NCERT Solutions for Exercise 1.3, Class 11, Maths

Exercise 1.3 Sets
Question and Answers
Class 11 – Maths

Class	Class 11
Subject	Mathematics
Chapter Name	Sets
Chapter No.	Chapter 1
Exercise	Exercise 1.3
Category	Class 11 Maths NCERT Solutions

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

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

(i) {2, 3, 4} ⊂ {1, 2, 3, 4, 5}

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

(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}

(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}

(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}

(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}

(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}

(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 }

(i) False. Here each element of {a, b} is an element of {b, c, a}.

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

(ii) True. We know that a, e are two vowels of the English alphabet.

(iii) { 1, 2, 3 } ⊂ { 1, 3, 5 }

(iii) False. 2 ∈ {1, 2, 3} where, 2∉ {1, 3, 5}

(iv) { a } ⊂ { a, b, c }

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

(v) { a } ∈ { a, b, c }

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

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

(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}

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

(i) {3, 4} ⊂ A

(i) {3, 4} ⊂ A is incorrect

Here 3 ∈ {3, 4}; where, 3∉A.

(ii) {3, 4} ∈ A

(ii) {3, 4} ∈A is correct

{3, 4} is an element of A.

(iii) {{3, 4}} ⊂ A

(iii) {{3, 4}} ⊂ A is correct

{3, 4} ∈ {{3, 4}} and {3, 4} ∈ A.

(iv) 1 ∈ A

(iv) 1∈A is correct

1 is an element of A.

(v) 1 ⊂ A

(v) 1⊂ A is incorrect

An element of a set can never be a subset of itself.

(vi) {1, 2, 5} ⊂ A

(vi) {1, 2, 5} ⊂ A is correct

Each element of {1, 2, 5} is also an element of A.

(vii) {1, 2, 5} ∈ A

(vii) {1, 2, 5} ∈ A is incorrect

{1, 2, 5} is not an element of A.

(viii) {1, 2, 3} ⊂ A

(viii) {1, 2, 3} ⊂ A is incorrect

3 ∈ {1, 2, 3}; where, 3 ∉ A.

(ix) φ ∈ A

(ix) Φ ∈ A is incorrect

Φ is not an element of A.

(x) φ ⊂ A

(x) Φ ⊂ A is correct

Φ is a subset of every set.

(xi) {φ} ⊂ A

(xi) {Φ} ⊂ A is incorrect

Φ∈ {Φ}; where, Φ ∈ A.

Question 4 Write down all the subsets of the following sets

(i) {a}

(i) Subsets of {a} are

Φ and {a}.

(ii) {a, b}

(ii) Subsets of {a, b} are

Φ, {a}, {b}, and {a, b}.

(iii) {1, 2, 3}

(iii) Subsets of {1, 2, 3} are

Φ, {1}, {2}, {3}, {1, 2}, {2, 3}, {1, 3}, and {1, 2, 3}.

(iv) φ 

(iv) Only subset of Φ is Φ.

Question 5 Write the following as intervals :

(i) {x : x ∈ R, – 4 < x ≤ 6}

(i) {x: x ∈ R, –4 < x ≤ 6} = (–4, 6]

(ii) {x : x ∈ R, – 12 < x < –10}

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

(iii) {x : x ∈ R, 0 ≤ x < 7}

(iii) {x: x ∈ R, 0 ≤ x < 7} = [0, 7)

(iv) {x : x ∈ R, 3 ≤ x ≤ 4}

(iv) {x: x ∈ R, 3 ≤ x ≤ 4} = [3, 4]

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

(i) (– 3, 0)

(i) (–3, 0) = {x: x ∈ R, –3 < x < 0}

(ii) [6, 12]

(ii) [6, 12] = {x: x ∈ R, 6 ≤ x ≤ 12}

(iii) (6, 12]

(iii) (6, 12] ={x: x ∈ R, 6 < x ≤ 12}

(iv) [–23, 5)

(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.

(i) For the set of right triangles, the universal set can be the set of all kinds of triangles or the set of polygons.

(ii) The set of isosceles triangles.

(ii) For the set of isosceles triangles, the universal set can be the set of all kinds 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}

(i) We know that A ⊂ {0, 1, 2, 3, 4, 5, 6}

B ⊂ {0, 1, 2, 3, 4, 5, 6}

So 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) φ

(ii) A ⊄ Φ, B ⊄ Φ, C ⊄ Φ

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

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

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

(iv) A ⊂ {1, 2, 3, 4, 5, 6, 7, 8}

B ⊂ {1, 2, 3, 4, 5, 6, 7, 8}

So 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.