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Short Intro

This article provides complete NCERT Solutions for Miscellaneous Exercise of Chapter 1 – Sets for Class 11 Mathematics. Students will learn important concepts such as sets, subsets, union, intersection, complement, intervals and Venn diagrams with easy step-by-step explanations. These solutions are highly useful for CBSE Board Exams and competitive exam preparation.

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Quick Information Box

Particular	Details
Class	11
Subject	Mathematics
Chapter	Sets
Exercise	Miscellaneous Exercise
Board	NCERT / CBSE
Main Topics	Set Operations & Venn Diagrams

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Concepts Used (Topics Covered)

• Representation of Sets
• Empty Set
• Finite & Infinite Sets
• Equal Sets
• Subsets
• Proper Subsets
• Universal Set
• Intervals
• Union & Intersection
• Difference of Sets
• Complement of Sets
• Venn Diagrams

The chapter introduces the fundamental theory of sets and their operations used in mathematics.

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Important Formulas

Union of Sets

$A\cup B=\{x:x\in A\text{ or }x\in B\}$A∪B={x:x∈A or x∈B}

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Intersection of Sets

$A\cap B=\{x:x\in A\text{ and }x\in B\}$A∩B={x:x∈A and x∈B}

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Difference of Sets

$A-B=\{x:x\in A\text{ and }x\notin B\}$A−B={x:x∈A and x∈/B}

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Complement of Sets

$A’=U-A$A′=U−A

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De Morgan’s Laws

$(A\cup B)’=A’\cap B’$(A∪B)′=A′∩B′

$(A\cap B)’=A’\cup B’$(A∩B)′=A′∪B′

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Questions & Step-by-step Solutions

Question 1

Write the following sets in roster form.

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(i)

$A=\{x:x\text{ is an integer and }-3\le x<7\}$A={x:x is an integer and −3≤x<7}

Solution

A = {−3, −2, −1, 0, 1, 2, 3, 4, 5, 6}

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(ii)

$B=\{x:x\text{ is a natural number less than }6\}$B={x:x is a natural number less than 6}

Solution

B = {1,2,3,4,5}

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(iii)

Two-digit natural numbers whose digit sum is 8.

Solution

C = {17,26,35,44,53,62,71,80}

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(iv)

Prime divisors of 60.

Solution

D = {2,3,5}

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Question 2

Find whether the following are finite or infinite sets.

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(i)

Set of months in a year.

Solution

Finite Set

because it contains 12 elements.

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(ii)

Set of natural numbers.

Solution

Infinite Set

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(iii)

Set of points on a line.

Solution

Infinite Set

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Question 3

Determine equal sets.

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(i)

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

Solution

Both contain same elements.

A = B

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(ii)

A = {a,b,c}
B = {a,b,c,d}

Solution

A ≠ B

because B has an extra element d.

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Question 4

Find subsets of the following sets.

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(i)

A = {a}

Solution

φ , {a}

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(ii)

B = {a,b}

Solution

φ , {a}, {b}, {a,b}

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(iii)

C = {1,2,3}

Solution

φ, {1}, {2}, {3}, {1,2}, {1,3}, {2,3}, {1,2,3}

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Question 5

Write intervals in set-builder form.

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(i)

(−3,0)

Solution

$\{x:-3<x<0\}${x:−3<x<0}

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(ii)

[6,12]

Solution

$\{x:6\le x\le12\}${x:6≤x≤12}

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(iii)

(6,12]

Solution

$\{x:6<x\le12\}${x:6<x≤12}

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Question 6

Find union and intersection.

Given:

A = {1,2,3,4}
B = {3,4,5,6}

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(i) Union

$A\cup B$A∪B

Solution

A ∪ B = {1,2,3,4,5,6}

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(ii) Intersection

$A\cap B$A∩B

Solution

A ∩ B = {3,4}

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Question 7

Find difference of sets.

Given:

A = {1,2,3,4,5}
B = {2,4}

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(i)

$A-B$A−B

Solution

A − B = {1,3,5}

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(ii)

$B-A$B−A

Solution

B − A = φ

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Question 8

Find complement of set.

Given:

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

Solution

$A’=U-A$A′=U−A

A′ = {1,3,5,7}

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Common Mistakes

• Confusing ∈ and ⊂
• Repeating elements in sets
• Incorrect interval notation
• Mixing union and intersection symbols
• Forgetting complement is taken with respect to universal set

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Exam Tips

• Learn all set symbols carefully.
• Practice Venn diagrams regularly.
• Remember:

A ∪ φ = A

and

A ∩ φ = φ

• Use step-by-step method in board exams.

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Practice MCQs

MCQ 1

Which set is subset of every set?

A. Universal Set
B. Empty Set
C. Singleton Set
D. Infinite Set

Answer

B. Empty Set

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MCQ 2

If:

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

then:

$A\cap B$A∩B

equals:

A. {1}
B. {2}
C. {3}
D. φ

Answer

B. {2}

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MCQ 3

If:

A = {1,2,3}

then number of subsets is:

A. 4
B. 6
C. 8
D. 16

Answer

C. 8

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FAQ Section

Q1. What is a set?

A well-defined collection of objects is called a set.

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Q2. What is empty set?

A set having no element is called empty set or null set.

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Q3. What is subset?

If every element of A belongs to B, then A is subset of B.

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Q4. What is complement of set?

Complement contains all elements of universal set which are not in the given set.

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