Set

2.0Formula for Power Set

3.0How to Write a Power Set?

4.0Properties of Power Set

5.0Examples of Power Set

6.0Applications of Power Set

7.0Practice Questions on Power Set

Table of Contents

Join ALLEN!

Power Set

1.0What is a Power Set (or) Definition of Power Set

Notation of Power Set

2.0Formula for Power Set

3.0How to Write a Power Set?

4.0Properties of Power Set

5.0Examples of Power Set

6.0Applications of Power Set

7.0Practice Questions on Power Set

Table of Contents