Operations on Sets-Union of Sets, Intersection of Sets, Difference of Sets, Complement of Sets | Class 11 Math Notes Study Material Download Free PDF

What is Set?

A set is a well-defined collection of objects.

The objects may be numbers, alphabets, names of people, etc. Sets are represented using upper-case letters such as A, B, etc. For Example,

A = {a, e, i, o, u} OR A is a set of vowels in the English alphabet.

Note: β€œB = collection of good students” is not a set because we don’t know the criteria for good students. Thus there is some ambiguity as to which students belong to the set and which do not.

Operations on Sets

  • Union of Sets
  • Intersection of Sets  
  • Difference of Sets
  • Complement of Sets

Intersection of Sets  

The intersection of two sets A and B is a set that contains all the elements that are common to both A and B. Formally it is written as 

[Tex]A\cap B = \{ x: x \in A \ and \ x \in B \}[/Tex]

In the following image, the shaded area is the intersection of sets A and B

Sets are fundamental in mathematics and are collections of distinct objects, considered as a whole. In this topic, we will explore the basic operations you can perform on sets, such as union, intersection, difference, and complement. These operations help us understand how sets interact with each other and allow us to solve various problems in mathematics and beyond.

Example: 

If A = {2, 3, 5, 7} and B = {1, 2, 3, 4, 5}
then the intersection of set A and B is the set A ∩ B = {2, 3, 5} 

In this example 2, 3, and 5 are the only elements that belong to both sets A and B. 

Union of Sets

Union of two sets A and B is a set that contains all the elements that are in A or in B or in both A and B. Formally it is written as

[Tex]A\cup B = \{ x: x\in A \ or \  x \in B \}[/Tex]

In the following image, the shaded area is the union of sets A and B.

Union

Example: 

If A = {2, 4, 8} and B = {2, 6, 8}
then the union of A and B is the set A βˆͺ B = {2, 4, 6, 8}

In this example, 2, 4, 6, and 8 are the elements that are found in set A or in set B or in both sets A and B

Complement or Difference Between Sets

The relative complement or set difference of two sets A and B is the set containing all the elements that are in A but not in B. Formally this is written as 

[Tex]A – B = \{ x: x \in A \  and \  x \notin B \}   [/Tex]

Sometimes this is also written as A \ B. In the following image, the shaded area represents the difference set of set A and set B

Difference Between Sets

Note:  A – B is equivalent to A ∩ B’ i.e.,  A – B = A ∩ B’

Example: 

If A = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} and B = {2, 3, 5, 7}
then A – B = {1, 4, 6, 8, 9, 10}
and further B – A = βˆ…

Universal Set and Absolute Complement

Universal set

A universal set is the set of all objects currently under consideration. It is usually denoted by the upper-case letter U. For example for a set of vowels, the universal set may be the set of alphabets.

Note: A set is always a subset of the universal set. 

[Tex] A\subseteq U  [/Tex]

Absolute complement

The absolute complement of a set A is the set of all elements that are in U but not in A. It is denoted as A’. In the following image, the shaded area represents the complement of set A

Absolute complement

The absolute complement is sometimes just called complement.

Note: A’ is equivalent to U – A i.e.,  A’ = U – A

Example: 

If U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} and A = {1, 2, 3}
then A’ = {4, 5, 6, 7, 8, 9, 10} = U – A

[Tex] A \subseteq B, \  if \  \forall x \  \{ x\in A \Rightarrow x \in B \}   [/Tex]

Subset and Proper Subset

Subset

For two set A and B, A is a subset of B if every element in A is also in B. A can be equal to B. This is formally written as  

Subset

In the following image, set A is a subset of B

[Tex]\phi \subseteq A   [/Tex]

Example: 

If A = {2, 4} and B = {1, 2, 3, 4, 5, 6, 7, 8}
then A is a subset of B

In this example, A is a subset of B, because all the elements in A are also in B

Notes: 

1. An empty set (or null set) is a subset of every set.

[Tex]A \subset B, \  if \  \forall x \  \{ x\in A \Rightarrow x \in B \} \  and \  A\neq B   [/Tex]

Example: 

βˆ… is a subset of the set {1, 2, 3, 4}

2. For a set A, the number of possible subsets is 2|A|. Where |A| = number of elements in A.

Example: 

For the set C = {1, 2, 3}, there are 23 = 8 possible subsets
they are βˆ…, {1}, {2}, {3}, {1, 2}, {2, 3}, {1, 3}, {1, 2, 3}

Proper subset (also called strict subset)

For two sets A and B, A is a proper subset of B, if A is a subset of B and A is not equal to B. Formally it is written as 

[Tex]B \supseteq A \  \ if \  A \subseteq B  [/Tex]

Example: 

For a set B = {1, 2, 3},
βˆ…, {1}, {2}, {3}, {1, 2}, {2, 3}, {1, 3} are all proper subsets of B
Note that {1, 2, 3} is not a proper subset of B, because they are equal

Superset and Proper Superset

Superset

For two sets A and B, if A is a subset of B then B is the superset of A. A can be equal to B. Formally it is denoted as 

Superset

In the following image, set B is the superset of set A

[Tex]B \supset A, \  \  if \  A\subseteq B \ and \  A\neq B    [/Tex]

Examples: 

  • If A = {2, 4} and B = {1, 2, 3, 4, 5, 6, 7, 8}
    then B is the superset of A, because A is a subset of B
     
  • If A = {11, 12} and B = {11, 12 } then B is the super set of A

Proper superset (also called strict superset)

For two sets A and B, if A is a subset of B and A is not equal to B, then B is the proper superset of A. Formally it is written as 

Examples: 

  • If A = {1, 2, 3} and B = {0, 1, 2, 3, 4, 5}
    then B is a proper superset of A, because A is a subset of B and A β‰  B  
     
  • If A = {2, 4, 6} and B = {2, 4, 6} then B is not a proper superset of A, because A = B 

Bringing the Set Operations Together

De Morgan’s laws

  1. The complement of the union of two sets is equal to the intersection of their complements
    i.e.,  (A βˆͺ B)’ = A’ ∩ B’
  2. The complement of the intersection of two sets is equal to the union of  their complements
    i.e.,  (A ∩ B)’ = A’ βˆͺ B’

Formula for the Cardinality of Union and Intersection

Formula for the Cardinality of Union and Intersection

The formula for the Cardinality of Union and Intersection is given below:

∣A βˆͺ B∣ = ∣A∣ + ∣B∣ βˆ’ ∣A ∩ B∣ 

Proof:

We can write
    |A βˆͺ B| = |A – B| + |A ∩ B| + |B – A|               β€”- by the sum of disjoint sets, refer to the Venn diagram above
    |A βˆͺ B| = (|A| – |A ∩ B|) + |A ∩ B| + |B – A|    β€”- Substitute |A – B| = |A| – |A ∩ B|
    |A βˆͺ B| = |A| + |B – A|                                    β€”- Simplify
    |A βˆͺ B| = |A| + |B| – |A ∩ B|                            β€”- Substitute |B – A| = |B| – |A ∩ B|)

Problems on Operations on Sets – Union & Intersection

Problem 1: There are 100 students in a class, 45 students said that they liked apples, and 30 of the students said that they liked both apples and oranges. Every student has to choose at least one of the two fruits. Find how many students like oranges.

Solution:

Let U = set of all students in the class
      A = set of students that like apples
      B = set of students that like oranges

Given: 
    |A| = 45
    |A ∩ B| = 30
    |U| = |A βˆͺ B| = 100 (because every student has to choose)

We need to find how many like oranges. i.e., |B|

The formula to be used is,
    |A βˆͺ B| = |A| + |B| – |A ∩ B|        β€”-(i)

Subtract |A| – |A ∩ B| from both sides in (i) to get
    |A βˆͺ B| – (|A| – |A ∩ B|) = |B|
or |B| = |A βˆͺ B| – (|A| – |A ∩ B|)

Substitute the given values and simplify,
    |B| = |A βˆͺ B| – (|A| – |A ∩ B|)
         = 100 – ( 45 -30 )
         = 85

Thus the number of students that like oranges is 85.

Problem 2: There are a total of 120 students in a class. 70 of them study mathematics, 40 study science, and 10 students study both mathematics and science. Find the number of students who
    i) Study mathematics but not science
    ii) Study science but not mathematics
    iii) Study mathematics or science

Solution:  

Let,
   U = set of all students in the class
   M = set of students that study mathematics
   S = set of students that study science

Our universal set here has 120 student i.e, |U| = 120

Given,
    |M| = 70
    |S| = 40
    |M ∩ S| = 10  (number of students that study both mathematics and science)

i) Finding the number of students that study mathematics but not science. In the following image, the shaded area represents the set of students that study mathematics but not science.

Sets are fundamental in mathematics and are collections of distinct objects, considered as a whole. In this topic, we will explore the basic operations you can perform on sets, such as union, intersection, difference, and complement. These operations help us understand how sets interact with each other and allow us to solve various problems in mathematics and beyond.

We are required to find |M – S|
By the Venn diagram, we can see that |M – S| can be written as |M| – |M ∩ S|
thus,
    |M – S| = |M| – |M ∩ S|
               = 70 – 10
               = 60

Thus the number of students who study mathematics but not science is 60

ii)  Finding the number of students that study science but not mathematics.  In the following image, the shaded area represents the set of students that study science but not mathematics

Sets are fundamental in mathematics and are collections of distinct objects, considered as a whole. In this topic, we will explore the basic operations you can perform on sets, such as union, intersection, difference, and complement. These operations help us understand how sets interact with each other and allow us to solve various problems in mathematics and beyond.

We are required to find |S – M|
By the Venn diagram, we can see that |S – M| can be written as |S| – |M ∩ S|
thus,    
    |S – M| = |S| – |M ∩ S|
               = 40 – 10
               = 30

Thus the number of students who study science but not mathematics is 30

iii) Finding the number of students who study mathematics or science.  In the following image, the shaded area represents the set of students that study mathematics or science.

Sets are fundamental in mathematics and are collections of distinct objects, considered as a whole. In this topic, we will explore the basic operations you can perform on sets, such as union, intersection, difference, and complement. These operations help us understand how sets interact with each other and allow us to solve various problems in mathematics and beyond.

We are required to find |M βˆͺ S|
By using the formula, |M βˆͺ S| = |M| + |S| – |M ∩ S|
    |M βˆͺ S| = |M| + |S| – |M ∩ S|
                 = 70 + 40 – 10
                 = 100

Thus the number of students who study science or mathematics is 100.

Operations on Sets – Solved Examples

Problem 1: Let A = {1, 2, 3, 4, 5}, B = {3, 4, 5, 6, 7}, and C = {4, 5, 6, 7, 8}. Find (A β–³ B) ∩ (B β–³ C), where β–³ represents symmetric difference.

Solution:

First, let’s find A β–³ B:

A β–³ B = (A βˆͺ B) – (A ∩ B)

A βˆͺ B = {1, 2, 3, 4, 5, 6, 7}

A ∩ B = {3, 4, 5}

A β–³ B = {1, 2, 6, 7}

Now, let’s find B β–³ C:

B β–³ C = (B βˆͺ C) – (B ∩ C)

B βˆͺ C = {3, 4, 5, 6, 7, 8}

B ∩ C = {4, 5, 6, 7}

B β–³ C = {3, 8}

Finally, we find the intersection of these results:

(A β–³ B) ∩ (B β–³ C) = {1, 2, 6, 7} ∩ {3, 8} = βˆ… (empty set)

Problem 2 : Given U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}, A = {1, 3, 5, 7, 9}, B = {2, 4, 6, 8}, and C = {1, 2, 3, 4, 5}.

Verify that (A βˆͺ B)’ = A’ ∩ B’, where β€˜ denotes complement with respect to U.

Solution:

A βˆͺ B = {1, 2, 3, 4, 5, 6, 7, 8, 9}

Find (A βˆͺ B)’:

(A βˆͺ B)’ = {10}

Find A’:

A’ = {2, 4, 6, 8, 10}

Find B’:

B’ = {1, 3, 5, 7, 9, 10}

Find A’ ∩ B’:

A’ ∩ B’ = {10}

Verify that (A βˆͺ B)’ = A’ ∩ B’:

Both sets equal {10}, so the equality holds.

Example 3 : Given sets A, B, and C, prove that A – (B βˆͺ C) = (A – B) ∩ (A – C).

Solution:

We’ll prove this by showing that an element x belongs to the left side if and only if it belongs to the right side.

Let x ∈ A – (B βˆͺ C)

This means x ∈ A and x βˆ‰ (B βˆͺ C)

x βˆ‰ (B βˆͺ C) implies x βˆ‰ B and x βˆ‰ C

Therefore:

x ∈ A and x βˆ‰ B, so x ∈ (A – B)

x ∈ A and x βˆ‰ C, so x ∈ (A – C)

Since x is in both (A – B) and (A – C), we can conclude:

x ∈ (A – B) ∩ (A – C)

Conversely, if x ∈ (A – B) ∩ (A – C):

x ∈ A and x βˆ‰ B

x ∈ A and x βˆ‰ C

This implies:

x ∈ A and x βˆ‰ B and x βˆ‰ C

Which is equivalent to: x ∈ A and x βˆ‰ (B βˆͺ C)

Therefore, x ∈ A – (B βˆͺ C)

Thus, we’ve shown that an element belongs to A – (B βˆͺ C) if and only if it belongs to (A – B) ∩ (A – C), proving the equality.

Problem 4 : Cartesian Product and Power Set

Let A = {1, 2} and B = {a, b}. Find |P(A Γ— B)|, where P denotes the power set and Γ— denotes the Cartesian product.

Solution:

First, find A Γ— B:

A Γ— B = {(1,a), (1,b), (2,a), (2,b)}

|A Γ— B| = 4

For any set S, |P(S)| = 2^|S|

Therefore, |P(A Γ— B)| = 2^4 = 16

Problem 5 : Set Equality

Prove that (A – B) βˆͺ (B – A) = (A βˆͺ B) – (A ∩ B)

Solution:

Let x be an arbitrary element. We’ll show x is in the left side if and only if it’s in the right side.

x ∈ (A – B) βˆͺ (B – A)

⇔ x ∈ (A – B) or x ∈ (B – A)

⇔ (x ∈ A and x βˆ‰ B) or (x ∈ B and x βˆ‰ A)

⇔ (x ∈ A or x ∈ B) and (x βˆ‰ A or x βˆ‰ B)

⇔ x ∈ (A βˆͺ B) and x βˆ‰ (A ∩ B)

⇔ x ∈ (A βˆͺ B) – (A ∩ B)

Thus, the two sets are equal.

Problem 6: Set Cardinality Let A and B be finite sets. Prove that |A βˆͺ B| = |A| + |B| – |A ∩ B|.

Solution:

Consider elements in A βˆͺ B:

Elements in A but not in B: |A| – |A ∩ B|

Elements in B but not in A: |B| – |A ∩ B|

Elements in both A and B: |A ∩ B|

Sum these up:

|A βˆͺ B| = (|A| – |A ∩ B|) + (|B| – |A ∩ B|) + |A ∩ B|

= |A| + |B| – |A ∩ B|

Problem 7 : Set Operations and Functions

Let f: A β†’ B be a function. Prove that f(A – B) βŠ† f(A) – f(B) for any subset B of A.

Solution:

Let y ∈ f(A – B). We need to show y ∈ f(A) – f(B).

Since y ∈ f(A – B), there exists x ∈ A – B such that f(x) = y.

x ∈ A – B implies x ∈ A and x βˆ‰ B.

Since x ∈ A, we know y = f(x) ∈ f(A).

We need to show y βˆ‰ f(B). If y ∈ f(B), there would exist z ∈ B such that f(z) = y.

But f(z) = y = f(x), and x βˆ‰ B. This contradicts x ∈ A – B.

Therefore, y ∈ f(A) and y βˆ‰ f(B), so y ∈ f(A) – f(B).

Thus, f(A – B) βŠ† f(A) – f(B).

Problem 8: Principle of Inclusion-Exclusion ,For finite sets A, B, and C, prove:

|A βˆͺ B βˆͺ C| = |A| + |B| + |C| – |A ∩ B| – |B ∩ C| – |A ∩ C| + |A ∩ B ∩ C|

Solution:

Start with |A βˆͺ B βˆͺ C|.

Add |A|, |B|, and |C|. This counts elements in A, B, C, but overcounts elements in intersections.

Subtract |A ∩ B|, |B ∩ C|, and |A ∩ C| to correct for double counting.

However, elements in A ∩ B ∩ C have now been subtracted too many times, so add |A ∩ B ∩ C| back.

This gives the formula: |A βˆͺ B βˆͺ C| = |A| + |B| + |C| – |A ∩ B| – |B ∩ C| – |A ∩ C| + |A ∩ B ∩ C|

Problem 9: Countable and Uncountable Sets

Prove that the set of all infinite binary sequences is uncountable.

Solution:

We’ll use Cantor’s diagonalization argument:

Assume the set is countable. Then we can list all sequences:

s1: a11, a12, a13, …

s2: a21, a22, a23, …

s3: a31, a32, a33, …

…

Construct a new sequence t: t1, t2, t3, … where:

ti = 1 if aii = 0

ti = 0 if aii = 1

This new sequence t differs from every sequence in the list:

It differs from s1 in the 1st digit

It differs from s2 in the 2nd digit

It differs from s3 in the 3rd digit

…

Therefore, t is not in the list, contradicting our assumption that the list contained all sequences.

Thus, the set of all infinite binary sequences is uncountable.

Problem 10: Axiom of Choice

Using the Axiom of Choice, prove that every vector space has a basis.

Solution:

Let V be a vector space. Let S be the set of all linearly independent subsets of V.

Define a partial order ≀ on S by inclusion: A ≀ B if and only if A βŠ† B.

Let C be a chain in S (i.e., a totally ordered subset).

Let U = βˆͺ{A : A ∈ C}. We claim U is an upper bound for C in S.

U is linearly independent: If not, some finite subset {u1, …, un} βŠ† U would be linearly dependent.

But each ui is in some Ai ∈ C, and since C is a chain, all ui are in the largest of these Ai.

This contradicts Ai being linearly independent.

By Zorn’s Lemma (equivalent to Axiom of Choice), S has a maximal element M.

M is a basis for V:

M is linearly independent by definition of S.

M spans V: If not, there exists v ∈ V not in span(M).

Then M βˆͺ {v} would be linearly independent and strictly larger than M, contradicting maximality.

Therefore, M is a basis for V.

Practice Problems on Operations on Sets

1. Let A = {1, 2, 3, 4, 5}, B = {2, 4, 6, 8}, and C = {3, 4, 5, 6}. Find:

  • (A ∩ B) βˆͺ (B ∩ C)
  • (A βˆͺ B) – C

2. If |A| = 5, |B| = 7, and |A βˆͺ B| = 10, find |A ∩ B|.

3. Prove or disprove: For any sets A, B, and C, (A – B) – C = A – (B βˆͺ C).

4. Let U be the universal set. Prove that for any sets A and B:

(A’ ∩ B’)’ = A βˆͺ B

5. If A and B are finite sets with |A| = m and |B| = n, Prove that:

|P(A) Γ— P(B)| = 2m+n

6. Let f: A β†’ B be a function. Prove or disprove:

For any subsets X and Y of A, f(X ∩ Y) = f(X) ∩ f(Y)

7. Given that A, B, and C are sets, prove or disprove:

8. If A βŠ† B and B βŠ† C, then P(A) βŠ† P(B) βŠ† P(C)

9. Let A be a finite set with n elements. How many different pairs of subsets (X, Y) are there such that X βˆͺ Y = A

10. Let A be an infinite set and B be a finite set. Prove that |A| = |A – B|.

Set theory forms the foundation of modern mathematics, encompassing fundamental concepts like unions, intersections, complements, and Cartesian products. It deals with the properties of collections of objects, exploring relationships between sets through operations and identities. Key areas include understanding set cardinality, power sets, and the nuances of infinite sets. Set theory problems often involve proving equalities, working with functions on sets, and applying principles like inclusion-exclusion. These concepts are crucial in various mathematical fields and have practical applications in computer science, particularly in areas like database design and algorithm analysis. Mastering set theory requires a blend of logical reasoning, algebraic manipulation, and abstract thinking, providing a powerful toolset for tackling complex mathematical and computational challenges.

FAQs on Operations on Sets

How can operations be performed on sets?

In a set theory, there are three major types of operations performed on sets, such as:

  • Union of sets (βˆͺ)
  • Intersection of sets (∩)
  • Difference of sets ( – )

What is the rule of set operations?

For two given sets A and B, AβˆͺB (read as A union B) is the set of distinct elements that belong to set A and set B or both. The number of elements in A βˆͺ B is given by n(AβˆͺB) = n(A) + n(B) βˆ’ n(A∩B), where n(X) is the number of elements in set X.

Why are set operations important?

These operations are fundamental in set theory and are used to manipulate sets, define relationships between sets, and solve problems involving collections of objects. There are three major types of operations performed on sets, such as: Union of sets. Intersection of sets

What is associative law of set operation?

Associative law in sets asserts that grouping of sets when conducting set operations has no effect on the end output. In other words, it makes no difference how we arrange the sets; the result will be the same. We can better understand it by the example mentioned below: a βˆͺ ( b βˆͺ c ) = ( a βˆͺ b ) βˆͺ c.

Er. Neeraj K.Anand is a freelance mentor and writer who specializes in Engineering & Science subjects. Neeraj Anand received a B.Tech degree in Electronics and Communication Engineering from N.I.T Warangal & M.Tech Post Graduation from IETE, New Delhi. He has over 30 years of teaching experience and serves as the Head of Department of ANAND CLASSES. He concentrated all his energy and experiences in academics and subsequently grew up as one of the best mentors in the country for students aspiring for success in competitive examinations. In parallel, he started a Technical Publication "ANAND TECHNICAL PUBLISHERS" in 2002 and Educational Newspaper "NATIONAL EDUCATION NEWS" in 2014 at Jalandhar. Now he is a Director of leading publication "ANAND TECHNICAL PUBLISHERS", "ANAND CLASSES" and "NATIONAL EDUCATION NEWS". He has published more than hundred books in the field of Physics, Mathematics, Computers and Information Technology. Besides this he has written many books to help students prepare for IIT-JEE and AIPMT entrance exams. He is an executive member of the IEEE (Institute of Electrical & Electronics Engineers. USA) and honorary member of many Indian scientific societies such as Institution of Electronics & Telecommunication Engineers, Aeronautical Society of India, Bioinformatics Institute of India, Institution of Engineers. He has got award from American Biographical Institute Board of International Research in the year 2005.

CBSE Class 11 Maths Syllabus for 2023-24 with Marking Scheme

CBSE syllabus for class 11 Maths is divided into 5 units. The table below shows the units, number of periods and marks allocated for maths subject. The maths theory paper is of 80 marks and the internal assessment is of 20 marks.

No.UnitsMarks
I.Sets and Functions23
II.Algebra25
III.Coordinate Geometry12
IV.Calculus08
V.Statistics and Probability12
Total Theory80
Internal Assessment20
Grand Total100

2025-26 CBSE Class 11 Maths Syllabus

Below you will find the CBSE Class Maths Syllabus for students.

Unit-I: Sets and Functions

1. Sets

Sets and their representations, empty sets, finite and infinite sets, equal sets, subsets, and subsets of a set of real numbers, especially intervals (with notations), universal set, Venn diagrams, union and intersection of sets, difference of sets, complement of a set and properties of complement.

2. Relations & Functions

Ordered pairs, Cartesian product of sets, number of elements in the Cartesian product of two finite sets, Cartesian product of the set of reals with itself (upto R x R x R), definition of relation, pictorial diagrams, domain, co-domain and range of a relation. Function as a special type of relation. Pictorial representation of a function, domain, co-domain and range of a function. Real valued functions, domain and range of these functions, constant, identity, polynomial, rational, modulus, signum, exponential, logarithmic and greatest integer functions, with their graphs. Sum, difference, product and quotients of functions.

3. Trigonometric Functions

Positive and negative angles, measuring angles in radians and in degrees and conversion from one measure to another, definition of trigonometric functions with the help of unit circle, truth of the identity, signs of trigonometric functions, domain and range of trigonometric functions and their graphs, expressing sin (xΒ±y) and cos (xΒ±y) in terms of sinx, siny, cosx & cosy and their simple applications.

Unit-II: Algebra

1. Complex Numbers and Quadratic Equations

Need for complex numbers, especiallyβˆšβˆ’1, to be motivated by the inability to solve some of the quadratic equations. Algebraic properties of complex numbers, Argand plane.

2. Linear Inequalities

Linear inequalities, algebraic solutions of linear inequalities in one variable and their representation on the number line.

3. Permutations and Combinations

The fundamental principle of counting. Factorial n. (n!) Permutations and combinations, derivation of Formulae for nPr and nCr and their connections, simple applications.

4. Binomial Theorem

Historical perspective, statement and proof of the binomial theorem for positive integral indices, Pascal’s triangle, simple applications.

5. Sequence and Series

Sequence and series, arithmetic progression (A. P.), arithmetic mean (A.M.),  geometric progression (G.P.), general term of a G.P., sum of n terms of a G.P., infinite G.P. and its sum, geometric mean (G.M.), relation between A.M. and G.M.

Unit-III: Coordinate Geometry

1. Straight Lines

Brief recall of two-dimensional geometry from earlier classes. Slope of a line and angle between two lines. Various forms of equations of a line: parallel to axis, point-slope form, slope-intercept form, two-point form, intercept form and normal form. General equation of a line. Distance of a point from a line.

2. Conic Sections

Sections of a cone: circles, ellipse, parabola, hyperbola, a point, a straight line and a pair of intersecting lines as a degenerated case of a conic section. Standard equations and simple properties of parabola, ellipse and hyperbola. Standard equation of a circle.

3. Introduction to Three-Dimensional Geometry

Coordinate axes and coordinate planes in three dimensions. Coordinates of a point. Distance between two points.

Unit-IV: Calculus

1. Limits and Derivatives

Derivative introduced as rate of change both as that of distance function and geometrically, intuitive idea of limit, limits of polynomials and rational functions trigonometric, exponential and logarithmic functions, definition of derivative relate it to the slope of the tangent of the curve, derivative of sum, difference, product and quotient of functions. Derivatives of polynomial and trigonometric functions.

Unit-V: Statistics and Probability

1. Statistics

Measures of Dispersion: Range, mean deviation, variance and standard deviation of ungrouped/grouped data.

2. Probability

Events; occurrence of events, β€˜not’, β€˜and’ and β€˜or’ events, exhaustive events, mutually exclusive events, Axiomatic (set theoretic) probability, connections with other theories of earlier classes. Probability of an event, probability of β€˜not’, β€˜and’ and β€˜or’ events.

Students can also get the syllabus of all the subjects by visiting CBSE Class 11 Syllabus page. Learn Maths & Science in an interactive & fun-loving way with Anand Classes App/Tablet.

Frequently Asked Questions on CBSE Class 11 Maths Syllabus 2025-26

Q1

What is the marks distribution for internals and theory exams according to the CBSE Maths Syllabus for Class 11?

The marks distribution for internals is 20 marks and the theory exam is 80 marks based on the CBSE Class 11 Maths Syllabus.

Q2

Which is the most important chapter in the CBSE Class 11 Maths Syllabus?

The important chapter in the CBSE Class 11 Maths Syllabus is Algebra which is for 25 marks in the overall weightage.

Q3

What are the chapters covered in Unit III of the CBSE Class 11 Maths Syllabus?

The chapters covered in Unit III of the CBSE Class 11 Maths Syllabus are straight lines, conic sections and an introduction to three-dimensional geometry.