NCERT Solutions Class 11 Maths Chapter-1 (Sets)

NCERT Solutions Class 11 Maths from class 11th Students will get the answers of Chapter-1 (Sets) . This chapter will help you to learn the basics and you should expect at least one question in your exam from this chapter.
We have given the answers of all the questions of NCERT Board Mathematics Textbook in very easy language, which will be very easy for the students to understand and remember so that you can pass with good marks in your examination.

NCERT Question-Answer

Class 11 Mathematics

Chapter-1 (Sets)

Questions and answers given in practice

Chapter-1 (Sets)

Exercise 1.1

 Q.1: Which of the following are sets. Explain your answer.
(a).  The collection of all day in a week which have the first letter S.
(b).  The collection of ten most famous singers of India.
(c).  A group of best football strikers of the world.
(d).  The collection of all girls in your school.
(e).  The collection of all odd numbers below 50.
(f).  A collection of poems written by the poet Shakespeare.
(g).  The collection of all prime numbers.
(h).  The collection of questions in science book.
(i).  A collection of most dangerous reptiles in India.

Solution:
(i) The collection of all months of a year beginning with the letter J is a well-defined collection of objects because one can definitely identity a month that belongs to this collection. Hence, this collection is a set.
(ii) The collection of ten most talented writer of India is not a well-defined collection because the criteria for determining a writer’s talent vary from person to person. Hence, this collection is not a set.
(iii) A team of eleven best cricket batsmen of the world is not a well-defined collection because the criteria for determining a batsman’s talent may vary from person to person. Hence, this collection is not a set.
(iv) The collection of all boys in your class is a well-defined collection because you can definitely identify a boy who belongs to this collection. Hence, this collection is a set.
(v) The collection of all natural numbers less than 100 is a well-defined collection because one can definitely identify a number that belongs to this collection. Hence, this collection is a set.
(vi) A collection of novels written by the writer Munshi Prem Chand is a well-defined collection because one can definitely identify a book that belongs to this collection. Hence, this collection is a set.
(vii) The collection of all even integers is a well-defined collection because one can definitely identify an even integer that belongs to this collection.
Hence, this collection is a set.
(viii) The collection of questions in this chapter is a well-defined collection because one can definitely identify a question that belongs to this chapter. Hence, this collection is a set.
(ix) The collection of most dangerous animals of the world is not a well-defined collection because the criteria for determining the dangerousness of an animal can vary from person to person. Hence, this collection is not a set.

Q.2: Let P = {2, 3, 4, 5, 6, 7}. Insert the correct symbol ∈or∉ inside the given blank spaces below:
(a).  2 . . . . . . . . . . P
(b).  9 . . . . . . . . . P
(c).  11 . . . . . . . . P
(d).  4 . . . . . . . . P
(e).  0 . . . . . . . . P
(f).  7 . . . . . . . . P

Solution:
(i) 5 ∈ A
(ii) 8 ∉ A
(iii) 0 ∉ A
(iv) 4 ∈ A
(v) 2 ∈ A
(vi) 10 ∉ A

Q.3: Write the given sets in roster form:
(a).  P = {y: y is an integer and -4 < y < 6}.
(b).  Q = {y: y is a natural number which is <8}
(c).  R = {y: y is a 2 digit natural number in which the sum of its digits is 9}
(d).  S = {y: y is a prime number which is a divisor of 70}
(e).  T = The set of all letters in the word ELEPHANT
(f).  U = The set of all letters in the word DIVISION

Solution:
(i) A = {x: x is an integer and –3 < x < 7}
The elements of this set are –2, –1, 0, 1, 2, 3, 4, 5, and 6 only.
Therefore, the given set can be written in roster form as
A = {–2, –1, 0, 1, 2, 3, 4, 5, 6}

(ii) B = {x: x is a natural number less than 6}
The elements of this set are 1, 2, 3, 4, and 5 only.
Therefore, the given set can be written in roster form as
B = {1, 2, 3, 4, 5}

(iii) C = {x: x is a two-digit natural number such that the sum of its digits is 8}
The elements of this set are 17, 26, 35, 44, 53, 62, 71, and 80 only.
Therefore, this set can be written in roster form as
C = {17, 26, 35, 44, 53, 62, 71, 80}

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

2	60
2	30
3	15
	5

∴60 = 2 × 2 × 3 × 5
The elements of this set are 2, 3, and 5 only.
Therefore, this set can be written in roster form as D = {2, 3, 5}.

Q.4: Write the given sets in set-builder form:
(a).  {4, 8, 12, 16, 20}
(b).  {3, 9, 27, 81}
(c).  {4, 16, 64, 256, 1024}
(d).  {1, 3, 5, 7…}
(e).  {1, 8, 27, 64….1000}

Solution:

Q.5: List all the elements of the following sets :
(i) A = {x : x is an odd natural number}

(iii) C = {x : x is an integer, x2 < 4}
(iv) D = {x : x is a letter in the word “LOYAL”}
(v) E = {x : x is a month of a year not having 31 days}
(vi) F = {x : x is a consonant in the English alphabet which precedes k}.

Solution:

Q.6: Match the following:

(A)	{1, 2, 3, 6}	(i)	{y: y is a divisor of 6 and also a prime number}
(B)	{T, R, I, G, O, N, M, E, Y}	(ii)	{y: y is less than 10 and also an odd number}
(C)	{2, 3}	(iii)	{y: y is natural number divisor of 6}
(D)	{1, 3, 5, 7, 9}	(iv)	{y: y is a letter of the word TRIGONOMETRY}

Solution:

Exercise 1.2 

Q.1: Which of the following given below is null set?
(i).  Set of odd natural numbers which is divisible by 2.
(ii).  Set of even numbers which are prime
(iii).  {x: x is a natural number, x<5 and x>7}
(iv).  {y: y is a point common to any two parallel lines}

Solution:

Q.2:  State whether the following sets are infinite or finite:
(i).  A set of months of a year.
(ii).  {1, 2, 3 ….}
(iii).  {1, 2, 3…99, 100}
(iv).  The set of positive integers which are greater than 100.
(v).  The set of prime numbers which are less than 99

Solution:

Q.3: State whether the following sets are infinite or finite:
(i).  The set of lines parallel to the x – axis.
(ii).  The set of letters in the vowels.
(iii).  The set of numbers multiple of 10.
(iv).  The set of humans living on Earth.
(v).  The set of circles passing through the origin (0, 0).

Solution:

Q.4: In the following set given below, state whether A = B or not:
(i).  A = {w, x, y, z}
B = {z, y, x, w}
(ii).  A = {5, 9, 13, 17}
B = {9, 5, 17, 19}
(iii).  A = {4, 2, 6, 10, 8}
B = {x: x is positive even integer and x≤10 }
(iv).   A = {x: x is a multiple of 10}
B = {10, 15, 20, 25, 30 …}

Solution:

Q.5 In the following set given below, is the pair of sets equal?
(i). A = {3, 4}
B = {y: y is solution of y²+5y+6=0}
(ii).  A = {a: a is a letter in the word FOLLOW}
B = {b: b is a letter in the word WOLF}

Solution:

Q.6: From the following sets, select equal sets:
A = {2, 4, 8, 12}
B = {1, 2, 3, 4}
C = {4, 8, 12, 14}
D = {3, 1, 4, 2}
E = {–1, 1}
F = {0, a}
G = {1, –1}
H = {0, 1}

Solution:

Exercise 1.3

Q.1: Fill in the blanks properly using ⊂ and ⊄.

(i).  {3, 4, 5} ____ {2, 3, 4, 5, 6}
(ii).  {a, b, c} ____ {d, c, d}
(iii).  {y: y is a pupil of Class 11 of the school} ____ {y: y is students of the school}
(iv).  {y: y is a circle in the plane} ____ {y: y is a circle in the same plane with radius 2 unit}
(v).  {y: y is an equilateral triangle in a plane} ____ {y: y is a rectangle in the same plane}
(vi).  {y: y is an equilateral triangle in a plane} ____ {y: y is a triangle in the plane}
(vii).  {y: y is an odd natural number} ____ {x: x is an integers}

Solution:

Q.2: State whether the given statements are true or false:
(i).  {b, c} ⊄ {c, d, e}
(ii).  {a, e, i} ⊂ {x: x is a vowel in the English alphabets}
(iii).  {1, 2, 3} ⊂{1, 2, 4, 5}
(iv).  {c} ⊂ {b, c, d}
(v).  {b} ∈ {a, b, c, d}
(vi).  {y: y is an even natural no. less than 6} ⊂ {y: y is a natural no. which can divide 36}

Solution:

Q.3: Let X = {11, 12, {13, 14}, 15}. According to the given set which of the given statements are false? Explain why.
(i).  {13, 14} ⊂ X
(ii).  {13, 14} ∈ X
(iii).  {{13, 14}} ⊂ X
(iv).  11 ∈ X
(v).  11 ⊂ X
(vi).  {11, 12, 15} ⊂ X
(vii).  {11, 12, 15} ∈ X
(viii).  {11, 12, 13} ⊂ X
(ix).  Ø ∈ X
(x).  Ø ⊂ X
(xi).  {Ø} ⊂ X

Solution:

S 

Q.4: Write all the subsets of the given sets:
(i).  {b}
(ii).  {b, c}
(iii).  {2, 3, 4}
(iv).  Ø

Solution:

Q.5: How many elements has P(X), if X = Ø ?

Solution:

Q.6: Write the given in the form of intervals:
(i).  {y: y ∈ R, –5 < y ≤ 7}
(ii).  {y: y ∈ R, –13 < y < –11}
(iii).  {y: y ∈ R, 1 ≤ y < 8}
(iv).  {y: y ∈ R, 4 ≤ y ≤ 5}

Solution:

Q.7:  Write the given intervals in the form of set – builder:
(i).  (–4, 1)
(ii).  [7, 13]
(iii).  (7, 13]
(iv).  [–24, 6)

Solution:

Q.8: What universal set/ sets would you propose for the given sets?

(i).  The set of right triangles
(ii).  The set of isosceles triangles

Solution:

Q.9: X = {1, 3, 5}, Y = {2, 4, 6} and Z = {0, 2, 4, 6, 8}

Which of the given sets can be considered as the universal set for the given sets X, Y and Z?
(i).  {0, 1, 2, 3, 4, 5, 6}
(ii).  Ø
(iii).  {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10}
(iv).  {1, 2, 3, 4, 5, 6, 7, 8}

Solution:

Exercise 1.4

Q.1: Find the union of each of the following pairs of sets:

(i). P = {1, 4, 6} and Q = {1, 3, 4}
(ii). X = {a, e, i, o, u} and Y = {x, y, z}
(iii). X = {x: x is a natural number and multiple of 3} and Y = {x: x is a natural number less than 6}
(iv). X = {x: x is a natural number and 1 < x ≤ 5} and Y = {x: x is a natural number and 5 < x < 10}
(v). X = {4, 5, 6} and Y = Φ

Solution:

Q.2:  Let A = {x, y} and B = {x, y, z}
Is A ⊂ B? What is A ∪ B?

Solution:

Q.3: If X and Y are two sets such that X ⊂ Y, then what is X ∪ Y?

Solution:

Q.4: If P = {1, 2, 3, 4}, Q = {3, 4, 5, 6}, R = {5, 6, 7, 8} and S = {7, 8, 9, 10}
Find the following:
(i).  P ∪ Q
(ii).  P ∪ R
(iii).  Q ∪ R
(iv).  Q ∪ S
(v).  P ∪ Q ∪ R
(vi).  P ∪ Q ∪ S
(vii).  Q ∪ R ∪ S

Solution:

Q.5:  Find the intersection of each of the following pairs of sets:
(i).  P = {1, 4, 6} and Q = {1, 3, 4}
(ii).  X = {a, e, i, o, u} and Y = {x, y, z}
(iii).  X = {x: x is a natural number and multiple of 3} and Y = {x: x is a natural number less than 6}
(iv).  X = {x: x is a natural number and 1 < x ≤ 5} and Y = {x: x is a natural number and 5 < x < 10}
(v).  X = {4, 5, 6} and Y = Φ

Solution 5:

Q.6: If A = {3, 5, 7, 9, 11}, B = {7, 9, 11, 13}, C = {11, 13, 15} and D = {15, 17}
Find the following:
(i).  A ∩ B
(ii).  B ∩  C
(iii).  A ∩  C ∩  D
(iv).  A ∩  C
(v).   B ∩  D
(vi).  A ∩  (B ∪ C)
(vii).  A ∩  D
(viii).  A ∩  (B ∪ D)
(ix).  (A ∩  B) ∩  (B ∪ C)
(x).  (A∪  D) ∩  (B ∪ C)

Solution:

Q.7: If A = {y: y is a natural number}, B ={y: y is an even natural number}, C = {y: y is an odd natural number} and D = {y: y is a prime number}
Find the following:
(i). A ∩ B
(ii). A ∩ C
(iii). A ∩ D
(iv). B ∩ C
(v). B ∩ D
(vi). C ∩ D

Solution:

Q.8: Which of the given pairs of sets are disjoint?
(i).  A = {1, 2, 3, 4} and B = {x: x is a natural number and 4 ≤ x ≤ 6}
(ii).  A = {a, e, i, o, u} and B = {c, d, e, f}
(iii).  A = {x: x is an even integer} and B = {x: x is an odd integer}

Solution:

Q.9: If A = {3, 6, 9, 12, 15, 18, 21}, B = {4, 8, 12, 16, 20}, C = {2, 4, 6, 8, 10, 12, 14, 16} and D = {5, 10, 15, 20}
Find the following:
(i).  A – B
(ii).  A – C
(iii).  A – D
(iv).  B – A
(v).  C – A
(vi).  D – A
(vii).  B – C
(viii).  B – D
(ix).  C – B
(x).  D – B
(xi).  C – D
(xii).  D – C

Solution:

Q.10: If X = {a, b, c, d} and Y = {f, b, d, g}
Find the following:
(i).  X – Y
(ii).  Y – X
(iii).  X ∩ Y

Solution:

Q.11: What is R – Q, if R is the set of real numbers and Q is the set of rational?

Solution:

Q.12: State whether the following statements are true or false. Give reason.
(i).  A = {2, 3, 4, 5} and B = {3, 6} are disjoint sets.
(ii).  A = {a, e, i, o, u } and B = {a, b, c, d} are disjoint sets.
(iii).  A = {2, 6, 10, 14} and B = {3, 7, 11, 15} are disjoint sets.
(iv).  A = {2, 6, 10} and B = {3, 7, 11} are disjoint sets.

Solution:

Exercise 1.5

Q.1: Let U = {1, 2, 3, 4, 5, 6, 7, 8, 9}
X = {1, 2, 3, 4, 5}
Y = {2, 4, 6} and
Z = {2, 4, 5, 6}.
Find the following sets:
(i).   X’
(ii).  Y’
(iii).  (X∪Y)′
(iv).  (X∪Z)′
(v).  (X′)‘
(vi).  (Y–Z)′

Solution:

Q.2: If U = {a, b, c, d, e, f, g, h}, find the complements of the given sets:
(i).  W = {a, b, c}
(ii).  X = {d, e, f, g}
(iii).  Y = {a, c, e, g}
(iv).  Z = {f, g, h, a}

Solution:

Q.3: Take natural numbers as the universal set. Write the complements of the given sets:
(i).  A = {y: y is an even natural number}
(ii).  B = {y: y is an odd natural number}
(iii).  C = {y: y is a positive multiple of 3}
(iv).  D = {y: y is a prime number}
(v).  E = {y: y is a natural number divisible by 3 and 5}
(vi).  F = {y: y is a perfect square}
(vii).  G = {y: y is perfect cube}
(viii).  H = {y: y + 5 = 8}
(ix).  I = {y: 2y + 5 = 9}
(x).  J = {y: y ≥ 7}
(xi).  K = {y: y ∈ N and 2y + 1 > 10}

Solution:

Q.4: If U = {1, 2, 3, 4, 5,6,7,8, 9}
A = {2, 4, 6, 8} and, B = {2, 3, 5, 7}.
Verify that:
(i).  (A∪B)′ = A′∩B′
(ii).  (A∩B)′ = A′∪B′

Solution:

Q.5: Draw the Venn diagrams for the following:
(i).  (A∪B)′
(ii).  A′∩B′
(iii).  (A∩B)′
(iv).  A′∪B′

Solution:

Q.6: Let U be the universal set that is the set of all triangles in a plane. If X is the set of all triangles with at least one angle different from 60∘, what is X’ ?

Solution:

Q.7: Complete the given statements using proper symbols:
(i).  A∪A′
(ii).  Ø ∩A
(iii).  A∩A′
(iv).  U′∩A

Solution:

 

Exercise 1.6

Q.1: If A and B are two sets such that n(A) = 16, n(B) = 24 and n(A∪B) = 39. Find n(A∩B).

Solution:

Q.2: If A and B are two sets such that A∪B has 17 elements, A has 9 elements and B has 14 elements. How many elements does A∩B have?

Solution:

Q.3: In a group there are 450 people, 200 speaks Hindi and 270 can speak English. How many people can speak both English and Hindi?

Solution:

Q.4: If X and Y are two sets such that X has 22 elements, Y has 34 elements, and n(X∩Y) has 10 elements, how many elements does n(X∪Y) have?

Solution:

Q.5: If A and B are two sets such that A has 45 elements, n (A ∩ B) has 15 elements and n (A ∪ B) has 70 elements, how many elements does B have?

Solution:

Q.6:There are 70 people, out of which 35 like tea, 55 like coffee, and each person likes at least one of the two beverages. How many people like both tea and coffee?

Solution:

Q.7: There are 70 students in a group, 35 like cricket, 15 like both tennis and cricket. How many like tennis only and not cricket? How many like tennis?

Solution:

Q.8: In a committee, 60 people speak French, 30 speak Spanish and 20 speak both Spanish and French. How many speak at least one of these two languages?

Solution: