Class 11 NCERT Solutions Chapter 1 Sets Exercise 1.3 (original) (raw)

Last Updated : 23 Jul, 2025

Set Theory is a branch of logical mathematics that studies the collection of objects and operations based on it. A set is simply a collection of objects or a group of objects. For example a Set of letters in English Alphabet.

The words collection, aggregate, and class are synonymous with the set. On the other hand elements, members, and objects are synonymous and stand for the members of the set of which the set is comprised.

Types of Sets

Question 1. Make correct statements by filling in the symbols ⊂ or ⊄ in the blank spaces:

****(i) {2, 3, 4} . . . {1, 2, 3, 4, 5}**

****(ii) {a, b, c} . . . {b, c, d}**

****(iii) {x : x is a student of Class XI of your school}. . .{x : x student of your school}**

****(iv) {x: x is a circle in the plane} . . .{x: x is a circle in the same plane with radius 1 unit}**

****(v) {x : x is a triangle in a plane} . . . {x : x is a rectangle in the plane}**

****(vi) {x : x is an** **equilateral triangle **in a plane} . . . {x : x is a triangle in the same plane}

(vii) {x : x is an even natural number} . . . {x : x is an integer}

**Solution:

****(i)** {2, 3, 4} ⊂ {1, 2, 3, 4,5}

****(ii)** {a, b, c} ⊄ {b, c, d}

****(iii)** {x : x is a student of Class XI of your school} ⊂ {x : x student of your school}

****(iv)** {x : x is a circle in the plane} ⊄ {x : x is a circle in the same plane with radius 1 unit}

****(v)** {x : x is a triangle in a plane} ⊄ {x : x is a rectangle in the plane}

****(vi)** {x : x is an equilateral triangle in a plane} ⊂ {x : x is a triangle in the same plane}

****(vii)** {x : x is an even natural number} ⊂ {x : x is an integer}

Question 2. Examine whether the following statements are true or false:

****(i) {a, b} ⊄ {b, c, a}**

****(ii) {a, e} ⊂ {x : x is a vowel in the English alphabet}**

****(iii) {1, 2, 3} ⊂ {1, 3, 5}**

****(iv) {a} ⊂ {a, b, c}**

****(v) {a} ∈ {a, b, c}**

****(vi) {x : x is an even natural number less than 6} ⊂ {x : x is a natural number which divides 36}**

**Solution:

****(i)** False. Each element of {a, b} is an element of {b, c, a}.

****(ii)** True. Since a, e are two vowels of the English alphabet.

****(iii)** False. 2 is subset of {1, 2, 3} but not subset of {1, 3, 5}

****(iv)** True. Each element of {a} is also an element of {a, b. c} .

****(v)** False. Elements of {a, b, c} are a, b, c. Hence, {a} ⊂ {a, b, c}

****(vi)** True

{x : x is an even natural number less than 6} = {2, 4}

{x: x is a natural number which divides 36} = {1, 2, 3, 4, 6, 9, 12, 18, 36}

{2, 4} ⊂ {1, 2, 3, 4, 6, 9, 12, 18, 36}

Question 3. Let A = {1, 2, {3, 4}, 5}. Which of the following statements are incorrect and why?

****(i) {3, 4} ⊂ A (ii) {3, 4} ∈ A (iii) {{3, 4}} ⊂ A (iv) 1 ∈ A (v) 1 ⊂ A (vi) {1, 2, 5} ⊂ A**

****(vii) {1, 2, 5} ∈ A (viii) {1, 2, 3} ⊂ A (ix) ∅ ∈ A (x) ∅ ⊂ A (xi) {∅} ⊂ A**

**Solution:

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

****(i)** {3, 4} ⊂ A is incorrect. Here 3 ∈ {3, 4}, where 3 ∉ A.

****(ii)** {3, 4} ∈ A is correct. {3, 4} is an element of A.

****(iii)** {{3, 4}} ⊂ A is correct. {3, 4} ∈ {{3, 4}} and {3, 4} ∈ A.

****(iv)** 1 ∈ A is correct. 1 is an element of A.

****(v)** 1 ⊂ A is incorrect. An element of a set can never be a subset of itself.

****(vi)** {1, 2, 5} ⊂ A is correct. Each element of {1, 2, 5} is also an element of A.

****(vii)** {1, 2, 5} ∈ A is incorrect. { 1, 2, 5 } is not an element of A.

****(viii)** {1, 2, 3} ⊂ A is incorrect. 3 ∈ {1, 2, 3}; where, 3 ∉ A.

****(ix)** ∅ ∈ A is incorrect. ∅ is not an element of A.

****(x)** ∅ ⊂ A is correct. ∅ is a subset of every set.

****(xi)** {∅} ⊂ A is incorrect. {∅} is not present in A.

Question 4. Write down all the subsets of the following sets

****(i) {a} (ii) {a, b} (iii) {1, 2, 3} (iv) ∅**

**Solution:

****(i)** Subsets of {a} are ∅ and {a}.

****(ii)** Subsets of {a, b} are {a}, {b}, and {a, b}.

****(iii)** Subsets of {1, 2, 3} are ∅, {1}, {2}, {3}, {1, 2}, {2, 3}, {1, 3}, and {1, 2, 3}.

****(iv)** Only subset of ∅ is ∅.

Question 5. Write the following as intervals:

****(i) {x : x ∈ R, – 4 < x ≤ 6} (ii) {x : x ∈ R, – 12 < x < –10}**

****(iii) {x : x ∈ R, 0 ≤ x < 7} (iv) {x : x ∈ R, 3 ≤ x ≤ 4}**

**Solution:

****(i)** {x : x ∈ R, – 4 < x ≤ 6} = (-4, 6]

****(ii)** {x : x ∈ R, – 12 < x < –10} = (-12, -10)

****(iii)** {x : x ∈ R, 0 ≤ x < 7} = [0, 7)

****(iv)** {x : x ∈ R, 3 ≤ x ≤ 4} = [3, 4]

Question 6. Write the following intervals in set-builder form :

****(i) (– 3, 0) (ii) [6, 12] (iii) (6, 12] (iv) [–23, 5)**

**Solution:

****(i)** (– 3, 0) = {x : x ∈ R, -3 < x < 0}

****(ii)** [6, 12] = {x : x ∈ R, 6 ≤ x ≤ 12}

****(iii)** (6, 12] = {x : x ∈ R, 6 < x ≤ 12}

****(iv)** [–23, 5) = {x : x ∈ R, -23 ≤ x < 5}

Question 7. What universal set(s) would you propose for each of the following :

****(i) The set of right triangles**

****(ii) The set of isosceles triangles.**

**Solution:

****(i)** The universal set for the set of right triangles is the set of triangles or the set of polygons.

****(ii)** The universal set for the set of isosceles triangles is the set of triangles or the set of polygons or the set of two-dimensional figures.

Question 8. Given the sets A = {1, 3, 5}, B = {2, 4, 6} and C = {0, 2, 4, 6, 8}, which of the following may be considered as universal set (s) for all the three sets A, B and C

****(i) {0, 1, 2, 3, 4, 5, 6}**

****(ii) ∅**

****(iii) {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10}**

****(iv) {1, 2, 3, 4, 5, 6, 7, 8}**

**Solution:

****(i)** A ⊂ {0, 1, 2, 3, 4, 5, 6}

B ⊂ {0, 1, 2, 3, 4, 5, 6}

But, C ⊄ {0, 1, 2, 3, 4, 5, 6}

Hence, the set {0, 1, 2, 3, 4, 5, 6} cannot be the universal set for the sets A, B, and C.

****(ii)** A ⊄ ∅, B ⊄ ∅, C ⊄ ∅

Hence, ∅ cannot be the universal set for the sets A, B, and C.

****(iii)** A ⊂ {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10}

B ⊂ {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10}

C ⊂ {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10}

Hence, the set {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10} is the universal set for the sets A, B, and C.

****(iv)** A ⊂ {1, 2, 3, 4, 5, 6, 7, 8}

B ⊂ {1, 2, 3, 4, 5, 6, 7, 8}

But, C ⊄ {1, 2, 3, 4, 5, 6, 7, 8}

Hence, the set {1, 2, 3, 4, 5, 6, 7, 8} cannot be the universal set for the sets A, B, and C.

Deleted Questions from NCERT

How many elements has P(A), if A = ∅?

**Solution:

For a set A with n(A) = m, then it can be shown that

Number of elements of P(A) = n[P(A)] = 2m

If A = ∅, we get n (A) = 0

So, n[P(A)] = 2° = 1

Therefore, P(A) has one element.

Summary

A set is a collection of distinct objects that form a group. Sets can be made up of any group of items, such as numbers, days of the week, or types of vehicles. Each item in a set is called an element, and all elements are unique. Sets are represented using curly brackets and a capital letter symbol. For example, the set of counting numbers less than 6 could be written as {1, 2, 3, 4, 5}