NCERT Class 11 Maths Chapter 1 Notes Sets - Download PDF

Class 11 Math chapter 1 sets notes are provided here. These notes are created by expert team at Careers360 considering the need of students. These Notes covers all topics, sub topics, concepts, formulae, exercises, and solved problems that are very useful to score well in the boards and competitive exams. In chapter 1 we will be discussing topics such as sets, representation of sets, types of sets, subset, power set, Venn diagrams, operations of sets, Laws of algebra.

Sets are collections of objects with specific characteristics. An empty set contains no elements. A finite set holds a definite number of elements, while an infinite set has an indefinite number. If two sets, P and Q, contain the exact same elements, they are considered equal. Set P is a subset of set Q if all elements in P also exist in Q. The power set [A(P)] of a set P consists of all possible subsets of P. The union of sets P and Q contains all elements found in either set. The intersection of sets P and Q comprises elements common to both. Similarly, the difference between sets P and Q, when in the same order, contains elements from P but not Q.

Also, students can refer,

• NCERT Solutions for Class 11 Maths Chapter 1 Sets
• NCERT Exemplar Class 11 Maths Chapter 1 Sets

Sets:

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:

Roaster form or tabular form:

In this form, elements are separated with commas and within braces { }.

Eg: set of natural numbers below 10: { 1, 2, 3, 4, 5, 6, 7, 8, 9}

Set-builder form: In this form, each element contains a common character that any other element outside the set does not contain.
Eg: F = {x: x is a natural number that divides 24}

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: Sets that contain a fixed or finite number of elements are called finite sets.

Represented by: A= { , , , , , }

Eg: A={natural numbers <20}

A= {1, 2, 3, 4, 5, 6, 7, 8, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19}

NOTE: Empty set is also a finite set.

Infinite sets: Sets that are not finite or the sets that are infinite are called infinite sets.

Represented by: {, , , , , , , , ……………………………}

Eg: A={n natural numbers}

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

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.

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

Represented with 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 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 open interval, represented as (a, b).

The set that includes end points 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 includes 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}

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

Formula to find a number of subsets:

m= number of elements in set A

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:

1. A∪B = B∪A is called commutative property.

2. A∪(B∪C) = (A∪B)∪C is associative property.

3. A∪∅ = A is identity law

4. A∪A is an idempotent law

5. U∪A is union law

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:

a)A∩B = B∩A is called commutative property.

b) A∩(B∩C) = (A∩B)∩C is an associative property.

c) A∩∅ = ∅ is identity law

d) A∩A is idempotent law

e) U∩A is union law

f) A∩(B∪C) = (A∩B)∪(A∩C) is distribution property.

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 is 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

Theorems and Proofs:

Theorems 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 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 proved;

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

With this topic we conclude NCERT Class 11 chapter 1 notes.

The link for NCERT textbook pdf is given below:

URL: ncert.nic.in/ncerts/l/kemh101.pdf

Significance of NCERT Class 11 Maths Chapter 1 Notes:

NCERT Class 11 Maths chapter 1 notes will be very much helpful for students to score maximum marks in their 11 board exams. NCERT Class 11 Mathematics chapter 1 is also very useful to cover major topics of Class 11 CBSE Mathematics Syllabus. The CBSE Class 11 Maths chapter 1 will help students to understand the formulas, statements, and rules in detail. This pdf also contains previous year’s questions and NCERT textbook pdf. The next part includes FAQ’s or frequently asked questions along with topic-wise explanations. These topics can also be downloaded from Class 11 chapter 1 Sets pdf download.

Subject Wise NCERT Exemplar Solutions

Following list contains subject wise NCERT exemplar solutions. These solutions can be used to command 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

Following list contains subject wise NCERT solutions. These solutions covers all the concepts comprehensively and can be used to command the concepts and score well in the exam.

• 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