RD Sharma Class 11 Solutions Chapter - Sets | Mathematics (Maths) Class 11 - Commerce PDF Download

 Page 1


1. Sets
Exercise 1.1
1. Question
What is the difference between a collection and a set? Give reasons to support your answer.
Answer
A collection can include different types of elements. Whereas a set is a well-defined collection of distinct
elements and it is enclosed in “{}.”
E.g.:
The collection of good teachers is a collection but not a set
Whereas when we take all the teachers of the school, then it is set. Because when we take all the teachers,
then it is a definite number, so it called a set. But when we say, good teachers, it changes opinion wise, so
we can say it is not definite hence we can call it a collection as it is not well defined.
2. Question
Which of the following collections are sets? Justify your answer:
(i) A collection of all natural numbers less than 50.
(ii) The collection of good hockey players in India.
(iii) The collection of all the girls in your class.
(iv) The collection of all talented writers of India.
(v) The collection of difficult topics in Mathematics.
(vi) The collection of novels written by Munshi Prem Chand.
(vii) The collection of all questions of this chapter.
(viii) The collection of all months of a year beginning with the letter J.
(ix) A collection of most dangerous animals of the world.
(x) The collection of prime integers.
Answer
(i) It is a set. As we are saying all numbers less than 50 it is a definite quantity hence it is a set.
(ii) It is not a set. Because when we say good hockey players, it changes opinion wise and cannot be well
defined or is not countable.
(iii) It is a set. As we are saying a collection of all the girls in your class it is a definite quantity hence it is a
set.
(iv) It is not a set. Because when we say a collection of all talented writers of India, it changes opinion wise
and cannot be well defined or is not countable.
(v) It is not a set. Because when we say a collection of all talented writers of India, it changes opinion wise
and cannot be well defined or is not countable.
(vi) It is a set. As we are saying a collection of novels written by Munshi Prem Chand.it is a definite quantity
hence it is a set.
(vii) It is a set. As we are saying a collection of all the questions in this chapter. It is a definite quantity hence
it is a set.
(viii) It is a set. As we are saying a collection of all months of a year beginning with the letter J. It is a definite
quantity hence it is a set.
(ix) It is not a set. Because when we say a collection of most dangerous animals of the world it changes
opinion wise and we cannot define dangerousness of an animal.
Page 2


1. Sets
Exercise 1.1
1. Question
What is the difference between a collection and a set? Give reasons to support your answer.
Answer
A collection can include different types of elements. Whereas a set is a well-defined collection of distinct
elements and it is enclosed in “{}.”
E.g.:
The collection of good teachers is a collection but not a set
Whereas when we take all the teachers of the school, then it is set. Because when we take all the teachers,
then it is a definite number, so it called a set. But when we say, good teachers, it changes opinion wise, so
we can say it is not definite hence we can call it a collection as it is not well defined.
2. Question
Which of the following collections are sets? Justify your answer:
(i) A collection of all natural numbers less than 50.
(ii) The collection of good hockey players in India.
(iii) The collection of all the girls in your class.
(iv) The collection of all talented writers of India.
(v) The collection of difficult topics in Mathematics.
(vi) The collection of novels written by Munshi Prem Chand.
(vii) The collection of all questions of this chapter.
(viii) The collection of all months of a year beginning with the letter J.
(ix) A collection of most dangerous animals of the world.
(x) The collection of prime integers.
Answer
(i) It is a set. As we are saying all numbers less than 50 it is a definite quantity hence it is a set.
(ii) It is not a set. Because when we say good hockey players, it changes opinion wise and cannot be well
defined or is not countable.
(iii) It is a set. As we are saying a collection of all the girls in your class it is a definite quantity hence it is a
set.
(iv) It is not a set. Because when we say a collection of all talented writers of India, it changes opinion wise
and cannot be well defined or is not countable.
(v) It is not a set. Because when we say a collection of all talented writers of India, it changes opinion wise
and cannot be well defined or is not countable.
(vi) It is a set. As we are saying a collection of novels written by Munshi Prem Chand.it is a definite quantity
hence it is a set.
(vii) It is a set. As we are saying a collection of all the questions in this chapter. It is a definite quantity hence
it is a set.
(viii) It is a set. As we are saying a collection of all months of a year beginning with the letter J. It is a definite
quantity hence it is a set.
(ix) It is not a set. Because when we say a collection of most dangerous animals of the world it changes
opinion wise and we cannot define dangerousness of an animal.
(x) It is a set as we are saying a collection of prime integers. It is a definite quantity hence it is a set.
3. Question
If A={0,1,2,3,4,5,6,7,8,9,10}, then insert the appropriate symbol or in each of the following blank spaces:
i. 4 ……A ii. -4……A
iii. 12…..A iv. 9…..A
v. 0……A vi. -2……A
Answer
First of all it is important that you know what does (?) this symbol means. In terms of mathematics it is used
as belongs to () is used as does not belong to.
i.4 ……A
As 4 is there in Set A then we can say 4 belongs to A
? 4 ? A
ii. -4……A
As -4 is not there in Set A then we can say 4 does not belong to A
? 4 A
iii. 12…..A
As 12 is not there in Set A then we can say 12 does not belong to A
? 12 A
iv.9 ……A
As 9 is there in Set A then we can say 9 belongs to A
? 9 ? A
v.0 ……A
As 0 is there in Set A, then we can say 0 belongs to A
? 0 ? A
vi.-2 ……A
As -2 is not there in Set A then we can say -2 does not belong to A
? -2 A
Exercise 1.2
1 A. Question
Describe the following sets in Roster form:
{x:x is a letter before e in the English alphabet}.
Answer
Here x:x it is read as x is such that x
So now when we read whole sentence it becomes x is such that x is a letter before e in the English alphabet.
Now letters before e are a,b,c,d.
? Roster form will be {a,b,c,d}.
1 B. Question
Describe the following sets in Roster form:
Page 3


1. Sets
Exercise 1.1
1. Question
What is the difference between a collection and a set? Give reasons to support your answer.
Answer
A collection can include different types of elements. Whereas a set is a well-defined collection of distinct
elements and it is enclosed in “{}.”
E.g.:
The collection of good teachers is a collection but not a set
Whereas when we take all the teachers of the school, then it is set. Because when we take all the teachers,
then it is a definite number, so it called a set. But when we say, good teachers, it changes opinion wise, so
we can say it is not definite hence we can call it a collection as it is not well defined.
2. Question
Which of the following collections are sets? Justify your answer:
(i) A collection of all natural numbers less than 50.
(ii) The collection of good hockey players in India.
(iii) The collection of all the girls in your class.
(iv) The collection of all talented writers of India.
(v) The collection of difficult topics in Mathematics.
(vi) The collection of novels written by Munshi Prem Chand.
(vii) The collection of all questions of this chapter.
(viii) The collection of all months of a year beginning with the letter J.
(ix) A collection of most dangerous animals of the world.
(x) The collection of prime integers.
Answer
(i) It is a set. As we are saying all numbers less than 50 it is a definite quantity hence it is a set.
(ii) It is not a set. Because when we say good hockey players, it changes opinion wise and cannot be well
defined or is not countable.
(iii) It is a set. As we are saying a collection of all the girls in your class it is a definite quantity hence it is a
set.
(iv) It is not a set. Because when we say a collection of all talented writers of India, it changes opinion wise
and cannot be well defined or is not countable.
(v) It is not a set. Because when we say a collection of all talented writers of India, it changes opinion wise
and cannot be well defined or is not countable.
(vi) It is a set. As we are saying a collection of novels written by Munshi Prem Chand.it is a definite quantity
hence it is a set.
(vii) It is a set. As we are saying a collection of all the questions in this chapter. It is a definite quantity hence
it is a set.
(viii) It is a set. As we are saying a collection of all months of a year beginning with the letter J. It is a definite
quantity hence it is a set.
(ix) It is not a set. Because when we say a collection of most dangerous animals of the world it changes
opinion wise and we cannot define dangerousness of an animal.
(x) It is a set as we are saying a collection of prime integers. It is a definite quantity hence it is a set.
3. Question
If A={0,1,2,3,4,5,6,7,8,9,10}, then insert the appropriate symbol or in each of the following blank spaces:
i. 4 ……A ii. -4……A
iii. 12…..A iv. 9…..A
v. 0……A vi. -2……A
Answer
First of all it is important that you know what does (?) this symbol means. In terms of mathematics it is used
as belongs to () is used as does not belong to.
i.4 ……A
As 4 is there in Set A then we can say 4 belongs to A
? 4 ? A
ii. -4……A
As -4 is not there in Set A then we can say 4 does not belong to A
? 4 A
iii. 12…..A
As 12 is not there in Set A then we can say 12 does not belong to A
? 12 A
iv.9 ……A
As 9 is there in Set A then we can say 9 belongs to A
? 9 ? A
v.0 ……A
As 0 is there in Set A, then we can say 0 belongs to A
? 0 ? A
vi.-2 ……A
As -2 is not there in Set A then we can say -2 does not belong to A
? -2 A
Exercise 1.2
1 A. Question
Describe the following sets in Roster form:
{x:x is a letter before e in the English alphabet}.
Answer
Here x:x it is read as x is such that x
So now when we read whole sentence it becomes x is such that x is a letter before e in the English alphabet.
Now letters before e are a,b,c,d.
? Roster form will be {a,b,c,d}.
1 B. Question
Describe the following sets in Roster form:
{x ?N: x
2
< 25}.
Answer
First thing analyze the given data. x ? N that implies x is a natural number.
x
2
 < 25
? x < ±5
As x belongs to the natural number that means x < 5.
All numbers less than 5 are 1,2,3,4.
? Roster form will be {1,2,3,4}.
1 C. Question
Describe the following sets in Roster form:
{x N: x is a prime number, 10 < x < 20}
Answer
X is a natural number and is between 10 and 20.
X is such that x is a prime number between 10 and 20.
Prime numbers between 10 and 20 are 11,13,17,19.
? Roster form will be {11,13,17,19}.
1 D. Question
Describe the following sets in Roster form:
{x N: x = 2n, n N}.
Answer
X is a natural number also x = 2n
? Roster form will be {2,4,6,8…..}.
This an infinite set.
1 E. Question
Describe the following sets in Roster form:
{x R: x > x}.
Answer
Any real number is equal to its value it is neither less nor greater.
According to the question we have to write Roster form of such real numbers which has value less than itself.
As there are no such numbers.
? Roster form will be ?.
This is called as null set.
1 F. Question
Describe the following sets in Roster form:
{x:x is a prime number which is a divisor of 60}.
Answer
All numbers which are divisor of 60 are = 1,2,3,4,5,6,10,12,15,20,30,60.
Now numbers which are prime are = 2,3,5.
Page 4


1. Sets
Exercise 1.1
1. Question
What is the difference between a collection and a set? Give reasons to support your answer.
Answer
A collection can include different types of elements. Whereas a set is a well-defined collection of distinct
elements and it is enclosed in “{}.”
E.g.:
The collection of good teachers is a collection but not a set
Whereas when we take all the teachers of the school, then it is set. Because when we take all the teachers,
then it is a definite number, so it called a set. But when we say, good teachers, it changes opinion wise, so
we can say it is not definite hence we can call it a collection as it is not well defined.
2. Question
Which of the following collections are sets? Justify your answer:
(i) A collection of all natural numbers less than 50.
(ii) The collection of good hockey players in India.
(iii) The collection of all the girls in your class.
(iv) The collection of all talented writers of India.
(v) The collection of difficult topics in Mathematics.
(vi) The collection of novels written by Munshi Prem Chand.
(vii) The collection of all questions of this chapter.
(viii) The collection of all months of a year beginning with the letter J.
(ix) A collection of most dangerous animals of the world.
(x) The collection of prime integers.
Answer
(i) It is a set. As we are saying all numbers less than 50 it is a definite quantity hence it is a set.
(ii) It is not a set. Because when we say good hockey players, it changes opinion wise and cannot be well
defined or is not countable.
(iii) It is a set. As we are saying a collection of all the girls in your class it is a definite quantity hence it is a
set.
(iv) It is not a set. Because when we say a collection of all talented writers of India, it changes opinion wise
and cannot be well defined or is not countable.
(v) It is not a set. Because when we say a collection of all talented writers of India, it changes opinion wise
and cannot be well defined or is not countable.
(vi) It is a set. As we are saying a collection of novels written by Munshi Prem Chand.it is a definite quantity
hence it is a set.
(vii) It is a set. As we are saying a collection of all the questions in this chapter. It is a definite quantity hence
it is a set.
(viii) It is a set. As we are saying a collection of all months of a year beginning with the letter J. It is a definite
quantity hence it is a set.
(ix) It is not a set. Because when we say a collection of most dangerous animals of the world it changes
opinion wise and we cannot define dangerousness of an animal.
(x) It is a set as we are saying a collection of prime integers. It is a definite quantity hence it is a set.
3. Question
If A={0,1,2,3,4,5,6,7,8,9,10}, then insert the appropriate symbol or in each of the following blank spaces:
i. 4 ……A ii. -4……A
iii. 12…..A iv. 9…..A
v. 0……A vi. -2……A
Answer
First of all it is important that you know what does (?) this symbol means. In terms of mathematics it is used
as belongs to () is used as does not belong to.
i.4 ……A
As 4 is there in Set A then we can say 4 belongs to A
? 4 ? A
ii. -4……A
As -4 is not there in Set A then we can say 4 does not belong to A
? 4 A
iii. 12…..A
As 12 is not there in Set A then we can say 12 does not belong to A
? 12 A
iv.9 ……A
As 9 is there in Set A then we can say 9 belongs to A
? 9 ? A
v.0 ……A
As 0 is there in Set A, then we can say 0 belongs to A
? 0 ? A
vi.-2 ……A
As -2 is not there in Set A then we can say -2 does not belong to A
? -2 A
Exercise 1.2
1 A. Question
Describe the following sets in Roster form:
{x:x is a letter before e in the English alphabet}.
Answer
Here x:x it is read as x is such that x
So now when we read whole sentence it becomes x is such that x is a letter before e in the English alphabet.
Now letters before e are a,b,c,d.
? Roster form will be {a,b,c,d}.
1 B. Question
Describe the following sets in Roster form:
{x ?N: x
2
< 25}.
Answer
First thing analyze the given data. x ? N that implies x is a natural number.
x
2
 < 25
? x < ±5
As x belongs to the natural number that means x < 5.
All numbers less than 5 are 1,2,3,4.
? Roster form will be {1,2,3,4}.
1 C. Question
Describe the following sets in Roster form:
{x N: x is a prime number, 10 < x < 20}
Answer
X is a natural number and is between 10 and 20.
X is such that x is a prime number between 10 and 20.
Prime numbers between 10 and 20 are 11,13,17,19.
? Roster form will be {11,13,17,19}.
1 D. Question
Describe the following sets in Roster form:
{x N: x = 2n, n N}.
Answer
X is a natural number also x = 2n
? Roster form will be {2,4,6,8…..}.
This an infinite set.
1 E. Question
Describe the following sets in Roster form:
{x R: x > x}.
Answer
Any real number is equal to its value it is neither less nor greater.
According to the question we have to write Roster form of such real numbers which has value less than itself.
As there are no such numbers.
? Roster form will be ?.
This is called as null set.
1 F. Question
Describe the following sets in Roster form:
{x:x is a prime number which is a divisor of 60}.
Answer
All numbers which are divisor of 60 are = 1,2,3,4,5,6,10,12,15,20,30,60.
Now numbers which are prime are = 2,3,5.
? Roster form will be {2,3,5}.
1 G. Question
Describe the following sets in Roster form:
{x:x is a two digit number such that the sum of its digits is 8}
Answer
Numbers which have sum as 8 are = 17,26,35,44,53,62,71,80
? Roster form will be {17,26,35,44,53,62,71,80}.
1 H. Question
Describe the following sets in Roster form:
The set of all letters in the word ‘Trigonometry’
Answer
All letters means no letter should be repeated
Trigonometry = t,r,i,g,o,n,m,e,y
? Roster form will be {t,r,i,g,o,n,m,e,y}
1 I. Question
Describe the following sets in Roster form:
The set of all letters in the word ‘Better.’
Answer
All letters means no letter should be repeated
Better = b,e,t,r
? Roster form will be {b,e,t,r}.
2. Question
Describe the following sets in set-builder form:
(i) A = {1,2,3,4,5,6}
(ii) B={1, 1/2 , 1/3, 1/4, 1/5,…..}
(iii) C={0,3,6,9,12,….}
(iv) D={10,11,12,13,14,15}
(v) E={0}
(vi) {1,4,9,16,…,100}
(vii) {2,4,6,8,….}
(viii) {5,25,125,625}
Answer
(i) {x:x ? N,x<7}
This is read as x is such that x belongs to natural number and x is less than 7. It satisfies all condition of
roster form.
(ii) {x ? z: x=1/n+1,n ? W}
This is read as x is such that x is an integer greater than or equal to 0. And it’s value is 1/x+1.
(iii) {x N: x = 3n, n W}
Page 5


1. Sets
Exercise 1.1
1. Question
What is the difference between a collection and a set? Give reasons to support your answer.
Answer
A collection can include different types of elements. Whereas a set is a well-defined collection of distinct
elements and it is enclosed in “{}.”
E.g.:
The collection of good teachers is a collection but not a set
Whereas when we take all the teachers of the school, then it is set. Because when we take all the teachers,
then it is a definite number, so it called a set. But when we say, good teachers, it changes opinion wise, so
we can say it is not definite hence we can call it a collection as it is not well defined.
2. Question
Which of the following collections are sets? Justify your answer:
(i) A collection of all natural numbers less than 50.
(ii) The collection of good hockey players in India.
(iii) The collection of all the girls in your class.
(iv) The collection of all talented writers of India.
(v) The collection of difficult topics in Mathematics.
(vi) The collection of novels written by Munshi Prem Chand.
(vii) The collection of all questions of this chapter.
(viii) The collection of all months of a year beginning with the letter J.
(ix) A collection of most dangerous animals of the world.
(x) The collection of prime integers.
Answer
(i) It is a set. As we are saying all numbers less than 50 it is a definite quantity hence it is a set.
(ii) It is not a set. Because when we say good hockey players, it changes opinion wise and cannot be well
defined or is not countable.
(iii) It is a set. As we are saying a collection of all the girls in your class it is a definite quantity hence it is a
set.
(iv) It is not a set. Because when we say a collection of all talented writers of India, it changes opinion wise
and cannot be well defined or is not countable.
(v) It is not a set. Because when we say a collection of all talented writers of India, it changes opinion wise
and cannot be well defined or is not countable.
(vi) It is a set. As we are saying a collection of novels written by Munshi Prem Chand.it is a definite quantity
hence it is a set.
(vii) It is a set. As we are saying a collection of all the questions in this chapter. It is a definite quantity hence
it is a set.
(viii) It is a set. As we are saying a collection of all months of a year beginning with the letter J. It is a definite
quantity hence it is a set.
(ix) It is not a set. Because when we say a collection of most dangerous animals of the world it changes
opinion wise and we cannot define dangerousness of an animal.
(x) It is a set as we are saying a collection of prime integers. It is a definite quantity hence it is a set.
3. Question
If A={0,1,2,3,4,5,6,7,8,9,10}, then insert the appropriate symbol or in each of the following blank spaces:
i. 4 ……A ii. -4……A
iii. 12…..A iv. 9…..A
v. 0……A vi. -2……A
Answer
First of all it is important that you know what does (?) this symbol means. In terms of mathematics it is used
as belongs to () is used as does not belong to.
i.4 ……A
As 4 is there in Set A then we can say 4 belongs to A
? 4 ? A
ii. -4……A
As -4 is not there in Set A then we can say 4 does not belong to A
? 4 A
iii. 12…..A
As 12 is not there in Set A then we can say 12 does not belong to A
? 12 A
iv.9 ……A
As 9 is there in Set A then we can say 9 belongs to A
? 9 ? A
v.0 ……A
As 0 is there in Set A, then we can say 0 belongs to A
? 0 ? A
vi.-2 ……A
As -2 is not there in Set A then we can say -2 does not belong to A
? -2 A
Exercise 1.2
1 A. Question
Describe the following sets in Roster form:
{x:x is a letter before e in the English alphabet}.
Answer
Here x:x it is read as x is such that x
So now when we read whole sentence it becomes x is such that x is a letter before e in the English alphabet.
Now letters before e are a,b,c,d.
? Roster form will be {a,b,c,d}.
1 B. Question
Describe the following sets in Roster form:
{x ?N: x
2
< 25}.
Answer
First thing analyze the given data. x ? N that implies x is a natural number.
x
2
 < 25
? x < ±5
As x belongs to the natural number that means x < 5.
All numbers less than 5 are 1,2,3,4.
? Roster form will be {1,2,3,4}.
1 C. Question
Describe the following sets in Roster form:
{x N: x is a prime number, 10 < x < 20}
Answer
X is a natural number and is between 10 and 20.
X is such that x is a prime number between 10 and 20.
Prime numbers between 10 and 20 are 11,13,17,19.
? Roster form will be {11,13,17,19}.
1 D. Question
Describe the following sets in Roster form:
{x N: x = 2n, n N}.
Answer
X is a natural number also x = 2n
? Roster form will be {2,4,6,8…..}.
This an infinite set.
1 E. Question
Describe the following sets in Roster form:
{x R: x > x}.
Answer
Any real number is equal to its value it is neither less nor greater.
According to the question we have to write Roster form of such real numbers which has value less than itself.
As there are no such numbers.
? Roster form will be ?.
This is called as null set.
1 F. Question
Describe the following sets in Roster form:
{x:x is a prime number which is a divisor of 60}.
Answer
All numbers which are divisor of 60 are = 1,2,3,4,5,6,10,12,15,20,30,60.
Now numbers which are prime are = 2,3,5.
? Roster form will be {2,3,5}.
1 G. Question
Describe the following sets in Roster form:
{x:x is a two digit number such that the sum of its digits is 8}
Answer
Numbers which have sum as 8 are = 17,26,35,44,53,62,71,80
? Roster form will be {17,26,35,44,53,62,71,80}.
1 H. Question
Describe the following sets in Roster form:
The set of all letters in the word ‘Trigonometry’
Answer
All letters means no letter should be repeated
Trigonometry = t,r,i,g,o,n,m,e,y
? Roster form will be {t,r,i,g,o,n,m,e,y}
1 I. Question
Describe the following sets in Roster form:
The set of all letters in the word ‘Better.’
Answer
All letters means no letter should be repeated
Better = b,e,t,r
? Roster form will be {b,e,t,r}.
2. Question
Describe the following sets in set-builder form:
(i) A = {1,2,3,4,5,6}
(ii) B={1, 1/2 , 1/3, 1/4, 1/5,…..}
(iii) C={0,3,6,9,12,….}
(iv) D={10,11,12,13,14,15}
(v) E={0}
(vi) {1,4,9,16,…,100}
(vii) {2,4,6,8,….}
(viii) {5,25,125,625}
Answer
(i) {x:x ? N,x<7}
This is read as x is such that x belongs to natural number and x is less than 7. It satisfies all condition of
roster form.
(ii) {x ? z: x=1/n+1,n ? W}
This is read as x is such that x is an integer greater than or equal to 0. And it’s value is 1/x+1.
(iii) {x N: x = 3n, n W}
(iv) {x:x ? N,9<x<16}
(v) {x:x=0}
(vi) {x ? N: x=n
2
,n=10, n ? N}
(vii) {x N: x = 2n, n N}
(viii) {x N: x = 5
n
,0<n<6 N}
3 A. Question
List all the elements of the following sets:
A={x:x
2
= 10, xZ}
Answer
First of all, x is an integer hence it can be positive and negative also.
X
2
 = 10
X= v10
X =±1,±2,±3
A={±1,±2,±3}
3 B. Question
List all the elements of the following sets:
Answer
Substituting the values of n we will get the solutions
At n=1, 
At n=1, 
At n=1, 
At n=1, 
At n=1, 
? 
3 C. Question
List all the elements of the following sets:
C=
Answer
? x is an integer between -1/2 and 9/2
So all integers between given values -0.5<x<4.5
0,1,2,3,4