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

∴ The roaster form of the set $C={x:x \;{\text {is a two-digit natural number such that the sum of its digits is 8}}}$ 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 $C=\left\{ x:x\text{In roaster form, the order in which the elements is an integer;}{{\text{x}}^{2}}\le 4 \right\}$ 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)

Conclusion

In conclusion, 11th Math 1 Exercise 1.1 has introduced students to the fundamental concepts of sets, including their definitions, types, and notations. Understanding these basics is crucial as they form the foundation for more complex topics in mathematics. By completing this class 11 maths exercise 1.1, students now have a solid grasp of what sets are and how they are used in mathematics. This knowledge will be invaluable as students progress through students studies, helping students tackle more advanced problems with confidence. 

Class 11 Maths Chapter 1: Exercises Breakdown

CBSE Class 11 Maths Chapter 1 Other Study Materials

Chapter-Specific NCERT Solutions for Class 11 Maths

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