NCERT Solutions for Class 11 Maths Chapter 1: Sets - Exercise 1.1

Exercise (1.1)

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

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

Ans:

To determine if the given statement is a set

A set is a collection of well-defined objects.

We can definitely identify the collection of months beginning with a letter J.

Thus, the collection of all months of a year beginning with the letter J is the set.

ii. The collection of ten most talented writers of India

Ans:

To determine if the given statement is a set

A set is a collection of well-defined objects.

The criteria for identifying the collection of ten most talented writers of India may vary from person to person. So it is not a well-defined object.

Thus, the collection of ten most talented writers of India is not a set.

iii. A team of eleven best cricket batsmen of the world.

Ans:

To determine if the given statement is a set

A set is a collection of well-defined objects.

The criteria for determining the eleven best cricket batsmen may vary from person to person. So it is not a well-defined object.

Thus, a team of eleven best cricket batsmen in the world is not a set.

iv. The collection of all boys in your class.

Ans:

To determine if the given statement is a set

A set is a collection of well-defined objects.

We can definitely identify the boys who are all studying in the class. So it is a well-defined object.

Thus, the collection of all boys in your class is a set.

v. The collection of all natural numbers less than $100$.

Ans:

To determine if the given statement is a set

A set is a collection of well-defined objects.

We can identify the natural numbers less than $100$ can easily be identified. So it is a well-defined object.

Thus, the collection of all natural numbers less than $100$ is a set.

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

Ans:

To determine if the given statement is a set

A set is a collection of well-defined objects.

We can identify the books that belong to the writer Munshi Prem Chand. So it is a well-defined object.

Thus, a collection of novels written by the writer Munshi Prem Chand is a set.

vii. The collection of all even integers.

Ans:

To determine if the given statement is a set

A set is a collection of well-defined objects.

We can identify integers that are all the collection of even integers. So it is not a well-defined object.

Thus, the collection of all even integers is a set.

viii. The collection of questions in this chapter.

Ans:

To determine if the given statement is a set

A set is a collection of well-defined objects.

We can easily identify the questions that are in this chapter. So it is a well-defined object.

Thus, the collection of questions in this chapter is a set.

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

Ans:

To determine if the given statement is a set

A set is a collection of well-defined objects.

The criteria for determining the most dangerous animals may vary according to the person. So it is not a well-defined object.

Thus, the collection of the most dangerous animals in the world is a set.

2. Let $A=\left\{ 1,2,3,4,5,6 \right\}$. Insert the appropriate symbol $\in $ or $\notin $ in the blank spaces:

i. $5...A$

Ans:

Given that,

$A=\left\{ 1,2,3,4,5,6, \right\}$

To insert the appropriate symbol $\in $ or $\notin $

The number $5$ is in the set.

$\therefore 5\in A$

ii. $8...A$

Ans:

Given that,

$A=\left\{ 1,2,3,4,5,6, \right\}$

To insert the appropriate symbol $\in $ or $\notin $

The number $8$ is not in the set.

$\therefore 8\notin A$

iii. $0...A$

Ans:

Given that,

$A=\left\{ 1,2,3,4,5,6, \right\}$

To insert the appropriate symbol $\in $ or $\notin $

The number $0$ is not in the set.

$\therefore 0\notin A$

iv. $4...A$

Ans:

Given that,

$A=\left\{ 1,2,3,4,5,6, \right\}$

To insert the appropriate symbol $\in $ or $\notin $

The number $4$ is in the set.

$\therefore 4\in A$

v. $2...A$

Ans:

Given that,

$A=\left\{ 1,2,3,4,5,6, \right\}$

To insert the appropriate symbol $\in $ or $\notin $

The number $2$ is in the set.

$\therefore 2\in A$

vi. $10...A$

Ans:

Given that,

$A=\left\{ 1,2,3,4,5,6, \right\}$

To insert the appropriate symbol $\in $ or $\notin $

The number $10$ is not in the set.

$\therefore 10\notin A$

3. Write the following sets in roster form:

i. $A=\left\{ x:x\text{ is an integer and -3x7} \right\}$

Ans:

Given that,

$A=\left\{ x:x\text{ is an integer and -3x7} \right\}$

To write the above expression in its roaster form

In roaster form, the order in which the elements are listed is immaterial.

The elements of the set are $-2,-1,0,1,2,3,4,5,6$.

$\therefore $ The roaster form of the set $A=\left\{ x:x\text{ is an integer and -3x7} \right\}$ is $A=\left\{ -2,-1,0,1,2,3,4,5,6 \right\}$.

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

Ans:

Given that,

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

To write the above expression in its roaster form

In roaster form, the order in which the elements are listed is immaterial.

The elements of the set are $1,2,3,4,5$.

$\therefore$ The roaster form of the set $B=\left\{ x:x\text{ is a natural number less than 6} \right\}$ is $B=\left\{ 1,2,3,4,5 \right\}$.

iii. $C=\left\{ x:x\text{ is a two-digit natural number such that sum of its digits is 8} \right\}$

Ans:

Given that,

$C=\left\{ x:x\text{ is a two-digit natural number such that sum of its digits is 8} \right\}$

To write the above expression in its roaster form

In roaster form, the order in which the elements are listed is immaterial.

The elements of the set are $17,26,35,44,53,62,71,80$ such that their sum is $8$

$\therefore $ The roaster form of the set $C=\left\{ x:x\text{ is a two-digit natural number such that sum of its digits is 8} \right\}$ is $\left\{ 17,26,35,44,53,62,71,80 \right\}$.

iv. $D=\left\{ x:x\text{ is a prime number which is divisor of 60} \right\}$

Ans:

Given that,

$D=\left\{ x:x\text{ is a prime number which is divisor of 60} \right\}$

To write the above expression in its roaster form

In roaster form, the order in which the elements are listed is immaterial.

The divisors of $60$ are $2,3,4,5,6$. Among these the prime numbers are $2,3,5$

The elements of the set are $2,3,5$.

$\therefore $ The roaster form of the set $D=\left\{ x:x\text{ is a prime number which is divisor of 60} \right\}$ is $D=\left\{ 2,3,5 \right\}$.

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

Ans:

Given that,

$E=$The set of all letters in the word TRIGONOMETRY

To write the above expression in its roaster form

In roaster form, the order in which the elements are listed is immaterial.

There are $12$ letters in the word TRIGONOMETRY out of which T, R and O gets repeated.

The elements of the set are T, R, I G, O, N, M, E, Y.

$\therefore $ The roaster form of the set $E=$The set of all letters in the word TRIGONOMETRY is $E=\left\{ T,R,I,G,O,N,M,E,Y \right\}$.

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

Ans:

Given that,

$F=$The set of all letters in the word BTTER

To write the above expression in its roaster form

In roaster form, the order in which the elements are listed is immaterial.

There are $6$ letters in the word BETTER out of which E and T are repeated.

The elements of the set are B, E, T, R.

$\therefore $ The roaster form of the set $F=$The set of all letters in the word BTTER

 is $F=\left\{ B,E,T,R \right\}$.

4. Write the following sets in the set builder form:

i. $\left( 3,6,9,12 \right)$

Ans:

Given that,

$\left\{ 3,6,9,12 \right\}$

To represent the given set in the set builder form

In set builder form, all the elements of a set possess a single common property which is not possessed by any element outside the set.

From the given set, we observe that the numbers in the set are multiple of $3$ from $1$ to $4$ such that $\left\{ x:x=3n,n\in N\text{ and 1}\le \text{n}\le \text{4} \right\}$

$\therefore \left\{ 3,6,9,12 \right\}=\left\{ x:x=3n,n\in N\text{ and 1}\le \text{n}\le \text{4} \right\}$

ii. $\left\{ 2,4,8,16,32 \right\}$

Ans:

Given that,

{2,4,8,16,32}

To represent the given set in the set builder form

In set builder form, all the elements of a set possess a single common property which is not possessed by any element outside the set.

From the given set, we observe that the numbers in the set are powers of $2$ from $1$ to $5$ such that $\left\{ x:x={{2}^{n}},n\in N\text{ and 1}\le \text{n}\le 5 \right\}$

$\therefore \left\{ 2,4,8,16,32 \right\}=\left\{ x:x={{2}^{n}},n\in N\text{ and 1}\le \text{n}\le 5 \right\}$

iii. $\left\{ 5,25,125,625 \right\}$

Ans:

Given that,

$\left\{ 5,25,125,625 \right\}$

To represent the given set in the set builder form

In set builder form, all the elements of a set possess a single common property which is not possessed by any element outside the set.

From the given set, we observe that the numbers in the set are powers of $5$ from $1$ to $4$ such that $\left\{ x:x={{5}^{n}},n\in N\text{ and 1}\le \text{n}\le \text{4} \right\}$

$\therefore \left\{ 5,25,125,625 \right\}=\left\{ x:x={{5}^{n}},n\in N\text{ and 1}\le \text{n}\le \text{4} \right\}$

iv. $\left\{ 2,4,6,... \right\}$

Ans:

Given that,

$\left\{ 2,4,6,... \right\}$

To represent the given set in the set builder form

In set builder form, all the elements of a set possess a single common property which is not possessed by any element outside the set.

From the given set, we observe that the numbers are the set of all even natural numbers.

$\therefore \left\{ 2,4,6,... \right\}=\left\{ x:x\text{ is an even natural number} \right\}$

v. $\left\{ 1,4,9,...100 \right\}$

Ans:

Given that,

$\left\{ 1,4,9,...100 \right\}$

To represent the given set in the set builder form

In set builder form, all the elements of a set possess a single common property which is not possessed by any element outside the set.

From the given set, we observe that the numbers in the set squares of numbers form $1$ to $10$ such that $\left\{ x:x={{n}^{2}},n\in N\text{ and 1}\le \text{n}\le 10 \right\}$

$\therefore \left\{ 1,4,9,...100 \right\}=\left\{ x:x={{n}^{2}},n\in N\text{ and 1}\le \text{n}\le 10 \right\}$

5. List all the elements of the following sets:

i. $A=\left\{ x:x\text{ is an odd natural number} \right\}$

Ans:

Given that,

$A=\left\{ x:x\text{ is an odd natural number} \right\}$

To list the elements of the given set

The odd natural numbers are $1,3,5,...$

$\therefore $ The set $A=\left\{ x:x\text{ is an odd natural number} \right\}$ has the odd natural numbers that are $\left\{ 1,3,5,... \right\}$

ii. $B=\left\{ x:x\text{ is an integer;-}\frac{1}{2}<x<\frac{9}{2} \right\}$

Ans:

Given that,

$B=\left\{ x:x\text{ is an integer;-}\frac{1}{2}<x<\frac{1}{2} \right\}$

To list the elements of the given set

$-\frac{1}{2}=-0.5$ and $\frac{9}{2}=4.5$

So the integers between $-0.5$ and $4.5$ are $0,1,2,3,4$

$\therefore $ The set $B=\left\{ x:x\text{ is an integer;-}\frac{1}{2}<x<\frac{1}{2} \right\}$ has an integers that are between $\left\{ 0,1,2,3,4 \right\}$

iii. $C=\left\{ x:x\text{ is an integer;}{{\text{x}}^{2}}\le 4 \right\}$

Ans:

Ans:

Given that,

$C=\left\{ x:x\text{ is an integer;}{{\text{x}}^{2}}\le 4 \right\}$

To list the elements of the given set

It is observed that,

${{x}^{2}}\le 4$

${{\left( -2 \right)}^{2}}=4\le 4$

${{\left( -1 \right)}^{2}}=1\le 4$

${{\left( 0 \right)}^{2}}=0\le 4$

${{\left( 1 \right)}^{2}}=1\le 4$

${{\left( 2 \right)}^{2}}=4\le 4$

$\therefore $The set $C=\left\{ x:x\text{ is an integer;}{{\text{x}}^{2}}\le 4 \right\}$ contains elements such as $\left\{ -2,-1,0,1,2 \right\}$

iv. $D=\left\{ x:x\text{ is a letter in the word ''LOYAL''} \right\}$

Ans:

Given that,

$D=\left\{ x:x\text{ is a letter in the word ''LOYAL''} \right\}$

To list the elements of the given set

There are $5$ total letters in the given word in which L gets repeated two times.

So the elements in the set are $\left\{ L,O,Y,A \right\}$

$\therefore $The set $D=\left\{ x:x\text{ is a letter in the word ''LOYAL''} \right\}$ consists the elements $\left\{ L,O,Y,A \right\}$.

v. $E=\left\{ x:x\text{ is a month of a year not having 31 days} \right\}$

Ans:

Given that,

$E=\left\{ x:x\text{ is a month of a year not having 31 days} \right\}$

To list the elements of the given set

The months that don’t have $31$ are as follows:

February, April, June, September, November

$\therefore $The set $E=\left\{ x:x\text{ is a month of a year not having 31 days} \right\}$ consist of the elements such that $\left\{ \text{February, April, June, September, November} \right\}$

vi. $F=\left\{ x:x\text{ is a consonant in the English alphabet which precedes k} \right\}$

Ans:

Given that,

$F=\left\{ x:x\text{ is a consonant in the English alphabet which precedes k} \right\}$

To list the elements of the given set

The consonants are letters in English alphabet other than vowels such as a, e, i, o, u and the consonants that precedes k include b, c, d, f, g, h, j

$\therefore $The set $F=\left\{ x:x\text{ is a consonant in the English alphabet which precedes k} \right\}$ consists of the set $\left\{ b,c,d,f,g,h,j \right\}$

6. Match each of the sets on the left in the roaster form with the same set on the right described inn set-builder form.

i. $\left\{ 1,2,3,6 \right\}$

Ans:

Given that,

$\left\{ 1,2,3,6 \right\}$

To match the roaster form in the left with the set builder form in the right

In roaster form, the order in which the elements are listed is immaterial.

In set builder form, all the elements of a set possess a single common property which is not possessed by any element outside the set.

It has been observed from the set that these set of numbers are the set of natural numbers which are also the divisors of $6$

Thus, $\left\{ 1,2,3,6 \right\}=\left\{ x:x\text{ is a natural number and is a divisor of 6} \right\}$ is the correct option which is option.

ii. $\left\{ 2,3 \right\}$

Ans:

Given that,

$\left\{ 2,3 \right\}$

To match the roaster form in the left with the set builder form in the right

In roaster form, the order in which the elements are listed is immaterial.

In set builder form, all the elements of a set possess a single common property which is not possessed by any element outside the set.

It has been observed from the set that these set of numbers are the set of prime numbers which are also the divisors of $6$

Thus, $\left\{ 2,3 \right\}=\left\{ x:x\text{ is a prime number and is a divisor of 6} \right\}$ is the correct option which is option (a)

iii. $\left\{ M,A,T,H,E,I,C,S \right\}$

Ans:

Given that,

$\left\{ M,A,T,H,E,I,C,S \right\}$

To match the roaster form in the left with the set builder form in the right

In roaster form, the order in which the elements are listed is immaterial.

In set builder form, all the elements of a set possess a single common property which is not possessed by any element outside the set.

It has been observed from the set of these letters of word MATHEMATICS.

Thus, $\left\{ M,A,T,H,E,I,C,S \right\}=\left\{ x:x\text{ is a letter of the word MATHEMATICS} \right\}$ is the correct option which is option (d)

iv. $\left\{ 1,3,5,7,9 \right\}$

Ans:

Given that,

$\left\{ 1,3,5,7,9 \right\}$

To match the roaster form in the left with the set builder form in the right

In roaster form, the order in which the elements are listed is immaterial.

In set builder form, all the elements of a set possess a single common property which is not possessed by any element outside the set.

It has been observed from the set that these sets of numbers are the set of odd numbers that are less than $10$.

Thus, $\left\{ 1,3,5,7,9 \right\}=\left\{ x:x\text{ is a odd number less than 10} \right\}$ is the correct option which is option (b)

Topics Covered in Exercise 1.1 Class 11 Maths NCERT

Exercise 1.1 is based on the following topics:

A. Introduction to Sets

In mathematics, Sets are simply a collection of distinct objects that form a group. A set can contain any number of items, such as numbers, days of the week, car types, and so on. An element of the set refers to each object in the set. When writing a set, curly brackets are used.

B. Sets and their Representations

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

Representation of Sets

There are 2 methods to represent a set:

• Roster or Tabular Form: In this form, we list all the members of the set within braces { } and separate them by commas.

• Set-builder Form: In the set-builder form, we list the property or properties satisfied by all the elements of the sets.

Importance of Sets for Class 11 Maths

Sets is a topic that is not only important in math but also useful in other subjects. Every student should learn about the elements of sets and the operations that can be performed on them. Students can quickly learn to execute operations on sets if they practise Class 11 Maths NCERT Solutions Chapter 1 on a regular practice. These solutions' sample problems and examples are effective at promoting a step-by-step grasp of this topic.

Class 11 Maths Chapter 1 – Exercise 1.1

In mathematics, a 'set' is a collection of distinct and well-defined objects. These objects are referred to like elements and these can be anything: people, numbers, alphabet, subsets, and so on. Sets are a significant part of mathematics as every field of it uses sets in some way. Sets can have topological or algebraic properties that are useful. If you want to pursue a career in the field of mathematics, then you must have a deep understanding of sets. It will help you in developing mathematical theories. It may sound a bit tricky here but our knowledgeable teachers ensure that you get every bit of this chapter and hold a strong command on sets. With Vedantu’s NCERT Solutions for Class 11 Maths Chapter 1, you can learn all the concepts of sets and their applications. With their smart learning methods, our experts have made the learning so simple that students won’t feel boredom while studying maths.

Coming on to the NCERT maths chapter 1, there are seven exercises in it including the miscellaneous one. Each one of them covers a crucial concept of sets. For your better understanding, a brief of every exercise is given below:-

Exercise 1.1

Exercise 1.1 of NCERT Class 11 Maths provides a basic introduction to sets and their representation. The students will learn about the difference between roaster form or builder form of sets. The knowledge gained from this exercise may help them in understanding other exercises and concepts discussed in this chapter. The students of class 11 will learn about recognizing a particular set type from a given cluster of data. This can only be done with a great understanding of appropriate concepts and knowledge. In addition to this, they will gain an understanding of forms of sets and one set to other alternative forms of the set. Exercise 1.1 of NCERT Class 11 is crucial as it tests the basic understanding of students. Moreover, they get to learn about various elements of a set through the question given in the Class 11 ex 1.1. Furthermore, the last question in exercise 1.1 asks students to match identical sets in various forms. The overall Exercise 1.1 of Class 11 NCERT Maths demands students to have knowledge of recognizing a set, determining appropriate symbols, transforming the forms, listing of set elements, etc. With the help of NCERT Solutions for Exercise 1.1 Class 11 Maths prepared by the experts of Vedantu, you can be able to solve every question with detailed reasons.

Exercise 1.2

Talking about exercise 1.2 of the chapter, it comprises many problems that will help you in gaining a thorough understanding of the null set. For instance, the first question asks whether a set of odd natural number divisible by 2 is a null set or not. As per the solutions prepared by our experts, it is a null set because no odd number is divisible by 2. In addition to this, you will get to learn about the finite and infinite sets. For example, a set of months of the year is a finite set because such a set will have only 12 elements only. On the other hand, a set of natural numbers is an infinite set. Furthermore, one will also learn about equal sets and their properties. The experts of Vedantu has prepared the solutions with detailed explanation and student-friendly content. These will help you in providing in-depth reasoning for every question and getting good marks in the exam.

Exercise 1.3

In this exercise, the primary motive of the NCERT is to build an understanding of equations and putting the right symbol in students studying in class 11. For that purpose, a few questions on forms of sets are placed in exercise 1.3 of class 11 maths. There are many symbols that have their own meaning in the set theory. It has been seen that students do not have a clear understanding of using the right symbol in the problems. This results in the deduction of marks in the exam. However, with our detailed NCERT Solutions for Class 11 Maths Chapter 1, you can get the highest marks as our experts have detailed out the exact meaning of each symbol with their proper usage in the problem. In exercise 1.3, you will also learn about a part of a set called 'subset' with the help of some problems mentioned in the exercise. For instance, a set of vowels is a subset of a set consisting of the English alphabets.

Exercise 1.4

This exercise covers the most crucial aspect of the set theory i.e., the union of two or more sets. The whole exercise consists of different types of questions on the union of sets. As a student, you will learn about the symbol of the 'union' which is 'U' and various properties of the union set. For example, you will learn that the commutative properties, associative properties, identity properties, distributive and intersection properties of a union set. For a better understanding of this concept, one needs to have a stronghold of these properties and their application. Many students get confused about these properties and their usage. With the tips given by Vedantu's experts, one can have a strong understanding of the aforementioned properties. Our detailed NCERT Solutions for Class 11 Maths Chapter 1 will assist you in getting the answers for all your doubts and that too in a very simple language.

Exercise 1.5

Coming on to the exercise 1.5 of class 11 maths, a lot of complex problems are given on the usage of union set. This exercise tests the overall knowledge and clarity of concepts of students they have gained till the previous exercise. The problems in exercise 1.5 are framed in such a way that only those students will be able to solve them those have a strong command over the properties of a Union and their applications. For instance, the value of the union is provided and you are asked to work out the value of different sections of the set. This requires a great understanding of the Venn diagram, complementary sets, and universal sets. However, you need not worry as our experts have solved each and every problem with a step-by-step explanation. They have prepared a solution for every question from scratch.

Exercise 1.6

The exercise 1.6 of class 11 maths introduces another aspect of sets which is called ‘intersection’ of two or more sets. It is denoted by ‘∩’ and represents the common elements between two or more sets. In this exercise, you will learn about the properties of an intersection set, such as commutative properties, associative properties, identity properties, distributive and intersection properties along with their application. Also, some complex problems that require the use of properties of a union and intersection are placed in exercise 1.6 of class 11 maths. A large number of students face difficulties in solving questions given in this exercise as it requires a strong understanding of all the concepts of sets. Remembering each concept is quite difficult but with our expert’s tips given in Class 11 Maths NCERT Solutions Chapter 1, you will be able to remember every concept in no time.

Miscellaneous Exercise

As a student of class 11, it is important for you to figure out whether you have learned all the concepts covered in the chapter. Miscellaneous exercise is the best way to revise and prepare for the exams. It covers complex problems on every concept you have learned in the whole chapter. It will help in making you more confident on sets. It has been seen that students usually skip miscellaneous exercise but our experts advise them not to do so. Just give it a try and in case, they face any difficulty, do refer Vedantu's NCERT Solutions for Class 11 Chapter 1. These will assist in getting a detailed explanation for each complex miscellaneous problem. Our solutions are comprehensive and well explained with the proper reasoning for the usage of any concept or theory. These solutions provide step-by-step guidance to students and assist them in solving their doubts in no time. 

Apart from the aforementioned concepts, there are other topics also on which our subject experts can assist you and clear your doubts. You can always reach them in case you face any problem and they will provide you with the best assistance related to your problems.

Access other exercise solutions of Class 11 Maths Chapter 1- Sets

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