MAK LABS — Std 9 Maths Part 1 Chapter 1 Sets — Practice Sets 1.1 to 1.4 and Problem Set 1 Solutions

1Write the following sets in roster form.

(i) Set of even natural numbers
A = {2, 4, 6, 8, 10, …}

(ii) Set of even prime numbers from 1 to 50
B = {2}
2 is the only even prime number, so this set has just one element (it is a singleton set).

(iii) Set of negative integers
C = {…, −4, −3, −2, −1}

(iv) Seven basic sounds of a sargam (sur)
D = {sa, re, ga, ma, pa, dha, ni}

2Write the following symbolic statements in words.

(i) ⁴⁄₃ ∈ Q
4/3 is an element of the set of rational numbers, that is, 4/3 is a rational number.

(ii) −2 ∉ N
−2 is not an element of the set of natural numbers, that is, −2 is not a natural number.

(iii) P = {p | p is an odd number}
P is the set of all p such that p is an odd number.

3Write any two sets by listing method and by rule method.

Sample answers (any correct sets are acceptable):

Set 1
Listing method: A = {2, 3, 5, 7}
Rule method: A = {x | x is a prime number less than 10}

Set 2
Listing method: B = {a, e, i, o, u}
Rule method: B = {y | y is a vowel of the English alphabet}

4Write the following sets using listing method.

(i) All months in the Indian solar year
A = {Chaitra, Vaishakh, Jyeshtha, Ashadh, Shravan, Bhadrapad, Ashwin, Kartik, Margashirsha, Paush, Magh, Phalgun}

(ii) Letters in the word ‘COMPLEMENT’
B = {C, O, M, P, L, E, N, T}
M and E appear twice in the word, but each is written only once in the set.

(iii) Set of human sensory organs
C = {eyes, ears, nose, tongue, skin}

(iv) Set of prime numbers from 1 to 20
D = {2, 3, 5, 7, 11, 13, 17, 19}

(v) Names of continents of the world
E = {Asia, Africa, North America, South America, Antarctica, Europe, Australia}

5Write the following sets using rule method.

(i) A = {1, 4, 9, 16, 25, 36, 49, 64, 81, 100}
A = {x | x = n², n ∈ N, n ≤ 10}
(the set of squares of the first ten natural numbers)

(ii) B = {6, 12, 18, 24, 30, 36, 42, 48}
B = {x | x = 6n, n ∈ N, n ≤ 8}
or B = {x | x is a multiple of 6 and x < 50}

(iii) C = {S, M, I, L, E}
C = {x | x is a letter of the word ‘SMILE’}

(iv) D = {Sunday, Monday, Tuesday, Wednesday, Thursday, Friday, Saturday}
D = {y | y is a day of the week}

(v) X = {a, e, t}
X = {x | x is a letter of the word ‘eat’}
(the words ‘tea’ or ‘ate’ give the same set)

1Decide which of the following are equal sets and which are not? Justify your answer.
A = {x | 3x − 1 = 2}, B = {x | x is a natural number but x is neither prime nor composite}, C = {x | x ∈ N, x < 2}

Set A: 3x − 1 = 2 ⇒ 3x = 3 ⇒ x = 1. ∴ A = {1}

Set B: 1 is the only natural number that is neither prime nor composite. ∴ B = {1}

Set C: The only natural number less than 2 is 1. ∴ C = {1}

All three sets have exactly the same element, 1. A = B = C — all three are equal sets.

2Decide whether set A and B are equal sets. Give reason for your answer.
A = Even prime numbers, B = {x | 7x − 1 = 13}

Set A: 2 is the only even prime number. ∴ A = {2}

Set B: 7x − 1 = 13 ⇒ 7x = 14 ⇒ x = 2. ∴ B = {2}

Every element of A is in B and every element of B is in A. A = B

3Which of the following are empty sets? Why?

(i) A = {a | a is a natural number smaller than zero}
Natural numbers begin at 1, so no natural number is smaller than zero. A = { }. Empty set

(ii) B = {x | x² = 0}
x² = 0 gives x = 0, so B = {0}. B has one element. Not empty (it is a singleton set)

(iii) C = {x | 5x − 2 = 0, x ∈ N}
5x − 2 = 0 gives x = 2/5, which is not a natural number. C = { }. Empty set

4Write with reasons, which of the following sets are finite or infinite.

(i) A = {x | x < 10, x is a natural number}
A = {1, 2, 3, 4, 5, 6, 7, 8, 9}. It has 9 elements, which can be counted. Finite

(ii) B = {y | y < −1, y is an integer}
B = {−2, −3, −4, …}. The list never ends. Infinite

(iii) C = Set of students of class 9 from your school
The number of students is limited and can be counted. Finite

(iv) Set of people from your village
However large, the population of a village can be counted. Finite

(v) Set of apparatus in laboratory
A laboratory has a limited, countable number of apparatus. Finite

(vi) Set of whole numbers
W = {0, 1, 2, 3, …}. There is no largest whole number. Infinite

(vii) Set of rational numbers
Between any two rational numbers there are more rational numbers, and they have no end. Infinite

1If A = {a, b, c, d, e}, B = {c, d, e, f}, C = {b, d}, D = {a, e}, then which of the following statements are true and which are false?

(i) C ⊆ B — b ∈ C but b ∉ B. False

(ii) A ⊆ D — b, c, d ∈ A but are not in D. False

(iii) D ⊆ B — a ∈ D but a ∉ B. False

(iv) D ⊆ A — both a and e are elements of A. True

(v) B ⊆ A — f ∈ B but f ∉ A. False

(vi) C ⊆ A — both b and d are elements of A. True

2Take the set of natural numbers from 1 to 20 as universal set and show set X and Y using Venn diagram.
(i) X = {x | x ∈ N, and 7 < x < 15} (ii) Y = {y | y ∈ N, y is prime number from 1 to 20}

U = {1, 2, 3, …, 20}

(i) X = {8, 9, 10, 11, 12, 13, 14}

X inside U; the other 13 numbers of U lie outside the circle.

(ii) Y = {2, 3, 5, 7, 11, 13, 17, 19}

Y (prime numbers) inside U; 1 and the composite numbers lie outside.

3U = {1, 2, 3, 7, 8, 9, 10, 11, 12}, P = {1, 3, 7, 10}, then (i) show the sets U, P and P′ by Venn diagram. (ii) Verify (P′)′ = P

P′ = elements of U that are not in P = {2, 8, 9, 11, 12}

(i)

The shaded region outside circle P (but inside U) is P′.

(ii) (P′)′ = elements of U that are not in P′
P′ = {2, 8, 9, 11, 12}, so (P′)′ = {1, 3, 7, 10}

But P = {1, 3, 7, 10}. (P′)′ = P — verified.

4A = {1, 3, 2, 7}, then write any three subsets of A.

Every set whose elements are all taken from A is a subset of A. Any three, for example:

P = {1, 3}, Q = {2, 7}, R = {1, 2, 3}

{ } (the empty set) and A itself are also subsets of A. In all, A has 2⁴ = 16 subsets.

5(i) Write the subset relation between the sets. P: residents in Pune, M: residents in Madhya Pradesh, I: residents in Indore, B: residents in India, H: residents in Maharashtra. (ii) Which set can be the universal set for above sets?

(i) Pune is in Maharashtra and Indore is in Madhya Pradesh, and both states are in India.

P ⊆ H ⊆ B (Pune → Maharashtra → India)

I ⊆ M ⊆ B (Indore → Madhya Pradesh → India)

(ii) Every other set is a subset of B. B, the set of all residents in India, can be the universal set.

6*Which set of numbers could be the universal set for the sets given below?

(i) A = set of multiples of 5, B = set of multiples of 7, C = set of multiples of 12
All multiples of 5, 7 and 12 are natural numbers. N, the set of natural numbers
(If negative multiples are also taken, the set of integers I is the universal set.)

(ii) P = set of integers which are multiples of 4, T = set of all even square numbers
P contains negative numbers such as −4, −8, so N cannot hold it. Both sets are subsets of the integers. I, the set of integers

Every even square is of the form (2k)² = 4k², which is a multiple of 4. So in fact T ⊆ P.

7Let all the students of a class be the universal set. Let set A be the students who secure 50% or more marks in Maths. Then write the complement of set A.

A′ contains the students of the class who are not in A.

A′ = set of students of the class who secure less than 50% marks in Maths

1If n(A) = 15, n(A ∪ B) = 29, n(A ∩ B) = 7, then n(B) = ?

n(A ∪ B) = n(A) + n(B) − n(A ∩ B)

29 = 15 + n(B) − 7

29 = 8 + n(B) ⇒ n(B) = 29 − 8 = 21

n(B) = 21

2In a hostel there are 125 students, out of which 80 drink tea, 60 drink coffee and 20 drink tea and coffee both. Find the number of students who do not drink tea or coffee.

Let T = students who drink tea, C = students who drink coffee.
n(T) = 80, n(C) = 60, n(T ∩ C) = 20

n(T ∪ C) = n(T) + n(C) − n(T ∩ C) = 80 + 60 − 20 = 120

Students who drink neither = 125 − 120 = 5

5 students do not drink tea or coffee.

3In a competitive exam 50 students passed in English. 60 students passed in Mathematics. 40 students passed in both the subjects. None of them fail in both the subjects. Find the number of students who passed at least in one of the subjects.

Let E = students who passed in English, M = students who passed in Mathematics.
n(E) = 50, n(M) = 60, n(E ∩ M) = 40

Passed in at least one subject = n(E ∪ M)

n(E ∪ M) = n(E) + n(M) − n(E ∩ M) = 50 + 60 − 40 = 70

70 students passed in at least one of the subjects.

4*A survey was conducted to know the hobby of 220 students of class IX. 130 students informed about their hobby as rock climbing and 180 students as sky watching. There are 110 students who follow both the hobbies. Then how many students do not have any of the two hobbies? How many of them follow the hobby of rock climbing only? How many students follow the hobby of sky watching only?

Let R = rock climbing, S = sky watching.
n(R) = 130, n(S) = 180, n(R ∩ S) = 110, total = 220

n(R ∪ S) = 130 + 180 − 110 = 200

Neither hobby = 220 − 200 = 20

Rock climbing only = n(R) − n(R ∩ S) = 130 − 110 = 20

Sky watching only = n(S) − n(R ∩ S) = 180 − 110 = 70

Numbers of students in each region: 20 + 110 + 70 + 20 = 220.

No hobby: 20 Rock climbing only: 20 Sky watching only: 70

5Observe the given Venn diagram and write the following sets. (i) A (ii) B (iii) A ∪ B (iv) U (v) A′ (vi) B′ (vii) (A ∪ B)′

From the textbook diagram: x, y, z are only in A; m, n are in both A and B; p, q, r are only in B; s, t are outside both circles.

(i) A = {x, y, z, m, n}
(ii) B = {m, n, p, q, r}
(iii) A ∪ B = {x, y, z, m, n, p, q, r}
(iv) U = {x, y, z, m, n, p, q, r, s, t}
(v) A′ = {p, q, r, s, t}
(vi) B′ = {x, y, z, s, t}
(vii) (A ∪ B)′ = {s, t}

1Choose the correct alternative answer for each of the following questions.

(i) If M = {1, 3, 5}, N = {2, 4, 6}, then M ∩ N = ? — (C) φ
M and N have no common element.
(ii) P = {x | x is an odd natural number, 1 < x ≤ 5}. Roster form? — (D) {3, 5}
x must be greater than 1, so 1 is not included; 3 and 5 are odd and at most 5.
(iii) P = {1, 2, …, 10}. What type of set is P? — (C) Finite set
P has exactly 10 elements.
(iv) M ∪ N = {1, 2, 3, 4, 5, 6} and M = {1, 2, 4}. Which set can be N? — (B) {3, 4, 5, 6}
3, 5 and 6 are not in M, so N must contain all three. Only option (B) does.
(v) If P ⊆ M, which set represents P ∩ (P ∪ M)? — (A) P
P ⊆ M gives P ∪ M = M, and then P ∩ M = P.
(vi) Which of the following sets are empty sets? — (A) Set of intersecting points of parallel lines
Parallel lines never meet. (B) = {2}, (C) = {February}, (D) = {0} are all non-empty.

2Find the correct option for the given question.

(i) Which of the following collections is a set? — (A) Colours of the rainbow
Its members are clearly defined. ‘Tall’, ‘rich’ and ‘easy’ are relative words.
(ii) Which of the following set represents N ∩ W? — (A) {1, 2, 3, …}
N ⊆ W, so N ∩ W = N.
(iii) P = {x | x is a letter of the word ‘indian’}. Listing form? — (B) {i, n, d, a}
Repeated letters i and n are written only once.
(iv) If T = {1, 2, 3, 4, 5} and M = {3, 4, 7, 8}, then T ∪ M = ? — (C) {1, 2, 3, 4, 5, 7, 8}

3Out of 100 persons in a group, 72 persons speak English and 43 persons speak French. Each one out of 100 persons speaks at least one language. Then how many speak only English? How many speak only French? How many of them speak English and French both?

Let E = persons who speak English, F = persons who speak French.
n(E) = 72, n(F) = 43, n(E ∪ F) = 100 (everyone speaks at least one language)

n(E ∩ F) = n(E) + n(F) − n(E ∪ F) = 72 + 43 − 100 = 15

Only English = 72 − 15 = 57

Only French = 43 − 15 = 28

57 + 15 + 28 = 100 persons.

Only English: 57 Only French: 28 Both: 15

470 trees were planted by Parth and 90 trees were planted by Pradnya on the occasion of Tree Plantation Week. Out of these, 25 trees were planted by both of them together. How many trees were planted by Parth or Pradnya?

Let A = trees planted by Parth, B = trees planted by Pradnya.
n(A) = 70, n(B) = 90, n(A ∩ B) = 25

n(A ∪ B) = n(A) + n(B) − n(A ∩ B) = 70 + 90 − 25 = 135

135 trees were planted by Parth or Pradnya.

5If n(A) = 20, n(B) = 28 and n(A ∪ B) = 36, then n(A ∩ B) = ?

n(A ∩ B) = n(A) + n(B) − n(A ∪ B) = 20 + 28 − 36 = 12

n(A ∩ B) = 12

6In a class, 8 students out of 28 have only dog as their pet animal at home, 6 students have only cat as their pet animal. 10 students have dog and cat both, then how many students do not have a dog or cat as their pet animal at home?

Students having a dog or a cat or both = 8 + 10 + 6 = 24

Students having neither = 28 − 24 = 4

D = students with a dog, C = students with a cat; 4 students lie outside both circles.

4 students do not have a dog or a cat.

7Represent the union of two sets by Venn diagram for each of the following.

(i) A = {3, 4, 5, 7}, B = {1, 4, 8}
A ∪ B = {1, 3, 4, 5, 7, 8} (common element: 4)

The whole shaded region is A ∪ B.

(ii) P = {a, b, c, e, f}, Q = {l, m, n, e, b}
P ∪ Q = {a, b, c, e, f, l, m, n} (common elements: b, e)

The whole shaded region is P ∪ Q.

(iii) X = {x | x is a prime number between 80 and 100}, Y = {y | y is an odd number between 90 and 100}
X = {83, 89, 97}, Y = {91, 93, 95, 97, 99}
X ∪ Y = {83, 89, 91, 93, 95, 97, 99} (common element: 97)

The whole shaded region is X ∪ Y.

8Write the subset relations between the following sets. X = set of all quadrilaterals, Y = set of all rhombuses, S = set of all squares, T = set of all parallelograms, V = set of all rectangles.

Every square is both a rhombus and a rectangle; every rhombus and every rectangle is a parallelogram; every parallelogram is a quadrilateral.

S ⊆ Y ⊆ T ⊆ X (square → rhombus → parallelogram → quadrilateral)

S ⊆ V ⊆ T ⊆ X (square → rectangle → parallelogram → quadrilateral)

9If M is any set, then write M ∪ φ and M ∩ φ.

The empty set adds no element to M and has no element in common with M.

M ∪ φ = M M ∩ φ = φ

10*Observe the Venn diagram and write the given sets U, A, B, A ∪ B and A ∩ B.

From the textbook diagram: 2, 3, 7 are only in A; 1, 5 are in both; 8, 9, 10 are only in B; 4, 11, 13 are outside both circles.

U = {1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 13}
A = {1, 2, 3, 5, 7}
B = {1, 5, 8, 9, 10}
A ∪ B = {1, 2, 3, 5, 7, 8, 9, 10}
A ∩ B = {1, 5}

11If n(A) = 7, n(B) = 13, n(A ∩ B) = 4, then n(A ∪ B) = ?

n(A ∪ B) = n(A) + n(B) − n(A ∩ B) = 7 + 13 − 4 = 16

n(A ∪ B) = 16

1. Sets

Practice Set 1.1

Practice Set 1.2

Practice Set 1.3

Practice Set 1.4

Problem Set 1