Maharashtra Board Class 9 Math Solution Chapter 1 – Sets
Balbharati Maharashtra Board Class 9 Math Solution Chapter 1: Sets. Marathi or English Medium Students of Class 9 get here Sets full Exercise Solution.
Std |
Maharashtra Class 9 |
Subject |
Math Solution |
Chapter |
Sets |
Practice Set – 1.1
(1) Write the following sets in roaster form.
(i) Set of even natural numbers
Ans: Set, A be the set of even natural numbers, then, A = {2, 4, 6, 8, 10, ——}
(ii) Set of even prime numbers from 1 to 50
Ans: Let, B be the set even prime numbers from 1 to 50, then, B = {2}
(iii) Set of negative integers
Ans: Let, C be the set of negative integers, then C = {-1, -2, -3, -4, -5, ——}
(iv) Seven basic sounds of a Sargam (Sur).
Ans: Let, D be the set of seven basic sounds of a sargam (Sur) Then, D = {SA, RE, GA, MA, PA, DHA, NI}
(2) Write the following symbolic statements in words.
(i) 4/3 ∈ Q
Ans: 4/3 belongs to set Q (The set of rational numbers)
(ii) – 2 ∉ N
Ans: – 2 does not belong to set N (The set of natural numbers)
(iii) P = {P|P is an odd numbers}
Ans: Let P is the set of all P such that P is an odd number.
(3) Write any two sets by listing method and by rule method.
Ans: (i) A ={x|x is a real number between 0 to 5}
A = {0.2, a, √2, e, π, 3.6, 4.9, ——-}
(ii) B = {x|x is a letter of the EAT}
B = {E, A, T}
(4) Write the following sets using listing method:
(i) All months in this Indian solar year.
Ans: Let, A be the set of all months in the Indian solar year, then
A = {Vaisakha, Jyaistha, Asadha, Sravana, Bhadra, Aswina, Kartika, Agrahayana, Pausa, Magha, Phalguna, Caitra}
(ii) Letters in the word ‘COMPLEMENT’.
Ans: Let, B be the set of letters in the word ‘COMPLEMENT’,
Then, B = {C, E, L, M, N, O, P, T}
(iii) Set of all sensory organs.
Ans: Let, C be the set of all sensory organs, then,
C = {Eyes, Ears, Nose, Tongue, Skin}
(iv) Set of prime numbers from 1 to 20.
Ans: Let, D be the set of prime numbers from 1 to 20, then,
D = {2, 3, 5, 7, 11, 13, 17, 19}
(v) Names of continents of the world.
Ans: Let, E be the set of names of continents of the world, then
E = {Asia, Africa, Europe, America, South America, Australia, Antarctica}
(5) Write the following sets using rule method:
(i) A = {1, 4, 9, 16, 25, 36, 49, 81, 100}
Ans: A = {x|x = n^{2}, 1 ≤ n ≤ 10}
(ii) B = {6, 12, 18, 24, 30, 42, 48}
Ans: B = {x|x is a multiple of 6 less than 50}
(iii) C = {S, M, I, L, E}
Ans: C = {x|x is a letter of the word SMILE}
(iv) D = [Sunday, Monday, Tuesday, Wednesday, Thursday, Friday, Saturday}
Ans: D = {x|x is a day of the week}
(v) x = {a, e, t}
Ans: x = {x|x is a letter of the word tea}
Page 6:
EX – 3
If A = {1, 2, 3} and B = {1, 2, 3, 4} Then A≠B, verify it.
Ans: Clearly, 4 ∈ B but 4 ∉ A
So, A and B are not equal sets
∴ A≠B
Ex – 4
A = {x|x is prime number and 10 <x <20} and
B = {11, 13, 17, 19} there A = B verify
Ans: Given, A = {x|x is a prime number and 10 <x <20}
= {11, 13, 17, 19}
= B
∴ A = B
Practice Set – 1.2
(1) Decide 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}
Ans: Given, A = {x|3x – 1 = 2}
= {x|3x = 2+1}
= {x|x = 3/3}
= {x|x = 1}
= {1}
Also, B = {x|x is a natural number but x is neither prime nor composite}
= {1}
And C = {x|x ∈ N, x <2}
= {1}
∴ Clearly, A, B and C are equal sets.
(2) Decide whether set A and B are equal sets. Give reason for your answer.
A = even prime minster
B = {x|7x – 1 = 13}
Ans: Given, A = even prime numbers
= {2}
And B = {x|7x – 1 = 13}
∴ 7x – 1 = 13 => 7x = 13+1 => x = 14/7 = 2
∴ B = {2}
∴ Clearly, A = B
(3) Which of the following are empty sets? Why?
(i) A = {a|a is a natural number smaller than zero}
Ans: ∵ there is no natural number less than zero
∴ A = ϕ
∴ A is an empty set
(ii) B = {x|x^{2} = 0}
Ans: ∵ x^{2} = 0 => x = 0
∴ B = {0}
∴ B is not an empty set.
(iii) C = {x|5x – 2 = 0, x ∈ N}
Ans: ∵ 5x – 2 = 0
=> 5x = 2
=> x = 2/3 ∉ N
∴ C = ϕ
∴ C is an empty set.
(4) Write with reasons, which of the following sets are finite or infinite.
(i) A = {x|x <10, x is a natural number}
Ans: A = {x|x <10, x is a natural number}
= {1, 2, 3, 4, 5, 6,7, 8, 9} which is finite
∴ A is a finite set
(ii) B = {y|y <-1, y is an integer}
Ans: B = {y|y <-1, y is an integer}
= {-2, -3, -4, -5, ——-} which is infinite
∴ B is an infinite set
(iii) C = set of students of class 9 from your school
Ans: Number of without in a particular class from a particular school is finite.
∴ C is a finite set.
(iv) Set of people from your village.
Ans: Number of people from village is finite.
Set, D = Set of people from your village
∴ D is finite set
(v) Set of apartments in laboratory
Ans: Set, E = Set of apartments in laboratory
Number of apartments in laboratory is finite
∴ E is a finite set
(vi) Set of whole members.
Ans: Set, F = set of whole numbers
= {0, 1, 2, 3, ——-} which is infinite
∴ F is infinite set
(vii) Set of rational numbers.
Ans: Set, G = Set of rational numbers
= {0, 0.2, 1, 3, 2.6, ——-} which is infinite
∴ G is infinite set.
Page – 8
Ex.
If A = {1, 3, 4, 7, 8} then write all possible subsets of A
i.e, P = {1, 3}, T = {4, 7, 8}, V = {1, 4, 8}, S = {1, 4, 7, 8}
In this way many subsets can be written. Write fine more subsets of set A.
Ans: The subsets of A are:
B = {4}, C = {1, 8}, D = {1, 4, 8}, E = {1, 3, 4, 7}, F = {4, 8}
Page – 9
Ex: Some sets are given below:
A = {——, -4, -2, 0, 2, 4, 6 —–}
B = {1, 2, 3, ——-}
C = {——- -12, -6, 0, 12, 18, ——}
D = {——-, -8, -4, 0, 4, 8 ——}
I = {——-, -3, -2, -1, 0, 1, 2, 3, 4, —–}
Ans: Here, (i) A, B, C, D are subsets of I
(2) C, D are subsets of A
Decide and discuss which of the following statements are true
(i) A is a subset of set B, C and D
(ii) B is a subset of all the sets which are given above
Ans: None of the statements are true.
Page – 10:
Ex – 2
Suppose, U = {1, 3, 9, 11, 13, 18, 19}
B = {3, 9, 11, 13}
∴ B ’ = {1, 18, 19}
Find (B ’) ’ and draw the inference.
(B ’) ’ is the set of elements which are not in B^{‘} but in U, is (B^{‘})^{‘} = B?
Ans: Given, U = {1, 3, 9, 11, 13, 18, 19}
B = {3, 9, 11, 13}
B^{‘} = {1, 18, 19}
∴ (B^{‘})^{‘} = {3, 9, 11, 13} (All the elements of U which are not in B^{1})
∴ Complement of a complement is the given set itself.
Practice Set – 1.3
(1) If 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
Ans: False because b ∈ c but b ∉ B.
(ii) A ≤ D
Ans: False because b, c, d ∈ A but b, c, d ∉ D
(iii) D ≤ B
Ans: False because a ∈ D but a ∉ B
(iv) D ≤ A
Ans: True
(v) B ≤ A
Ans: False because f ∈ B but f ∉ A
(vi) C ≤ A
Ans: True
(2) Take 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 a prime number from 1 to 20}
Ans: Here, x = {8, 9, 10, 11, 12, 13, 14}
y = {2, 3, 5, 7, 11, 13, 17, 19}
(3) U = {1, 2, 3, 7, 8, 9, 10, 11, 12}
P = {1, 3, 7, 10}
Then (i) show the U, P and P^{’} by Venn diagram
(ii) Verify (p ’) ’ = P
Ans: (i)
(ii) Given, P = {1, 3, 7, 10} and U = {1, 2, 3, 7, 8, 9, 10, 11, 12}
∴ P^{’} = elements of U which are not P.
= {2, 8, 9, 11, 12}
∴ (P ’) ’ = Elements of U which are not P^{’}
= {1, 3, 7, 10}
= P
∴ (P ’) ’ = P
(4) A = {1, 3, 2, 7} Then write any three subsets of A.
Ans: The subsets of A are –
B = {1, 3}, C = {2}, D = {1, 2, 7}
(5) (i) Write the subset relation between the sets.
P is the set of all residents in Pune
M is the set of all residents in Madhya Pradesh
I is the set of all residents in Indore
B is the set of all residents in India
H is the of all residents in Maharashtra
(ii) Which set can be the universal set for above sets?
Ans: (i) Since, Pune is a city in Maharashtra which is in India.
∴ P<H<B
Since, Indore is a city in Madhya Pradesh which is in India.
∴ I<M<B
(ii) B, The set of all residents in India is the universal set for above sets.
(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
Ans: The set of natural numbers is an universal set for A, B and C.
(ii) P = set of integers which are multiples of 4
T = set of all even square numbers
Ans: The set of integers is an universal set for P and T
(7) Let all the students of a class is an universal set. Let set A be the students who secure 50% or more marks in Maths.
Then write the complement of set A.
Ans: Given, A = Set of students who secure 50% or more marks in maths
∴ A ’ = set of students who secure less than 50% marks in maths
Page – 14
Remember this:
Let, A = {1, 2, 3, 5, 7, 9, 11, 13}, B = {1, 2, 4, 6, 8, 12, 13}
Verify n (A ∪ B) = n (A) + n (B) – n(A ∩ B) for A and B.
Ans: Given, A = {1, 2, 3, 5, 7, 9,11, 13}
B = {1, 2, 4, 6, 8, 12, 13}
∴ A ∪ B = {1, 2, 4, 5, 6, 7, 8, 9, 11, 12, 13}
A ∩ B = {1, 2, 13}
∴ n(A) = 8, n(B) = 7, n(A ∪ B) = 12, n(A ∩ B) = 3
∴ n(A) + n(B) – n(A ∩ B) = 8+7-3 = 12 = n(A ∪ B)
∴ n(A ∪ B) = n(A) + n(B) – n(A ∩ B)
Practice Set – 1.4
(i) If n(A) = 15, n(A ∪ B) = 29, n(A ∩ B) = 7 then n(B) = ?
Ans: We know, n(A∪ B) = n(A) + n(B) – n(A ∩ B)
=> n(B) = n(A ∪ B) + n(A ∩ B) – n(A) ——- (i)
Given, n(A) = 15. n(A ∪ B) = 29, n(A ∩ B) = 7
Putting these values in (i) we get, n(B) = 29+7-15 = 21
(2) In 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.
Ans: Let, A be the set of students who drink tea and B be the set of students who drink coffee.
Here, n(∪) = 125, n(A) = 80, n(B) = 60, n(A ∩ B) = 20
Now, (A ∪ B)’ is the set of students who do not drink tea or coffee.
∴ We know, n(A ∪ B) = n(A) + n(B) – n(A ∩ B)
= 80 + 60 – 20
= 140 – 20 = 120
∴ Number of students who do not drink tea or coffee
= n (A ∪ B)’ = n(∪) – n(A ∪ B)
= 125 – 120
= 5
(3) In a competitive exam 50 students passed in English, 60 students passed in Mathematics, 40 students passed in both the subjects, none of them failed in both the subject. Find the number of students who passed at least in one of the subjects?
Ans: Let, A be the set of students who passed in English
B be the set of students who passed in Mathematics
Here, n(A) = 50, n(B) = 60, n(A ∩ B) = 40
∵ Number of students who passed at least one of the
Subjects = n(A ∪ B) = n(A) + n(B) – n(A ∩ B)
= 50 + 60 – 40
= 70
(4) A survey was conducted to know the hobby of 220 students of class (IX). Out of which 130 students informed about their hobby as rock climbing and 180 students informed about their hobby 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?
Ans: Let, A be the set of students with hobby as rock climbing
B be the set of students with hobby as sky watching
Here, n(∪) = 220, n(A) = 130, n(B) = 180, n(A ∩ B) = 110
Number of students who follow at least one of the hobbies = n(A ∪ B) = n(A) + n(B) – n(A ∩ B)
= 130 + 180 – 110
= 200
∴ Number of students who does not follow any of the two hobbies = n(A ∪ B)^{’} = n(u) – n(A ∩ B)
= 220 – 200
= 20
Again, Number of students who follow the hobby of rock climbing only = n(A) – n(A ∩ B)
= 130 – 110 = 20
Number of students who follow the hobby of sky watching only = n(B) – n(A ∩ B) = 180 – 110 = 70
(5) Observe the given Venn diagram and write the following sets.
(1) A
(ii) B
(iii) A ∪ B
(iv) U
(v) A^{’}
(vi) B^{’}
(vii) (A ∪ B)^{’}
Ans: (i) A = {x, y, z, m, n}
(ii) B = {P, q, r, m, n}
(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}
Problem set 1
Choose the correct alternative answer for each of the following questions:
1> M = {1,3,5}, N = {2,4,6} then M ∩ N = ?
(A) {1,2,3,4,5,6} (B) {1,3,5} (c) Φ (D) {2,4,6}
A: (c)
2> P = { x | x is an add natural number 1< x ≤ 5 }
How to write this set in roster form ?
(a) {1,3,5} , (B) {1,2,3,4,5} (c) {1,3} (d) {3,5)
A: (d) {3,5}
(III) P = {1,2,……….,10} What type of set P is ?
(a) Null set (B) Infinite Wet (c) Finite wet (D) None of these
A: Finite Wet
(Iv) M ∪ N = { 1,2,3,4,5,6} and M = { 1,2,4} then which the following represent set N ?
(a) {1,2,3} (b) {3,4,5,6} (c) {2,5,6} (d) {4,5,6}
A: (B) {3,4,5,6}
(v) If P ≤ M, then Which of the following wet represent p ∩ (p ∪ M) ?
(A) P , (B) M , (C) P u M , (d) P n M
A: (a) P
(vi) Which of the Following wets are empty wets ?
(A) set of intersecting Paints of Parallel Lines
(b) set of even prime numbers
(C) Month of an English calendar having less than 30.
(d) P = { x | x + 1 , -1 < x < 1 }
A: (A) Set of interacting paints of parallel lines
(2) Find the Correct option for the given question.
(1) Which of the following collections is a wet ?
(A) colours of the rainbow
(B) tall tress in the School campus
(c) Rich people in the village
(d) Easy examples in the books.
A: Colours of the rainbow
(II) Which of the following wet represent N n W ?
(A) {1,2,3,……..}
(B) {0,1,2,3,……}
(c) {0}
(d) {}
A: {1,2,3……}
(III) P = { X| x is a letter of the word ‘indian’ } then which one of the following is wet P in listing form ?
(a) {I,n,d} (b) {I,n,d,a} (c) {I,n,d,i,a} (d) {n,d,a}
A: (B) {I,n,d,a}
(iv) If T = { 1,2,3,4,5,} and M = {3,4,7,8} then T ∪ M = ?
(A) {1,2,3,4,5,7} , (B) { 1,2,3,7,8}
(c) {1,2,3,4,5,7,8}, (D) {3,4}
A: (c) 1,2,3,4,5,7,8}
(3) Out of 100 persons in a group, 72 persons speak English, 43 persons speak French. each one out of 100 perons speak at least one language. Then how many speak only English? How many speak only French? How ,any of them speak English and French both ?
Let,
A be Set of persons who speak english
B be Set of persons who speak French.
Here,
n(A) = 72, n(B)= 43
∵ Each one out of 100 Persons speak at least one language
therefore, n ( A ∪ B) = 100
Now,
Number of persons who speak only English = n (A) – n (A ∩ B)
Number of persons who speak only French = n (b) – n (A ∩ B)
Number of persons who speak both English and French
= n (A ∩ B)
= n (a) + n (B) – n (A ∪ B)
= 72 + 43 – 100
= 15
Number of students who speak only English
= n (A) – n (A ∩ B)
= 72 – 15
= 57
Number of Students who speak only French
= n (b) – n (A ∩ B)
= 43-15
= 28
(4) 70 tress were planted by parth and 90 trss were planted by pradmya on the occasion of tree plantation week. out of these, 25 tress planted by both of them together. How many tress planted by parth and pradmya.
A: Let, A be set of trees planted by parth.
B be set of Trees planted by pradmya
Here, n (A) = 70, n (B) = 90, n (A n B) = 25
therefore, Number of trees planted by parth or pradnya
= n (A u B)
= n (A) + n (B) – n (A n B)
= 70 + 90 – 25
= 135
(5) If n (A) = 20, n (B) and n (A ∪ B) = 36 then n ( A ∩ B) = ?
A: n (A ∪ B) = n (A) + n (B) – n (A ∩ B)
n (A ∩ B) = n (A) + n (B) – n (A ∪ B)
= 20 + 28 – 36
= 12
6> In a Class, 8 Students out of 28 home only dog as their put animal at home, 6 Students home only cat as their put animal. 10 students home dog and cat as their put animal at home ?
given:
Number of students having dog only = 8
Number of students having Cat only = 6
Number of students having both cat and dog = 10
Therefore, Number of Students having atleast one of the pet animals = 8 + 6 + 10
= 24
Now, Number of students who do not have a dog or cat as their pet animals
= total numbers of students – Number of students having at least one of them = 28 – 24 = 4
(iii) X = { x | x is a prime between 80 and 100}
y = { Y | y is an odd number between 90 and 100}
Ans.
X = { X | x is a Prime between 80 and 100}
= { 83, 89, 97}
y = { Y | y is an odd number between 90 and 100
= { 91, 93,95, 97, 99}
(8) Write the subset relations between the following sets
X = Set of all quadrilaterals
Y = Set of all rhombus
S = Set of all Square
T = Set of all Parallelograms
V = Set of all rectangles
Ans: since, all Square are rectangles, all rectangles are parallelograms and all parallelograms are Quadrilateral
Therefore, S ≤ V ≤ T ≤ X
And since, all squares are rhombus, all rhombuses are parallelograms and all parallelograms are Quadrilaterals.
Therefore, S ≤ V ≤ T ≤ X
Observe the vann diagram and write the given sets U, A,B , A U B, and, A n B
Ans: U = { 1,2,3,4,5,6,7,8,9,10,11,13}
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}
11> If n (A) = 7, n (B) = 13, n (A ∩ B) = 4 then n (A ∪ B) = ?
We know,
n (A ∪ B) = n (A) + n (B) – n (A ∩ B)
= 7 + 13 – 4
= 16
Activity 1: fill in the blanks with elements of the set.
A = { 1, 11, 13} , B = {8,5,10,11,15}
A= { } , B= {}
A ∩ B = {},
A’ ∩ B’ {}
A ∪ B = {}
(A ∪ B)’ = {}
(A ∪ B)’ = {}
Ans: given,
U = { 1,3,5,8,9,10,11,12,13,15}
A = {1,11,13}
B = {8,5,10,11,15}
Therefore, A’ = {3,5,8,9,10,12,15}
B’ = {1,3,9,12,13}
A ∩ B = {11}
A’ ∩ B’ = {3,9,12} …..(1)
A ∪ B = {1,5,8,10,11,13,15}
A’ ∪ B’ = { 1, 3,5,8,9,10,12,13,15}…..(2)
(A ∩ B)’ = {1,3,5,8,9,10,12,13,15} …..(3)
(A ∪ B)’ = {3,9,12} ….(4)
From (2) and (3), (A ∩ B)’ = A’ ∪ B’
From (4) and (1), ( A ∪ B)’ = A’ ∩ B’