**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.

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