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Class 11 Maths Chapter 1 Sets: Revision Notes

Every topic in Mathematics begins with understanding a collection of objects, numbers, or elements—and that's exactly where Sets comes in. Think of Sets as the language of Mathematics. From grouping numbers to solving probability questions and understanding functions and relations, the concepts introduced in this chapter appear throughout Class 11, Class 12, and even higher mathematics. A clear understanding of sets not only strengthens your fundamentals but also makes learning upcoming chapters much easier.

Whether you're revising before a school exam or preparing for JEE Main and JEE Advanced, ALLEN's Class 11 Maths Chapter 1 Sets Revision Notes are designed to help you revise the complete chapter in less time. Inside these notes, you'll find simplified explanations, visual representations, important formulas, standard results, shortcut techniques, and exam-oriented practice pointers that make revision quick, effective, and stress-free.

1.0Chapter Snapshot

Parameter	Details
Difficulty Level	Moderate
Revision Time	2-3 Hours
Key Concepts Covered	Types of Sets, Operations on Sets, De Morgan's Laws, Set Identities and Venn Diagram

Other Important details:

Topic	Overview
Introduction to Sets	Definition, notation, and representation of sets
Types of Sets	Empty, singleton, finite, infinite, equal, equivalent, and universal sets
Subsets	Proper subsets and power sets
Operations on Sets	Union, intersection, difference, and complement
Venn Diagrams	Graphical representation of set operations
Algebra of Sets	Important laws and identities
Intervals	Open, closed, and half-open intervals
Applications	Solving mathematical and real-life problems using sets

2.0Chapter 1 Smart Study Material by ALLEN Experts: Download PDFs

Resource	Status
Chapter Mock Test Paper
Chapter Mock Test Paper Solution

Must-Revise Concepts

finite and infinite sets

Set Theory and Number System

Sets

Sets, Relation and Function PYQ with solutions

Sets

1.0Definition of Sets

In Mathematics, a set is a well-defined collection of different objects, considered as an object in its own right. These objects can be anything: numbers, people, letters, or other abstract entities. Sets are denoted by curly braces, such as {1, 2, 3}, and each object in a set is called an element or member. Sets are fundamental in Mathematics and are used to study relationships, perform operations, and define various Mathematical structures.

2.0Representation of Sets in Set Theory

In set theory, sets are typically represented using curly braces { } to enclose their elements. The elements of a set can be anything, such as numbers, letters, or even other sets. Here are some common ways sets are represented and described in set theory:

Roster Form

In roster form, the elements of a set are listed explicitly within curly braces, separated by commas. For example: Set P = {1, 2, 3, 4, 5}

Set-Builder Form

In set-builder notation, a set is described by specifying a property or condition that its elements must satisfy. The notation is {x | P(x)}, where x represents the elements and P(x) is the property or condition. For example: Set B = {x | x is an even number}

Interval Notation

In Mathematics, intervals can be used to represent sets of real numbers. For example, Set C = {x | 0 < x < 10} can be represented as (0, 10), indicating all real numbers greater than 0 and less than 10.

Special Sets

Empty Set: Denoted by ϕ or { }, it is the set with no elements.
Universal Set: Denoted by U, it is the set containing all elements under consideration.
Singleton Set: A set containing only one element. For example, {7}.

These representations help in defining and describing sets in a clear and concise manner, allowing for precise Mathematical reasoning and operations within set theory.

3.0Elements of Set

Elements of a set are the individual objects or numbers that belong to the set. In set notation, elements are listed inside curly braces { } and separated by commas. The elements of a set can be letters, numbers, symbols, or even other sets. Each element in a set is unique, and there are no duplicate elements within a set.

For example, consider the set A = {1, 2, 3, 4}. In this set:

1, 2, 3, and 4 are the elements of set A.
Each element is distinct and appears only once in the set.
The order of elements in a set does not matter. Thus, {1, 2, 3, 4} is the same as {4, 3, 2, 1} as they contain the same elements.

In set theory, elements are crucial for defining the properties and relationships of sets, performing set operations, and analyzing mathematical concepts.

Also Read : Domain and Range of a Relation

1.0Types of Sets

Null Set or Empty Set or Void Set

A set having no elements is called a Null Set or Empty Set or Void set. It is denoted by ϕ or { }.

Singleton Set

A set consisting of one or a single element is called a Singleton Set.

Finite Set

A set that has only a finite number of elements is called a Finite Set.

Infinite Set

A set which has an infinite number of elements is called an infinite Set.

Subset

If every element in set A is also found in set B, then A is termed as a subset of B, denoted as A ⊆ B.

Proper subset

If set A is a subset of set B, but A is not equal to B, then A is considered a proper subset of B, represented as A ⊂ B.

Universal Set

A set consisting of all possible elements that occur in the discussion is called a Universal set and is denoted by U.

Power set

Let A be any set. The set of all subsets of A is called the power set of A and is denoted by P(A).

Equivalent Set or Equal Set

If every element of set A is also an element of set B, and vice versa, they are equal sets denoted as A = B.

For example, A = {3,4,5,6} and B = {6,5,4,3} are equal sets. However, A = {set of even numbers} and B = {set of natural numbers} are not equal as their elements differ.

Disjoint Sets

Two sets, X and Y are called disjoint set if they have no common elements, denoted as X ∩ Y = ϕ. This signifies that their intersection results in zero elements.

2.0Operations on Sets

Cardinality of a Set

The cardinality of a set A, denoted as |A|, represents the number of elements in the set.

Formula: ∣A∣ = n, where n is the count of elements in set A.

Union of Sets

The union of two sets, denoted as A ∪ B, comprises all the elements that are in either set A or set B, or both.

Formula : A ∪ B = { x : x ∈ A or x ∈ B}

Example: If A = {1,2,3} and B = {3,4,5}, then A ∪ B = {1,2,3,4,5}.

Intersection of Sets

The intersection of two sets, A and B, denoted as A ∩ B, is the set of all elements that are common to both A and B.

Formula : A ∩ B = { x : x ∈ A and x ∈ B}

Example: If A = {1,2,3} and B = {3,4,5}, then A∩B = {3}.

Complement of Sets

The complement of a set A, denoted as A′ or Ā, with respect to a universal set U, contains all elements in U that are not in A.

Formula : A′ = { x ∈ U : x ∉ A}

Example: If U = {1,2,3,4,5} is the universal set and A = {1,2}, then A′ = {3,4,5}.

Difference of Sets

The set difference between sets A and B, denoted as A−B, is the set of elements that are in A but not in B.

Formula : A - B = { x : x ∈ A and x ∉ B}

Example: If A = {1, 2, 3} and B = {3, 4, 5}, then A - B = {1, 2}.

Cartesian Product of Sets

The Cartesian product of Set A and Set B, denoted as A × B, represents the collection of all ordered pairs (a, b) where a is in Set A and b is in Set B.

Formula : A × B = { (a, b) : a ∈ A, b ∈ B}

Power Set

The power set of a set A, denoted as P(A), is the set of all subsets of A, including the empty set and A itself.

Formula : P(A) = { B : B ⊆ A}

These operations are fundamental in set theory and are used to manipulate sets, study relationships between sets, and solve various Mathematical problems.

3.0Set Formulas

Some of the Important Sets Formula are

n(A ∪ B) = n(A) + n(B) – n(A ∩ B)
n(A ∪ B) = n(A) + n(B) ⇔ A, B are disjoint sets.
n( A – B) + n( A ∩ B ) = n(A)
n( B – A) + n( A ∩ B ) = n(B)
n( A – B) + n ( A ∩ B) + n( B – A) = n ( A ∪ B )
n ( A ∪ B ∪ C ) = n(A) + n(B) + n(C) – n ( A ∩ B) – n ( B ∩ C) – n ( C ∩ A) + n ( A ∩ B ∩ C)

4.0Venn Diagrams

Clearly (A – B) ∪ (B – A) ∪ (A ∩ B) = A ∪ B

5.0Properties of Sets

Associative Property

A ∪ ( B ∪ C) = ( A ∪ B) ∪ C
A ∩ ( B ∩ C) = ( A ∩ B) ∩ C

Distributive Property

A ∪ ( B ∩ C) = ( A ∪ B) ∩ ( A ∪ C)
A ∩ ( B ∪ C) = ( A ∩ B) ∪ ( A ∩ C)

De Morgan’s Law

Law of union : ( A ∪ B)′ = A′ ∩ B′
Law of Intersection :( A ∩ B)′ = A′ ∪ B′

Complement Law

A ∪ A′ = A′ ∪ A = U
A ∩ A′ = ϕ

Idempotent Law and Law of a null and universal set

For any Finite set, A

A ∪ A = A
A ∩ A = A
Φ′ = U
ϕ = U′

6.0Solved Examples

Example 1: Write the given statement in Roster Form and Set Builder Form of representation of a set:

The set of all integers that lie between –3 and 7

Solution: Roaster Form: { –2, –1, 0, 1, 2, 3, 4, 5, 6}

Set Builder Form: {x : x ∈ Z, –3 < x < 7}

Example 2: Given the sets X = {1, 2, 3}, Y = {3, 4}, Z = {4, 5, 6}, then X ∩ ( Y ∪ Z) is

Solution: Y ∪ Z = {3, 4, 5, 6}

X ∩ ( Y ∪ Z) = {3}

Example 3: A group of members know at least one of the two languages, Hindi or Kannada. In the group, 150 members know Hindi, 80 members know Kannada and 70 of them know both. How many members are there in the group?

Solutions: Let H = Set of persons who know Hindi.

K = Set of persons who know Kannada.

n(H ∩ K) = the number of persons who know both Hindi and Kannada is 70.

n(H ∪ K) = n(H) + n(K) – n(H ∩ K)

= 150 + 80 – 70 = 160

Example 4: Find A ✕ B when A = {x | x is prime number less than 5} and B = {1, 2, 3}

Solution: A = {2, 3}; B = {1, 2, 3}

A ✕ B = {(2,1), (2,2), (2,3), (3,1), (3,2), (3,3)}

7.0Practice Problems

Question 1: Write the given statement in Roster Form and Set Builder Form of representation of a set:

The set of all integers that lie between 5 and 15.

Question 2: Given the sets A = {5, 2, 7}, B = {3, 8}, C = {3, 2, 6}, then find A ∩ ( B ∪ C).

Question 3: In a class of 60, students like Mathematics, 25 like science, and 15 like both. Then the number of students who like either Mathematics or Science is?

Question 4: Let A and B be the two sets in the universal set. Then A – B equals-

a.A ∩ B′

b. A′ ∩ B

c. A ∩ B

d. None of these

Also Read

Operation on Sets	Sets, Relation and Function Previous Year Questions with Solutions	De Morgan’s Laws
Finite and Infinite Sets	Domain Codomain Range Functions

1.0Learning Outcomes

After studying these revision notes, you will be able to:

Understand the concept and representation of sets with confidence.
Differentiate between various types of sets and their properties.
Solve problems involving subsets, proper subsets, and power sets.
Perform operations like union, intersection, difference, and complement accurately.
Interpret and solve questions using Venn diagrams.
Apply important laws and identities of set algebra in problem-solving.
Use interval notation correctly in mathematical problems.
Solve JEE Main, JEE Advanced, and CBSE-level questions with improved accuracy and speed.

2.05-Minute Quick Revision

Must-Revise Concepts

Definition and notation of sets
Roster and Set-builder forms
Types of sets
Universal set and empty set
Subsets and proper subsets
Power set
Union and intersection
Difference and complement of sets
De Morgan's Laws
Venn diagrams
Interval notation

Important Formula Recall

Number of subsets of a set containing n elements = 2ⁿ
Number of proper subsets = 2ⁿ − 1
n(A ∪ B) = n(A) + n(B) − n(A ∩ B)
n(A') = n(U) − n(A)
De Morgan's Laws

3.0High Weightage Topics

Topic	JEE Importance	Difficulty Level
Types of Sets	High	Easy
Subsets and Power Sets	High	Moderate
Operations on Sets	Very High	Moderate
Venn Diagram Problems	Very High	Moderate
Cardinality of Sets	High	Moderate
De Morgan's Laws	High	Moderate
Set Identities	Medium	Moderate

4.0Formula Handbook

The following formulas, identities, and standard results are essential for solving numerical and conceptual questions from Sets in JEE Main, JEE Advanced, and CBSE examinations.

5.0Common Mistakes & JEE Tips

Although Sets is considered one of the easier chapters in Class 11 Mathematics, students often lose marks due to incorrect interpretation of symbols, Venn diagram errors, and confusion between similar concepts. Developing conceptual clarity is more important than memorizing definitions.

Common Mistake	Correct Approach
Confusing ∈ with ⊂	Remember that ∈ represents an element, whereas ⊂ represents a subset.
Treating an empty set and the set containing zero as the same	∅ and {0} are completely different sets.
Incorrectly counting subsets	Use (2^n) only when the set contains n distinct elements.
Forgetting the intersection while applying the cardinality formula	Always subtract (n(A \cap B)) to avoid double counting.
Drawing inaccurate Venn diagrams	Label each region carefully before solving.
Mixing equal sets with equivalent sets	Equal sets have identical elements, whereas equivalent sets have the same number of elements.
Ignoring universal set while finding complements	Always identify the universal set before solving complement-based questions.
Memorizing identities without understanding them	Practice proving standard identities using Venn diagrams for better retention.

6.0ALLEN Faculty Tips

Learn the meaning of every set notation before attempting numerical problems.
Practice Venn diagram questions regularly to improve logical reasoning.
Memorize important set identities and De Morgan's Laws.
Focus on cardinality-based questions, as they are frequently asked in JEE Main.
Understand the difference between subsets, proper subsets, equal sets, and equivalent sets.
Revise interval notation carefully before moving to Relations and Functions.
Solve previous years' questions to understand the pattern of conceptual questions.

7.0PYQ Trend Analysis

Sets is a fundamental chapter in Class 11 Mathematics. While direct questions are generally straightforward, concepts from this chapter are frequently used in Relations & Functions, Probability, and Mathematical Reasoning, making it important from both conceptual and examination perspectives.

Note: The trend below is based on the analysis of previous years' JEE Main and JEE Advanced question papers. The exact number of questions may vary each year.

Topic	Questions Asked (Last 5 Years)	Average Difficulty
Types of Sets	1–2	Easy
Subsets & Power Sets	2–3	Easy–Moderate
Operations on Sets	2–3	Moderate
Venn Diagram Problems	2–4	Moderate
Cardinality of Sets	2–3	Moderate
De Morgan's Laws	1–2	Moderate
Mixed Concept Questions	1–2	Moderate

8.0Most Common Question Types

Identifying different types of sets.
Finding the number of subsets and proper subsets.
Solving problems using union, intersection, and complement.
Questions based on Venn diagrams involving two and three sets.
Applying cardinality formulas to determine unknown values.
Proving identities using set algebra.
Conceptual MCQs based on De Morgan's Laws and set operations.

9.0Smart Revision Strategy

For a quick revision before your examination, follow this sequence:

Revise all set symbols and notations.
Understand different types of sets with examples.
Practice subsets and power set questions.
Memorize important set operations and identities.
Revise De Morgan's Laws thoroughly.
Practice Venn diagram-based questions.
Solve cardinality problems involving two and three sets.
Revise interval notation.
Attempt Previous Years' Questions (PYQs).
Finish with mixed practice questions for confidence.

Class 11 Maths Chapters List:- JEE & NEET Revision Notes

Chapter 1 Sets

Chapter 2 Relations and Functions

Chapter 3 Trigonometric Functions

Chapter 4 Complex Numbers and Quadratic Equations

Chapter 5 Linear Inequalities

Chapter 6 Permutations and Combinations

Chapter 7 Binomial Theorem

Chapter 8 Sequence and Series

Chapter 9 Straight Lines

Chapter 10 Conic Sections

Chapter 11 Introduction to Three-dimensional Geometry

Chapter 12 Limits and Derivatives

Chapter 13 Statistics

Chapter 14 Probability

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Class 11 Maths Chapter 1 Sets: Revision Notes

Every topic in Mathematics begins with understanding a collection of objects, numbers, or elements—and that's exactly where Sets comes in. Think of Sets as the language of Mathematics. From grouping numbers to solving probability questions and understanding functions and relations, the concepts introduced in this chapter appear throughout Class 11, Class 12, and even higher mathematics. A clear understanding of sets not only strengthens your fundamentals but also makes learning upcoming chapters much easier.

Whether you're revising before a school exam or preparing for JEE Main and JEE Advanced, ALLEN's Class 11 Maths Chapter 1 Sets Revision Notes are designed to help you revise the complete chapter in less time. Inside these notes, you'll find simplified explanations, visual representations, important formulas, standard results, shortcut techniques, and exam-oriented practice pointers that make revision quick, effective, and stress-free.

1.0Chapter Snapshot

Parameter	Details
Difficulty Level	Moderate
Revision Time	2-3 Hours
Key Concepts Covered	Types of Sets, Operations on Sets, De Morgan's Laws, Set Identities and Venn Diagram

Other Important details:

Topic	Overview
Introduction to Sets	Definition, notation, and representation of sets
Types of Sets	Empty, singleton, finite, infinite, equal, equivalent, and universal sets
Subsets	Proper subsets and power sets
Operations on Sets	Union, intersection, difference, and complement
Venn Diagrams	Graphical representation of set operations
Algebra of Sets	Important laws and identities
Intervals	Open, closed, and half-open intervals
Applications	Solving mathematical and real-life problems using sets

2.0Chapter 1 Smart Study Material by ALLEN Experts: Download PDFs

Resource	Status
Chapter Mock Test Paper
Chapter Mock Test Paper Solution

Must-Revise Concepts

finite and infinite sets

Set Theory and Number System

Sets

Sets, Relation and Function PYQ with solutions

Sets

1.0Definition of Sets

In Mathematics, a set is a well-defined collection of different objects, considered as an object in its own right. These objects can be anything: numbers, people, letters, or other abstract entities. Sets are denoted by curly braces, such as {1, 2, 3}, and each object in a set is called an element or member. Sets are fundamental in Mathematics and are used to study relationships, perform operations, and define various Mathematical structures.

2.0Representation of Sets in Set Theory

In set theory, sets are typically represented using curly braces { } to enclose their elements. The elements of a set can be anything, such as numbers, letters, or even other sets. Here are some common ways sets are represented and described in set theory:

Roster Form

In roster form, the elements of a set are listed explicitly within curly braces, separated by commas. For example: Set P = {1, 2, 3, 4, 5}

Set-Builder Form

In set-builder notation, a set is described by specifying a property or condition that its elements must satisfy. The notation is {x | P(x)}, where x represents the elements and P(x) is the property or condition. For example: Set B = {x | x is an even number}

Interval Notation

In Mathematics, intervals can be used to represent sets of real numbers. For example, Set C = {x | 0 < x < 10} can be represented as (0, 10), indicating all real numbers greater than 0 and less than 10.

Special Sets

Empty Set: Denoted by ϕ or { }, it is the set with no elements.
Universal Set: Denoted by U, it is the set containing all elements under consideration.
Singleton Set: A set containing only one element. For example, {7}.

These representations help in defining and describing sets in a clear and concise manner, allowing for precise Mathematical reasoning and operations within set theory.

3.0Elements of Set

Elements of a set are the individual objects or numbers that belong to the set. In set notation, elements are listed inside curly braces { } and separated by commas. The elements of a set can be letters, numbers, symbols, or even other sets. Each element in a set is unique, and there are no duplicate elements within a set.

For example, consider the set A = {1, 2, 3, 4}. In this set:

1, 2, 3, and 4 are the elements of set A.
Each element is distinct and appears only once in the set.
The order of elements in a set does not matter. Thus, {1, 2, 3, 4} is the same as {4, 3, 2, 1} as they contain the same elements.

In set theory, elements are crucial for defining the properties and relationships of sets, performing set operations, and analyzing mathematical concepts.

Also Read : Domain and Range of a Relation

1.0Types of Sets

Null Set or Empty Set or Void Set

A set having no elements is called a Null Set or Empty Set or Void set. It is denoted by ϕ or { }.

Singleton Set

A set consisting of one or a single element is called a Singleton Set.

Finite Set

A set that has only a finite number of elements is called a Finite Set.

Infinite Set

A set which has an infinite number of elements is called an infinite Set.

Subset

If every element in set A is also found in set B, then A is termed as a subset of B, denoted as A ⊆ B.

Proper subset

If set A is a subset of set B, but A is not equal to B, then A is considered a proper subset of B, represented as A ⊂ B.

Universal Set

A set consisting of all possible elements that occur in the discussion is called a Universal set and is denoted by U.

Power set

Let A be any set. The set of all subsets of A is called the power set of A and is denoted by P(A).

Equivalent Set or Equal Set

If every element of set A is also an element of set B, and vice versa, they are equal sets denoted as A = B.

For example, A = {3,4,5,6} and B = {6,5,4,3} are equal sets. However, A = {set of even numbers} and B = {set of natural numbers} are not equal as their elements differ.

Disjoint Sets

Two sets, X and Y are called disjoint set if they have no common elements, denoted as X ∩ Y = ϕ. This signifies that their intersection results in zero elements.

2.0Operations on Sets

Cardinality of a Set

The cardinality of a set A, denoted as |A|, represents the number of elements in the set.

Formula: ∣A∣ = n, where n is the count of elements in set A.

Union of Sets

The union of two sets, denoted as A ∪ B, comprises all the elements that are in either set A or set B, or both.

Formula : A ∪ B = { x : x ∈ A or x ∈ B}

Example: If A = {1,2,3} and B = {3,4,5}, then A ∪ B = {1,2,3,4,5}.

Intersection of Sets

The intersection of two sets, A and B, denoted as A ∩ B, is the set of all elements that are common to both A and B.

Formula : A ∩ B = { x : x ∈ A and x ∈ B}

Example: If A = {1,2,3} and B = {3,4,5}, then A∩B = {3}.

Complement of Sets

The complement of a set A, denoted as A′ or Ā, with respect to a universal set U, contains all elements in U that are not in A.

Formula : A′ = { x ∈ U : x ∉ A}

Example: If U = {1,2,3,4,5} is the universal set and A = {1,2}, then A′ = {3,4,5}.

Difference of Sets

The set difference between sets A and B, denoted as A−B, is the set of elements that are in A but not in B.

Formula : A - B = { x : x ∈ A and x ∉ B}

Example: If A = {1, 2, 3} and B = {3, 4, 5}, then A - B = {1, 2}.

Cartesian Product of Sets

The Cartesian product of Set A and Set B, denoted as A × B, represents the collection of all ordered pairs (a, b) where a is in Set A and b is in Set B.

Formula : A × B = { (a, b) : a ∈ A, b ∈ B}

Power Set

The power set of a set A, denoted as P(A), is the set of all subsets of A, including the empty set and A itself.

Formula : P(A) = { B : B ⊆ A}

These operations are fundamental in set theory and are used to manipulate sets, study relationships between sets, and solve various Mathematical problems.

3.0Set Formulas

Some of the Important Sets Formula are

n(A ∪ B) = n(A) + n(B) – n(A ∩ B)
n(A ∪ B) = n(A) + n(B) ⇔ A, B are disjoint sets.
n( A – B) + n( A ∩ B ) = n(A)
n( B – A) + n( A ∩ B ) = n(B)
n( A – B) + n ( A ∩ B) + n( B – A) = n ( A ∪ B )
n ( A ∪ B ∪ C ) = n(A) + n(B) + n(C) – n ( A ∩ B) – n ( B ∩ C) – n ( C ∩ A) + n ( A ∩ B ∩ C)

4.0Venn Diagrams

Clearly (A – B) ∪ (B – A) ∪ (A ∩ B) = A ∪ B

5.0Properties of Sets

Associative Property

A ∪ ( B ∪ C) = ( A ∪ B) ∪ C
A ∩ ( B ∩ C) = ( A ∩ B) ∩ C

Distributive Property

A ∪ ( B ∩ C) = ( A ∪ B) ∩ ( A ∪ C)
A ∩ ( B ∪ C) = ( A ∩ B) ∪ ( A ∩ C)

De Morgan’s Law

Law of union : ( A ∪ B)′ = A′ ∩ B′
Law of Intersection :( A ∩ B)′ = A′ ∪ B′

Complement Law

A ∪ A′ = A′ ∪ A = U
A ∩ A′ = ϕ

Idempotent Law and Law of a null and universal set

For any Finite set, A

A ∪ A = A
A ∩ A = A
Φ′ = U
ϕ = U′

6.0Solved Examples

Example 1: Write the given statement in Roster Form and Set Builder Form of representation of a set:

The set of all integers that lie between –3 and 7

Solution: Roaster Form: { –2, –1, 0, 1, 2, 3, 4, 5, 6}

Set Builder Form: {x : x ∈ Z, –3 < x < 7}

Example 2: Given the sets X = {1, 2, 3}, Y = {3, 4}, Z = {4, 5, 6}, then X ∩ ( Y ∪ Z) is

Solution: Y ∪ Z = {3, 4, 5, 6}

X ∩ ( Y ∪ Z) = {3}

Example 3: A group of members know at least one of the two languages, Hindi or Kannada. In the group, 150 members know Hindi, 80 members know Kannada and 70 of them know both. How many members are there in the group?

Solutions: Let H = Set of persons who know Hindi.

K = Set of persons who know Kannada.

n(H ∩ K) = the number of persons who know both Hindi and Kannada is 70.

n(H ∪ K) = n(H) + n(K) – n(H ∩ K)

= 150 + 80 – 70 = 160

Example 4: Find A ✕ B when A = {x | x is prime number less than 5} and B = {1, 2, 3}

Solution: A = {2, 3}; B = {1, 2, 3}

A ✕ B = {(2,1), (2,2), (2,3), (3,1), (3,2), (3,3)}

7.0Practice Problems

Question 1: Write the given statement in Roster Form and Set Builder Form of representation of a set:

The set of all integers that lie between 5 and 15.

Question 2: Given the sets A = {5, 2, 7}, B = {3, 8}, C = {3, 2, 6}, then find A ∩ ( B ∪ C).

Question 3: In a class of 60, students like Mathematics, 25 like science, and 15 like both. Then the number of students who like either Mathematics or Science is?

Question 4: Let A and B be the two sets in the universal set. Then A – B equals-

a.A ∩ B′

b. A′ ∩ B

c. A ∩ B

d. None of these

Also Read

Operation on Sets	Sets, Relation and Function Previous Year Questions with Solutions	De Morgan’s Laws
Finite and Infinite Sets	Domain Codomain Range Functions

1.0Learning Outcomes

After studying these revision notes, you will be able to:

Understand the concept and representation of sets with confidence.
Differentiate between various types of sets and their properties.
Solve problems involving subsets, proper subsets, and power sets.
Perform operations like union, intersection, difference, and complement accurately.
Interpret and solve questions using Venn diagrams.
Apply important laws and identities of set algebra in problem-solving.
Use interval notation correctly in mathematical problems.
Solve JEE Main, JEE Advanced, and CBSE-level questions with improved accuracy and speed.

2.05-Minute Quick Revision

Must-Revise Concepts

Definition and notation of sets
Roster and Set-builder forms
Types of sets
Universal set and empty set
Subsets and proper subsets
Power set
Union and intersection
Difference and complement of sets
De Morgan's Laws
Venn diagrams
Interval notation

Important Formula Recall

Number of subsets of a set containing n elements = 2ⁿ
Number of proper subsets = 2ⁿ − 1
n(A ∪ B) = n(A) + n(B) − n(A ∩ B)
n(A') = n(U) − n(A)
De Morgan's Laws

3.0High Weightage Topics

Topic	JEE Importance	Difficulty Level
Types of Sets	High	Easy
Subsets and Power Sets	High	Moderate
Operations on Sets	Very High	Moderate
Venn Diagram Problems	Very High	Moderate
Cardinality of Sets	High	Moderate
De Morgan's Laws	High	Moderate
Set Identities	Medium	Moderate

4.0Formula Handbook

The following formulas, identities, and standard results are essential for solving numerical and conceptual questions from Sets in JEE Main, JEE Advanced, and CBSE examinations.

5.0Common Mistakes & JEE Tips

Although Sets is considered one of the easier chapters in Class 11 Mathematics, students often lose marks due to incorrect interpretation of symbols, Venn diagram errors, and confusion between similar concepts. Developing conceptual clarity is more important than memorizing definitions.

Common Mistake	Correct Approach
Confusing ∈ with ⊂	Remember that ∈ represents an element, whereas ⊂ represents a subset.
Treating an empty set and the set containing zero as the same	∅ and {0} are completely different sets.
Incorrectly counting subsets	Use (2^n) only when the set contains n distinct elements.
Forgetting the intersection while applying the cardinality formula	Always subtract (n(A \cap B)) to avoid double counting.
Drawing inaccurate Venn diagrams	Label each region carefully before solving.
Mixing equal sets with equivalent sets	Equal sets have identical elements, whereas equivalent sets have the same number of elements.
Ignoring universal set while finding complements	Always identify the universal set before solving complement-based questions.
Memorizing identities without understanding them	Practice proving standard identities using Venn diagrams for better retention.

6.0ALLEN Faculty Tips

Learn the meaning of every set notation before attempting numerical problems.
Practice Venn diagram questions regularly to improve logical reasoning.
Memorize important set identities and De Morgan's Laws.
Focus on cardinality-based questions, as they are frequently asked in JEE Main.
Understand the difference between subsets, proper subsets, equal sets, and equivalent sets.
Revise interval notation carefully before moving to Relations and Functions.
Solve previous years' questions to understand the pattern of conceptual questions.

7.0PYQ Trend Analysis

Sets is a fundamental chapter in Class 11 Mathematics. While direct questions are generally straightforward, concepts from this chapter are frequently used in Relations & Functions, Probability, and Mathematical Reasoning, making it important from both conceptual and examination perspectives.

Note: The trend below is based on the analysis of previous years' JEE Main and JEE Advanced question papers. The exact number of questions may vary each year.

Topic	Questions Asked (Last 5 Years)	Average Difficulty
Types of Sets	1–2	Easy
Subsets & Power Sets	2–3	Easy–Moderate
Operations on Sets	2–3	Moderate
Venn Diagram Problems	2–4	Moderate
Cardinality of Sets	2–3	Moderate
De Morgan's Laws	1–2	Moderate
Mixed Concept Questions	1–2	Moderate

8.0Most Common Question Types

Identifying different types of sets.
Finding the number of subsets and proper subsets.
Solving problems using union, intersection, and complement.
Questions based on Venn diagrams involving two and three sets.
Applying cardinality formulas to determine unknown values.
Proving identities using set algebra.
Conceptual MCQs based on De Morgan's Laws and set operations.

9.0Smart Revision Strategy

For a quick revision before your examination, follow this sequence:

Revise all set symbols and notations.
Understand different types of sets with examples.
Practice subsets and power set questions.
Memorize important set operations and identities.
Revise De Morgan's Laws thoroughly.
Practice Venn diagram-based questions.
Solve cardinality problems involving two and three sets.
Revise interval notation.
Attempt Previous Years' Questions (PYQs).
Finish with mixed practice questions for confidence.

Frequently Asked Questions

What is Class 11 Maths Chapter 1: Sets about?

Why is Sets important for JEE Main and JEE Advanced?

Which topics should I focus on while revising Sets?

How can I prepare Sets effectively for JEE?

Are ALLEN's Class 11 Maths Chapter 1 Sets Revision Notes sufficient for quick revision?

Join ALLEN!

Class 11 Maths Chapter 1 Sets: Revision Notes

1.0Chapter Snapshot

2.0Chapter 1 Smart Study Material by ALLEN Experts: Download PDFs

On this page

Sets

1.0Definition of Sets

2.0Representation of Sets in Set Theory

Roster Form

Set-Builder Form

Interval Notation

Special Sets

3.0Elements of Set

On this page

1.0Learning Outcomes

2.05-Minute Quick Revision

Must-Revise Concepts

Important Formula Recall

3.0High Weightage Topics

4.0Formula Handbook

5.0Common Mistakes & JEE Tips

6.0ALLEN Faculty Tips

7.0PYQ Trend Analysis

8.0Most Common Question Types

9.0Smart Revision Strategy

On this page

Frequently Asked Questions

What is Class 11 Maths Chapter 1: Sets about?

Why is Sets important for JEE Main and JEE Advanced?

Which topics should I focus on while revising Sets?

How can I prepare Sets effectively for JEE?

Are ALLEN's Class 11 Maths Chapter 1 Sets Revision Notes sufficient for quick revision?

Join ALLEN!

Class 11 Maths Chapter 1 Sets: Revision Notes

1.0Chapter Snapshot

2.0Chapter 1 Smart Study Material by ALLEN Experts: Download PDFs

On this page

Sets

1.0Definition of Sets

2.0Representation of Sets in Set Theory

Roster Form

Set-Builder Form

Interval Notation

Special Sets

3.0Elements of Set

On this page

1.0Learning Outcomes

2.05-Minute Quick Revision

Must-Revise Concepts

Important Formula Recall

3.0High Weightage Topics

4.0Formula Handbook

5.0Common Mistakes & JEE Tips

6.0ALLEN Faculty Tips

7.0PYQ Trend Analysis

8.0Most Common Question Types

9.0Smart Revision Strategy

On this page