NCERT Class 11 Maths Chapter 1 Notes Sets - Download PDF

Imagine you are in charge of compiling the participants' list for your college’s annual fest, which includes multiple events. Now, some participants' names are in more than one category, and to avoid any duplication, you need to compare and group these lists. Set theory, which is covered in Chapter 1 of Maths class 11, comes into play in these types of situations and proves to be the savior by comparing and organizing the data. In class 11 maths chapter 1 NCERT notes, we discuss some basic definitions and operations involving sets. These notes focus on making the learning process smooth for students.

Sets have many real-life applications, and strengthening the core concepts is very important to achieving good marks not only in the class 11 exam but also in other competitive exams. Students can use the set of class 11 notes as a revision tool after finishing the textbook exercises. These sets of class 11 questions and answers have been prepared by experienced Careers360 experts and are filled with important concepts and formulas. Students can also check the NCERT Exemplar Class 11 Maths Chapter 1 Sets for further references.

Sets

A set is a well-defined collection of elements. These elements may include numbers, symbols, variables, etc.

We generally represent them using capital letter(A, B, C, D, Z, N etc)

Elements in the sets are represented by small letters(a, b, c, d, e)

Elements of sets are represented using closed brackets { } .

Examples of sets:

• i) set of first n natural numbers: N = 1, 2, 3, 4, 5,.....,n
• ii) set of rational numbers: R.
• iii) set of whole numbers: W.
• iv) set of positive integers: Z+
• v) set of negative integers: Z-

Representation of Sets:

We have 2 methods to represent a set:

1. Roster or Tabular Form

In roster form, all the elements of a set are listed; the elements are separated by commas and are enclosed within braces $\{\}$.

Example: $\{\mathrm{a},\mathrm{e},\mathrm{i},\mathrm{o},\mathrm{u}\}$ represents the set of all the vowels in the English alphabet in roster form.
In roster form, the order in which the elements are listed is immaterial, i.e. the set of all natural numbers which divide 14 is $\{1,2,7,14\}$ and can also be represented as $\{1,14,7,2\}$.

An element is not generally repeated in the roster form of a set, i.e., all the elements are taken as distinct. For example, the set of letters forming the word 'SCHOOL' is $\{\mathrm{S},\mathrm{C},\mathrm{H},\mathrm{O},\mathrm{L}\}$ or $\{\mathrm{H},\mathrm{O},\mathrm{L},\mathrm{C},\mathrm{S}\}$. Here, the order of listing elements has no relevance.

2. Set-builder Form

In set-builder form, all the elements of a set possess a single common property that is not possessed by any element outside the set. If $Z$ contains all values of $x$ for which the condition $q(x)$ is true, then we write

$Z=\{x:q(x)\}\text{ or }Z=\{x\mid q(x)\}$
Where, ': ' or ' | ' is read as 'such that'
eg. The set $A=\{0,1,8,27,64,\dots$.$\}$ can be written in Set Builder form as
$\mathrm{A}=\{{\mathrm{X}}^3:\mathrm{X}$ is a non-negative integer $\}$

Types of Sets:

Empty set: A set that does not contain any element and is empty. Such sets are called empty (or) null (or) void sets.

Represented as: { } or ∅

Eg: A={x:1<x,x is a natural number}

So A is an empty set, as we don't have any natural element that is less than 1.

Finite set: A set that is empty or consists of a finite number of elements is called a finite set.
Examples: $\phi ,\{\mathrm{a}\},\{1,2,5,9\},\{\mathrm{x}:\mathrm{x}$ is a person of age more than 18$\}$

Infinite set: A set that has infinite elements is called an infinite set.
Examples:
a set of all the lines passing through a point,
set of all circles in a plane,
set of all points in a plane,
$N,Z,Q,Q^{\prime },R$
$\{x:2<x<2.1\}$
Note: All finite sets can be represented in set-builder form, but infinite sets cannot be represented in set-builder form because they have no fixed pattern.

Singleton set: A set which is having only one element is called a singleton set.
Example: $\{3\},\{b\}$,
$\{\{1,2,3\}\}$ is also a singleton as it has one element, which is a set.
$\{\phi \}$ is also a singleton set
Note: Empty and singleton sets are finite sets.

Equal sets: Sets where two sets have exactly the same elements.

Represented by the symbol: “ = “

Examples:

i) Let A, B be two sets.

A= set of natural even elements less than 10

A = {2, 4, 6, 8}

B = set of multiples of 2 less than 10 that are natural

B = {2, 4, 6, 8}

Here, A = B. Since every element of A is also an element of B and vice versa.

ii) A = {1, 2, 3, 4}

B = {1, 3, 2, 1, 4, 1}

Then also A = B, the set elements do not change even if repeated.

The order of the elements also doesn’t matter.

Subset:

Any Set that has every element of another set is called a subset.

Representation: A⊂B if a ∈A ⇒ a ∈B

The above representation means that A should have every element of set B, but B need not have every element of A.

NOTE: Every set is a subset of itself

Eg: A = {1, 3, 5, 7, 9}

B = {1, 2, 3, 4, 5, 6, 7, 8, 9}

Here, A⊂B because every element of A is an element of B and vice versa doesn’t matter

Venn diagram of a subset:

Proper set: If A⊂B and A≠B, then such set A is called a proper subset of B.

B is called the superset of A.

Eg: A = {1, 2, 3, 4, 5, 6}

B = {1, 2, 3, 4, 5, 6, 7, 8}

Here A⊂B and A≠B.

So, A is a proper subset of B.

B is a superset of A.

Intervals of subsets:

When a,b ∈ real numbers, a<b then,

Set of real numbers: {x: a<x<b} is an open interval, represented as (a, b).

The set that includes endpoints also then such intervals are closed intervals.

{y: a ≤ y ≤ b}, closed interval represented by [a, b].

Eg: (p, q] ⇒ {x: p < x ≤ q}

Here in the set elements include q and exclude p.

Pictorial representation of intervals:

Power set: It is a set that is a collection of all subsets including null or empty sets and itself denotes a power set.

Represented by p(A).

Eg: A = {1, 2, 3, 4}

Subsets: ∅, {1}, {2}, {3}, {4}, {1, 2}, {1, 3}, {1,4}, {2,3}, {2, 4}, {3, 4}, {1, 2, 3}, {1, 2, 4}, {1, 3, 4}, {2, 3, 4}, {1, 2, 3, 4}

Total subsets:16

Points:

1. First is a null set

2. Next singleton sets(sets with one element}

3. Next, with two elements

4. The set itself

Remember: {1, 2} = {2, 1}

(The order doesn’t matter, so it should be written only once.)

Formula to find a number of subsets:

$\mathrm{m}=$ number of elements in set A
Formula: $2^m$
$Eg:A=\{1,2,3,4\}t=2^4=16$. We found the same above.

$A=\{1,2,3,4\}$

Subsets: ∅, {1}, {2}, {3}, {4}, {1, 2}, {1, 3}, {1, 4}, {2, 3}, {2, 4}, {3, 4}, {1, 2, 3}, {1, 2, 4}, {1, 3, 4}, {2, 3, 4}, {1, 2, 3, 4}

Total subsets:16

Universal Set:

The set that includes all the elements is called a universal set.

Represented using U.

Venn Diagram:

{0, 1, 2, 3, 4, 5, 6, 7, 8, 9} all belong to the universal set.

Operation of Sets:

Union of sets: The set contains all the elements of A and B and repeating elements of both sets A, B is taken only once.

Represented by: ∪

Read as A union B.

Set builder form of union sets:

{y: y∈A or y∈B}

Venn diagram of union:

Eg: A = {1, 2, 3, 4, 5} B = {2, 4, 5, 6, 7, 8}

A∪B = {1, 2, 3, 4, 5, 6, 7, 8}

Properties of union

• $A\cup B=B\cup A$ (Commutative Property)
• $(A\cup B)\cup C=A\cup (B\cup C)$ (Associative property)
• $\mathrm{A}\cup \phi =\mathrm{A}$ (Law of identity element, $\phi$ is the identity of Null Set)
• $\mathrm{A}\cup \mathrm{A}=\mathrm{A}$ (Idempotent law)
• $U\cup A=U($ Law of $U)$
• If $A$ is a subset of $B$, then $A\cup B=B$

The Intersection of Sets:

This set contains the elements that are common in the sets.

Represented as: ∩

Read as A intersection B.

Set builder form of intersection:

{y: y ∈ A and y ∈ B}

Venn diagram of AB:

Eg: A = {1, 2, 3, 4, 5, 6} B = (2, 4, 6, 8, 10}

A∩B = {2, 4, 6}

Properties of intersection

• $\mathrm{A}\cap \mathrm{B}=\mathrm{B}\cap \mathrm{A}$ (Commutative law).
• $(A\cap B)\cap C=A\cap (B\cap C)$ (Associative law).
• $\mathrm{A}\cap \varphi =\varphi$,
• $\mathrm{A}\cap \mathrm{U}=\mathrm{A}$ (Law of $\varphi$ and U$)$.
• $\mathrm{A}\cap \mathrm{A}=\mathrm{A}$ (Idempotent law)
• If $A$ is subset of $B$, then $A\cap B=A$

Difference of Sets:

In the set, the elements belong to A but do not belong to B; such sets are called Differences of sets.

Represented as: A-B

Set builder form: {y: y ∈ A and y ∉ B}

Venn diagram of A - B:

Eg: A = {1, 2, 3, 4, 5}

B = {2, 3, 4, 8, 9}

A - B = {1, 5}

Here, even though we have {2, 3, 4} in A as they belong to B also do not include them in A - B.

Complement of set:

The elements other than A that are in the universe come under complement elements.

Represented by: A’

Set builder form: {y: y ∈ U and y ∉ A}

Venn diagram:

Eg: A = {y: y ∈ U and not divisor of 24}

A’ = {1, 2, 3, 4, 6, 8, 12, 24}

Properties of complement sets:

1. A∪A’ = U (complement laws)

2. A∩ A’ = ∅ (complement laws)

3. (A∪B)’ = A’∩B’ (Demorgan’s law)

4. (A∩B)’ = A’∪ B’ (Demorgan’s law)

5. (A’)’ = A (double complement)

6. U’ = ∅ and ∅’= U

De-Morgan’s Laws

1. (A ∪ B)′ = A′ ∩ B′

Let $x$ be any element in $(A\cup B{)}^{\prime }$

$x\in (A\cup B{)}^{\prime }\iff x\notin (A\cup B)$

$\iff x\notin A$ and $x\notin B$ (As $x$ does not belong to $A\cup B$, it cannot belong to both $A$ and B)
$\begin{array}{rl}& \iff x\in A^{\prime }\text{ and }x\in B^{\prime }\\ & \iff x\in (A^{\prime }\cap B^{\prime })\\ \therefore x\in (A\cup B{)}^{\prime }\iff & x\in (A^{\prime }\cap B^{\prime })\end{array}$
So, any element that belongs to $(A\cup B{)}^{\prime }$ also belongs to $(A^{\prime }\cap B^{\prime })$, and vice versa
So, these sets have exactly the same elements.
Hence, they are equal.

2. (A ∩ B)′ = A′ ∪ B′

Let $x$ be any element in $(A\cap B{)}^{\prime }$

$x\in (A\cap B{)}^{\prime }\iff x\notin (A\cap B)$

$kx\notin A$ or $x\notin B$ (as $x\notin (A\cap B)$, means it is not common in $A$ and $B$, and thus either it is not in $A$ or not in B)

$\begin{array}{rl}& \iff x\in A^{\prime }\text{ or }x\in B^{\prime }\\ & \iff x\in (A^{\prime }\cup B^{\prime })\\ & \therefore x\in (A\cap B{)}^{\prime }\iff x\in (A^{\prime }\cup B^{\prime })\end{array}$
So, any element that belongs to $(A\cap B{)}^{\prime }$ also belongs to $(A^{\prime }\cup B^{\prime })$, and vice versa
So, these sets have exactly the same elements.
Hence, they are equal.

Theorems and Proofs:

Theorem 1:

n(A∪B) = n(A) + n (B) - n(A∩B)

Proof:

n ( A – B) + n ( A ∩ B ) + n ( B – A )

n(A) - n ( A ∩ B ) + n(B) - n( A ∩ B ) + n( A ∩ B )

n(A) + n(B) - n( A ∩ B )

Hence, it is proved.

Theorem 2: n(A ∪ B ∪ C ) = n ( A ) + n ( B ) + n ( C ) – n ( A ∩ B ) – n ( B ∩ C) – n ( A ∩ C ) + n ( A ∩ B ∩ C )

Proof:

n ( A ∪ B ∪ C ) = n (A) + n ( B ∪ C ) – n [ A ∩ ( B ∪ C ) ]

n (A) + n ( B ) + n ( C ) – n ( B ∩ C ) – n [ A ∩ ( B ∪ C ) ]

n [ A ∩ ( B ∪ C ) ] = n ( A ∩ B ) + n ( A ∩ C ) – n [ ( A ∩ B ) ∩ (A ∩ C)] n ( A ∩ B ) + n ( A ∩ C ) – n (A ∩ B ∩ C)

Similarly for the other two:

Finally, it is proved.

n ( A ∪ B ∪ C ) = n (A) + n ( B ) + n ( C ) – n ( A ∩ B ) – n ( B ∩ C) – n ( A ∩ C ) + n ( A ∩ B ∩ C )

Importance of NCERT Class 11 Maths Chapter 1 Notes:

NCERT Class 11 Maths chapter 1 notes are convenient for students who aspire to achieve good marks in the Class 11 board exams as well as in other competitive exams. Here are some reasons why students should read these notes.

• The latest CBSE 2025-26 guidelines have been followed in these notes.
• These notes are well-structured and cover all the important concepts and formulas.
• These notes will give conceptual clarity to students, and students can use these notes for revision.
• Notes are written in easy language so that students can learn these concepts easily.

NCERT Class 11 Notes Chapter Wise

Subject Wise NCERT Exemplar Solutions

The following links contain subject-wise NCERT exemplar solutions. These solutions can be used to get acquainted with the concepts and score well in the exam.

• NCERT Exemplar Class 11 Solutions
• NCERT Exemplar Class 11 Maths
• NCERT Exemplar Class 11 Physics
• NCERT Exemplar Class 11 Chemistry
• NCERT Exemplar Class 11 Biology

Subject Wise NCERT Solutions

The following list contains subject-wise NCERT solutions. These solutions cover all the concepts comprehensively.

• NCERT Solutions for Class 11 Mathematics
• NCERT Solutions for Class 11 Chemistry
• NCERT Solutions for Class 11 Physics
• NCERT Solutions for Class 11 Biology

NCERT Books and Syllabus

Students can use the following links to check the latest NCERT syllabus and read some reference books.