Q.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 ten most talented writers of India.
(iii) A team of eleven bestcricket batsmen of 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 Premchand.
(vii) The collection of all even integers.
(viii) The collection of questions in this Chapter.
(ix) A collection of most dangerous animals of the world.
Ans.
(i) The collection of all months of a year beginning with the letter J is a welldefined collection of objects because one can definitely identify a month that belongs to this collection. Hence, this collection is a set.
(ii) The collection of ten most talented writers of India is not a welldefined collection because the criteria for determining a writer’s talent may 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 welldefined 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 welldefined 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 welldefined 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 welldefined 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 welldefined 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 welldefined 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 welldefined 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 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
Ans.
(i) 5 ∈ A
(ii) 8 ∉ A
(iii) 0 ∉ A
(iv) 4 ∈ A
(v) 2 ∈ A
(vi) 10 ∉ A
Q.3. Write the following sets in roster form:
(i) A = {x : x is an integer and –3 < x < 7}.
(ii) B = {x : x is a natural number less than 6}.
(iii) C = {x : x is a twodigit natural number such that the sum of its digits is 8}
(iv) D = {x : x 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.
Ans.
(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 twodigit 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}
∴ 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}.
(v) E = The set of all letters in the word TRIGONOMETRY
There are 12 letters in the word TRIGONOMETRY, out of which letters T, R, and O are repeated.
Therefore, this set can be written in roster form as
E = {T, R, I, G, O, N, M, E, Y}
(vi) F = The set of all letters in the word BETTER
There are 6 letters in the word BETTER, out of which letters E and T are repeated.
Therefore, this set can be written in roster form as
F = {B, E, T, R}
Q.4. Write the following sets in the setbuilder form:
(i) (3, 6, 9, 12)
(ii) {2, 4, 8, 16, 32}
(iii) {5, 25, 125, 625}
(iv) {2, 4, 6 …}
(v) {1, 4, 9 … 100}
Ans.
(i) {3, 6, 9, 12} = {x : x = 3n, n∈ N and 1 ≤ n ≤ 4}
(ii) {2, 4, 8, 16, 32}
It can be seen that 2 = 2^{1}, 4 = 2^{2}, 8 = 2^{3}, 16 = 2^{4}, and 32 = 2^{5}.
∴ {2, 4, 8, 16, 32} = {x : x = 2^{n}, n∈ N and 1 ≤ n ≤ 5}
(iii) {5, 25, 125, 625}
It can be seen that 5 = 5^{1}, 25 = 5^{2}, 125 = 5^{3}, and 625 = 5^{4}.
∴ {5, 25, 125, 625} = {x : x = 5^{n}, n∈N and 1 ≤ n ≤ 4}
(iv) {2, 4, 6 …}
It is a set of all even natural numbers.
∴ {2, 4, 6 …} = {x : x is an even natural number}
(v) {1, 4, 9 … 100}
It can be seen that 1 = 1^{2}, 4 = 2^{2}, 9 = 3^{2} …100 = 10^{2}.
∴ {1, 4, 9… 100} = {x : x = n^{2}, n∈N and 1 ≤ n ≤ 10}
Q.5. List all the elements of the following sets:
(i) A = {x : x is an odd natural number}
(ii) B = {x:x is an integer,  1/2<x<9/2}
(iii) C = {x : x is an integer, x^{2} ≤ 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 proceeds k}.
Ans.
(i) A = {x : x is an odd natural number} = {1, 3, 5, 7, 9 …
(ii) B = {x:x is an integer,  1/2<n<9/2}
It can be seen that 1/2 = 0.5 and 9/2 = 4.5
∴ B = {0, 1, 2, 3, 4}
(iii) C = {x : x is an integer; x^{2} ≤ 4}
It can be seen that
(–1)^{2} = 1 ≤ 4; (–2)^{2} = 4 ≤ 4; (–3)^{2} = 9 > 4
0^{2} = 0 ≤ 4
1^{2} = 1 ≤ 4
2^{2} = 4 ≤ 4
3^{2} = 9 > 4
∴ C = {–2, –1, 0, 1, 2}
(iv) D = (x : x is a letter in the word “LOYAL”) = {L, O, Y, A}
(v) E = {x : x is a month of a year not having 31 days}
= {February, April, June, September, November}
(vi) F = {x : x is a consonant in the English alphabet which precedes k}
= {b, c, d, f, g, h, j}
Q.6. Match each of the set on the left in the roster form with the same set on the right described in setbuilder form:
Ans.
(i) All the elements of this set are natural numbers as well as the divisors of 6. Therefore, (i) matches with (c).
(ii) It can be seen that 2 and 3 are prime numbers. They are also the divisors of 6.
Therefore, (ii) matches with (a).
(iii) All the elements of this set are letters of the word MATHEMATICS. Therefore, (iii) matches with (d).
(iv) All the elements of this set are odd natural numbers less than 10. Therefore,
(iv) matches with (b).
EXERCISE  1.2
Q.1. Which of the following are examples of the null set
(i) Set of odd natural numbers divisible by 2
(ii) Set of even prime numbers
(iii) {x : x is a natural numbers, x < 5 and x > 7}
(iv) {y : y is a point common to any two parallel lines}
Ans.
(i) A set of odd natural numbers divisible by 2 is a null set because no odd number is divisible by 2.
(ii) A set of even prime numbers is not a null set because 2 is an even prime number.
(iii) {x : x is a natural number, x < 5 and x > 7} is a null set because a number cannot be simultaneously less than 5 and greater than 7.
(iv) {y : y is a point common to any two parallel lines} is a null set because parallel lines do not intersect. Hence, they have no common point.
Q.2. Which of the following sets are finite or infinite
(i) The set of months of a year
(ii) {1, 2, 3 ...}
(iii) {1, 2, 3 ... 99, 100}
(iv) The set of positive integers greater than 100
(v) The set of prime numbers less than 99
Ans.
(i) The set of months of a year is a finite set because it has 12 elements.
(ii) {1, 2, 3 …} is an infinite set as it has infinite number of natural numbers.
(iii) {1, 2, 3 …99, 100} is a finite set because the numbers from 1 to 100 are finite in number.
(iv) The set of positive integers greater than 100 is an infinite set because positive integers greater than 100 are infinite in number.
(v) The set of prime numbers less than 99 is a finite set because prime numbers less than 99 are finite in number.
Q.3. State whether each of the following set is finite or infinite:
(i) The set of lines which are parallel to the xaxis
(ii) The set of letters in the English alphabet
(iii) The set of numbers which are multiple of 5
(iv) The set of animals living on the earth
(v) The set of circles passing through the origin (0, 0)
Ans.
(i) The set of lines which are parallel to the xaxis is an infinite set because lines parallel to the xaxis are infinite in number.
(ii) The set of letters in the English alphabet is a finite set because it has 26 elements.
(iii) The set of numbers which are multiple of 5 is an infinite set because multiples of 5 are infinite in number.
(iv) The set of animals living on the earth is a finite set because the number of animals living on the earth is finite (although it is quite a big number).
(v) The set of circles passing through the origin (0, 0) is an infinite set because infinite number of circles can pass through the origin.
Q.4. In the following, state whether A = B or not:
(i) A = {a, b, c, d}; B = {d, c, b, a}
(ii) A = {4, 8, 12, 16}; B = {8, 4, 16, 18}
(iii) A = {2, 4, 6, 8, 10}; 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 ...}
Ans.
(i) A = {a, b, c, d}; B = {d, c, b, a}
The order in which the elements of a set are listed is not significant.
∴ A = B
(ii) A = {4, 8, 12, 16}; B = {8, 4, 16, 18}
It can be seen that 12 ∈ A but 12 ∉ B.
∴ A ≠ B
(iii) A = {2, 4, 6, 8, 10}
B = {x : x is a positive even integer and x ≤ 10}
= {2, 4, 6, 8, 10}
∴ A = B
(iv) A = {x : x is a multiple of 10}
B = {10, 15, 20, 25, 30 …}
It can be seen that 15 ∈ B but 15 ∉ A.
∴ A ≠ B
Q.5. Are the following pair of sets equal? Give reasons.
(i) A = {2, 3}; B = {x : x is solution of x^{2} +5x +6 = 0}
(ii) A = {x : x is a letter in the word FOLLOW}; B = {y : y is a letter in the word WOLF}
Ans.
(i) A = {2, 3}; B = {x : x is a solution of x^{2} + 5x + 6 = 0}
The equation x^{2} + 5x + 6 = 0 can be solved as:
x(x + 3) 2(x + 3) = 0
(x + 2)(x + 3) = 0
x = –2 or x = –3
∴ A = {2, 3}; B = {–2, –3}
∴ A ≠ B
(ii) A = {x : x is a letter in the word FOLLOW} = {F, O, L, W}
B = {y : y is a letter in the word WOLF} = {W, O, L, F}
The order in which the elements of a set are listed is not significant.
∴ A = B
Q.6. From the sets given below, 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}
Ans. 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}; A = {0, 1}
It can be seen that
8 ∈ A, 8 ∉ B, 8 ∉ D, 8 ∉ E, 8 ∉ F, 8 ∉ G, 8 ∉ H
⇒ A ≠ B, A ≠ D, A ≠ E, A ≠ F, A ≠ G, A ≠ H
Also, 2 ∈ A, 2 ∉ C
∴ A ≠ C
3 ∈ B, 3 ∉ C, 3 ∉ E, 3 ∉ F, 3 ∉ G, 3 ∉ H
∴ B ≠ C, B ≠ E, B ≠ F, B ≠ G, B ≠ H
12 ∈ C, 12 ∉ D, 12 ∉ E, 12 ∉ F, 12 ∉ G, 12 ∉ H
∴ C ≠ D, C ≠ E, C ≠ F, C ≠ G, C ≠ H
4 ∈ D, 4 ∉ E, 4 ∉ F, 4 ∉ G, 4 ∉ H
∴ D ≠ E, D ≠ F, D ≠ G, D ≠ H
Similarly, E ≠ F, E ≠ G, E ≠ H
F ≠ G, F ≠ H, G ≠ H
The order in which the elements of a set are listed is not significant.
∴ B = D and E = G
Hence, among the given sets, B = D and E = G.
Q.1. Make correct statements by filling in the symbols ⊂ or ⊄ in the blank spaces:
(i) {2, 3, 4} … {1, 2, 3, 4, 5}
(ii) {a, b, c} … {b, c, d}
(iii) {x : x is a student of Class XI of your school} … {x : x student of your school}
(iv) {x : x is a circle in the plane} … {x : x is a circle in the same plane with radius 1 unit}
(v) {x : x is a triangle in a plane} … {x : x is a rectangle in the plane}
(vi) {x : x is an equilateral triangle in a plane} … {x : x is a triangle in the same plane}
(vii) {x : x is an even natural number} … {x : x is an integer}
Ans.
(i) {2, 3, 4} ⊂ {1, 2, 3, 4, 5}
(ii) {a, b, c} ⊄ {b, c, d}
(iii) {x : x is a student of class XI of your school} ⊂ {x : x is student of your school}
(iv) {x : x is a circle in the plane} ⊄ {x : x is a circle in the same plane with radius 1 unit}
(v) {x : x is a triangle in a plane} ⊄ {x : x is a rectangle in the plane}
(vi) {x : x is an equilateral triangle in a plane} ⊂ {x : x in a triangle in the same plane}
(vii) {x : x is an even natural number} ⊂ {x : x is an integer}
Q.2. Examine whether the following statements are true or false:
(i) {a, b} ⊄ {b, c, a}
(ii) {a, e} ⊂ {x: x is a vowel in the English alphabet}
(iii) {1, 2, 3} ⊂ {1, 3, 5}
(iv) {a} ⊂ {a. b, c}
(v) {a} ∈ (a, b, c)
(vi) {x : x is an even natural number less than 6} ⊂ {x : x is a natural number which divides 36}
Ans.
(i) False. Each element of {a, b} is also an element of {b, c, a}.
(ii) True. a, e are two vowels of the English alphabet.
(iii) False. 2∈ {1, 2, 3}; however, 2∉ {1, 3, 5}
(iv) True. Each element of {a} is also an element of {a, b, c}.
(v) False. The elements of {a, b, c} are a, b, c. Therefore, {a} ⊂ {a, b, c}
(vi) True. {x : x is an even natural number less than 6} = {2, 4}
{x : x is a natural number which divides 36} = {1, 2, 3, 4, 6, 9, 12, 18, 36}
Q.3. Let A = {1, 2, {3, 4,}, 5}. Which of the following statements are incorrect and why?
(i) {3, 4} ⊂ A
(ii) {3, 4} ∈ A
(iii) {{3, 4}} ⊂ A
(iv) 1 ∈ A
(v) 1⊂ A
(vi) {1, 2, 5} ⊂ A
(vii) {1, 2, 5} ∈ A
(viii) {1, 2, 3} ⊂ A
(ix) Φ ∈ A
(x) Φ ⊂ A
(xi) {Φ} ⊂ A
Ans. A = {1, 2, {3, 4}, 5}
(i) The statement {3, 4} ⊂ A is incorrect because 3 ∈ {3, 4}; however, 3∉A.
(ii) The statement {3, 4} ∈A is correct because {3, 4} is an element of A.
(iii) The statement {{3, 4}} ⊂ A is correct because {3, 4} ∈ {{3, 4}} and {3, 4} ∈ A.
(iv) The statement 1∈A is correct because 1 is an element of A.
(v) The statement 1⊂ A is incorrect because an element of a set can never be a subset of itself.
(vi) The statement {1, 2, 5} ⊂ A is correct because each element of {1, 2, 5} is also an element of A.
(vii) The statement {1, 2, 5} ∈ A is incorrect because {1, 2, 5} is not an element of A.
(viii) The statement {1, 2, 3} ⊂ A is incorrect because 3 ∈ {1, 2, 3}; however, 3 ∉ A.
(ix) The statement Φ ∈ A is incorrect because Φ is not an element of A.
(x) The statement Φ ⊂ A is correct because Φ is a subset of every set.
(xi) The statement {Φ} ⊂ A is incorrect because Φ∈ {Φ}; however, Φ ∈ A.
Q.4. Write down all the subsets of the following sets:
(i) {a}
(ii) {a, b}
(iii) {1, 2, 3}
(iv) Φ
Ans.
(i) The subsets of {a} are Φ and {a}.
(ii) The subsets of {a, b} are Φ, {a}, {b}, and {a, b}.
(iii) The subsets of {1, 2, 3} are Φ, {1}, {2}, {3}, {1, 2}, {2, 3}, {1, 3}, and{1, 2, 3}
(iv) The only subset of Φ is Φ.
Q.5. How many elements has P(A), if A = Φ?
Ans. We know that if A is a set with m elements i.e., n(A) = m, then n[P(A)] = 2^{m}.
If A = Φ, then n(A) = 0.
∴ n[P(A)] = 2^{0} = 1
Hence, P(A) has one element.
Q.6. Write the following as intervals:
(i) {x : x ∈ R, –4 < x ≤ 6}
(ii) {x : x ∈ R, –12 < x < –10}
(iii) {x : x ∈ R, 0 ≤ x < 7}
(iv) {x : x ∈ R, 3 ≤ x ≤ 4}
Ans.
(i) {x : x ∈ R, –4 < x ≤ 6} = (–4, 6]
(ii) {x : x ∈ R, –12 < x < –10} = (–12, –10)
(iii) {x : x ∈ R, 0 ≤ x < 7} = [0, 7)
(iv) {x : x ∈ R, 3 ≤ x ≤ 4} = [3, 4]
Q.7. Write the following intervals in setbuilder form:
(i) (–3, 0)
(ii) [6, 12]
(iii) (6, 12]
(iv) [–23, 5)
Ans.
(i) (–3, 0) = {x : x ∈ R, –3 < x < 0}
(ii) [6, 12] = {x : x ∈ R, 6 ≤ x ≤ 12}
(iii) (6, 12] = {x : x ∈ R, 6 < x ≤ 12}
(iv) [–23, 5) = {x : x ∈ R, –23 ≤ x < 5}
Q.8. What universal set (s) would you propose for each of the following:
(i) The set of right triangles
(ii) The set of isosceles triangles
Ans.
(i) For the set of right triangles, the universal set can be the set of triangles or the set of polygons.
(ii) For the set of isosceles triangles, the universal set can be the set of triangles or the set of polygons or the set of twodimensional figures.
Q.9. Given the sets A = {1, 3, 5}, B = {2, 4, 6} and C = {0, 2, 4, 6, 8}, which of the following may be considered as universals set (s) for all the three sets A, B and C
(i) {0, 1, 2, 3, 4, 5, 6}
(ii) Φ
(iii) {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10}
(iv) {1, 2, 3, 4, 5, 6, 7, 8}
Ans.
(i) It can be seen that A ⊂ {0, 1, 2, 3, 4, 5, 6}
B ⊂ {0, 1, 2, 3, 4, 5, 6}
However, C ⊄ {0, 1, 2, 3, 4, 5, 6}
Therefore, the set {0, 1, 2, 3, 4, 5, 6} cannot be the universal set for the sets A, B, and C.
(ii) A ⊄ Φ, B ⊄ Φ, C ⊄ Φ
Therefore, Φ cannot be the universal set for the sets A, B, and C.
(iii) A ⊂ {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10}
B ⊂ {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10}
C ⊂ {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10}
Therefore, the set {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10} is the universal set for the sets A, B, and C.
(iv) A ⊂ {1, 2, 3, 4, 5, 6, 7, 8}
B ⊂ {1, 2, 3, 4, 5, 6, 7, 8}
However, C ⊄ {1, 2, 3, 4, 5, 6, 7, 8}
Therefore, the set {1, 2, 3, 4, 5, 6, 7, 8} cannot be the universal set for the sets A, B, and C.
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