# Sets Chapter 1 Class 11 Maths NCERT Textbook With Solutions PDF

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### Chapter 1: Sets

#### 1.1 Introduction

The concept of set serves as a fundamental part of present-day mathematics. Today this concept is being used in almost every branch of mathematics.

Sets are used to define the concepts of relations and functions. The study of geometry, sequences, probability, etc. requires the knowledge of sets.

The theory of sets was developed by German mathematician Georg Cantor (1845-1918). He first
encountered sets while working on “problems on trigonometric series”.

In this chapter, we discuss some basic definitions and operations involving sets

#### 1.2 Sets and their Representations

In everyday life, we often speak of collections of objects of a particular kind, such as a pack of cards, a crowd of people, a cricket team, etc.

In mathematics also, we come across collections, for example, of natural numbers, points, prime numbers, etc. More specially, we examine the following collections:

(i) Odd natural numbers less than 10, i.e., 1, 3, 5, 7, 9
(ii) The rivers of India
(iii) The vowels in the English alphabet, namely, a, e, i, o, u
(iv) Various kinds of triangles
(v) Prime factors of 210, namely, 2,3,5 and 7
(vi) The solution of the equation: x2 – 5x + 6 = 0, viz, 2 and 3.

We note that each of the above examples is a well-defined collection of objects in the sense that we can definitely decide whether a given particular object belongs to a given collection or not.

For example, we can say that the river Nile does not belong to the collection of rivers of India. On the other hand, the river Ganga does belong to this collection.

#### 1.5 Equal Sets

Given two sets A and B, if every element of A is also an element of B and if every element of B is also an element of A, then the sets A and B are said to be equal. Clearly, the two sets have exactly the same elements.

### NCERT Solutions Class 11 Maths Chapter 1 Sets

1. Which of the following are sets? Justify our answer.

(i) The collection of all months of a year beginning with the letter J.

(ii) The collection of the ten most talented writers of India.

(iii) A team of eleven best-cricket batsmen in the world.

(iv) The collection of all boys in your class.

(v) The collection of all-natural numbers less than 100.

(vi) A collection of novels written by the writer Munshi Prem Chand.

(vii) The collection of all even integers.

(viii) The collection of questions in this Chapter.

(ix) A collection of the most dangerous animals in the world.

Solution:

(i) The collection of all months of a year beginning with the letter J is a well-defined collection of objects as one can identify a month that belongs to this collection.

Therefore, this collection is a set.

(ii) The collection of the ten most talented writers of India is not a well-defined collection as the criteria to determine a writer’s talent may differ from one person to another.

Therefore, this collection is not a set.

(iii) A team of eleven best-cricket batsmen of the world is not a well-defined collection as the criteria to determine a batsman’s talent may vary from one person to another.

Therefore, this collection is not a set.

(iv) The collection of all boys in your class is a well-defined collection as you can identify a boy who belongs to this collection.

Therefore, this collection is a set.

(v) The collection of all-natural numbers less than 100 is a well-defined collection as one can find a number that belongs to this collection.

Therefore, this collection is a set.

(vi) A collection of novels written by the writer Munshi Prem Chand is a well-defined collection as one can find a book that belongs to this collection.

Therefore, this collection is a set.

(vii) The collection of all even integers is a well-defined collection as one can find an integer that belongs to this collection.

Therefore, this collection is a set.

(viii) The collection of questions in this Chapter is a well-defined collection as one can find a question that belongs to this chapter.

Therefore, this collection is a set.

(ix) A collection of the most dangerous animals of the world is not a well-defined collection as the criteria to find the dangerousness of an animal can differ from one animal to another.

Therefore, this collection is not a set.

2. Let A = {1, 2, 3, 4, 5, 6}. Insert the appropriate symbol ∈or ∉ in the blank spaces:

(i) 5…A (ii) 8…A  (iii) 0…A

(iv) 4…A (v) 2…A (vi) 10…A

Solution:

(i) 5 ∈ A

(ii) 8 ∉ A

(iii) 0 ∉ A

(iv) 4 ∈ A

(v) 2 ∈ A

(vi) 10 ∉ A

3. Write the following sets in roster form:

(i) A = {xx is an integer and –3 < < 7}.

(ii) B = {xx is a natural number less than 6}.

(iii) C = {xx is a two-digit natural number such that the sum of its digits is 8}

(iv) D = {xx is a prime number which is divisor of 60}.

(v) E = The set of all letters in the word TRIGONOMETRY.

(vi) F = The set of all letters in the word BETTER.

Solution:

(i) A = {xx is an integer and –3 < < 7}

–2, –1, 0, 1, 2, 3, 4, 5, and 6 only are the elements of this set.

Hence, the given set can be written in roster form as

A = {–2, –1, 0, 1, 2, 3, 4, 5, 6}

(ii) B = {xx is a natural number less than 6}

1, 2, 3, 4, and 5 only are the elements of this set

Hence, the given set can be written in roster form as

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

(iii) C = {xx is a two-digit natural number such that the sum of its digits is 8}

17, 26, 35, 44, 53, 62, 71, and 80 only are the elements of this set

Hence, the given set can be written in roster form as

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

(iv) D = {xx is a prime number which is divisor of 60}

Here 60 = 2 × 2 × 3 × 5

2, 3 and 5 only are the elements of this set

Hence, the given set can be written in roaster form as

D = {2, 3, 5}

(v) E = The set of all letters in the word TRIGONOMETRY

TRIGONOMETRY is a 12 letters word out of which T, R, and O are repeated.

Hence, the given set can be written in roaster form as

E = {T, R, I, G, O, N, M, E, Y}