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 Written as per the revised syllabus prescribed by the Maharashtra State Board
of Secondary and Higher Secondary
Education, Pune.
STD. IX
Algebra
Fifth Edition: March 2016
Salient Features
• Written as per the new textbook.
• Exhaustive coverage of entire syllabus. • Topic-wise distribution of all textual questions and practice problems at
the beginning of every chapter.
• Covers answers to all textual exercises and problem set.
• Includes additional problems for practice.
• Multiple choice questions for effective preparation.
• Comprehensive solution to Question Bank.
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P.O. No. 15194
10040_10510_JUP
PREFACE
Algebra is the branch of mathematics which deals with the study of rules of operations and relations, and the
concepts arising from them. It has wide applications in different fields of science and technology. It deals
with concepts like linear equations, quadratic equations etc. Its application in statistics deals with measures
of central tendency, representation of statistical data etc.
The study of Algebra requires a deep and intrinsic understanding of concepts, terms and formulae. Hence, to
ease this task, we present “Std. IX: Algebra”, a complete and thorough guide, extensively drafted to boost
the students confidence. The question answer format of this book helps the student to understand and grasp
each and every concept thoroughly. The book is based on the new text book and covers the entire syllabus.
At the beginning of every chapter, topic-wise distribution of all textual questions and practice problems has
been provided for simpler understanding of different types of questions. It contains answers to textual
exercises, problems sets and Question bank. It also includes additional questions and multiple choice
questions for practice. Graphs are drawn with proper scale. Another feature of the book is its layout which is
attractive and inspires the student to read.
Lastly, I would like to thank all those who have helped me in preparing this book. There is always room for
improvement and hence I welcome all suggestions and regret any errors that may have occurred in the
making of this book.
A book affects eternity; one can never tell where its influence stops.
Best of luck to all the aspirants!
Yours faithfully,
Publisher
No.
Topic Name
Page No.
1
Sets
1
2
Real Numbers
19
3
Algebraic Expressions
62
4
Linear Equations in Two Variables
97
5
Graphs
133
6
Ratio and Proportion
194
7
Statistics
224
8
Question Bank
259
01 Sets Type of Problems
Definition of Sets
Chapter 01: Sets
Exercise
1.1
Q.1
Problem set-1
Q.1
1.1
Method of Writing Sets
Types of Sets
Practice Problems
(Based on Exercise 1.1)
1.2
Q.1, 2, 3, 4
Practice Problems
(Based on Exercise 1.2)
Practice Problems
(Based on Exercise 1.3)
Q.1, 2, 4, 5
Q.1, 3, 4
1.4
Q.1, 2, 3, 4, 5
Practice Problems
(Based on Exercise 1.4)
Practice Problems
(Based on Exercise 1.5)
Problem set-1
1.5
Practice Problems
(Based on Exercise 1.5)
Problem set-1
1.3
Practice Problems
Draw a Venn Diagram
Q.4, 5, 9
Q.11, 12, 22
1.5
Word Problems on Sets
Q.1, 2
Problem set-1
Problem set-1
Number of elements in a the Set
Q.1, 2
Q.2, 3, 10
1.3
Operations on Sets and their Properties
Q.2, 3, 4
Problem set-1
Problem set-1
Subset and Universal Set
Q. Nos.
(Based on Exercise 1.3)
Practice Problems
Q.1, 2, 3, 4
Q.6, 7, 8, 13, 14, 21, 23
Q.1, 2, 5
Q.2
Q.15, 18
Q. 3, 4
Q.3, 4
Q.16, 17, 19
Q.3
Q.2
Q.1, 5
(Based on Exercise 1.5)
Problem set-1
Q.20
1
Std. IX : Algebra Introduction
Consider the following examples:
i.
Collection of books in a library.
ii.
Collection of cloths in a shop.
Objects in each of these examples can be seen clearly.
Such collections are well defined collections.
Consider the following examples:
i.
Brilliant students in a class.
ii.
Happy people in the city.
The term “brilliant” and “happy” are relative terms.
A person may be brilliant or happy according to one
person but he may not be so according to the other
person.
It is important to determine whether a given
collection is well defined or not. Well defined
collections or groups are termed as “Sets”. George
Cantor, (1845-1918) a German Mathematician is a
creator of “Set theory” which has become a
fundamental theory in Mathematics.
1.1
Definition of Sets
Set: A well defined collection of objects is called a
“set”.
Example:
i.
Collection of odd natural numbers.
ii.
Collection of whole numbers.
Each object in the set is called as an “element” or a
“member” of the set.
Example:
i.
For a set containing odd natural numbers,
elements are 1, 3, 5, 7, …
ii.
For a set of whole numbers, elements are
0, 1, 2, 3, …
Collection of elements which are not well defined,
do not form a set. Such sets usually contain relative
terms like easy, good, favourite, etc.
Example:
The collection of good books in a library.
Here, ‘good’ is a relative term whose meaning will
vary from person to person.
Important Points to Remember:
1.
Sets are denoted by capital alphabets. e.g. A,
B, C, X, Y, Z, etc.
But the elements of a set are denoted by small
alphabets e.g. a, b, p, q, r, etc.
If ‘r’ is an element of set P, then it is written
as r  P and is read as:
i.
‘r’ belongs to set P or
ii.
‘r’ is a member of set P or
iii. ‘r’ is an element of set P.
2.
2
3.
4.
1.2
Symbol ‘’ stands for ‘belongs to’, ‘is a
member of’ or ‘is an element of’.
If ‘r’ is not an element of set P, then it is
written as r  P and it is read as:
i.
‘r’ does not belong to set P or
ii.
‘r’ is not a member of set P or
iii. ‘r’ is not an element of set P .
The symbol  stands for ‘does not belong to’
or ‘not a member of’ or ‘not an element of’.
The set of Natural numbers, Whole numbers,
Integers, Rational numbers, Real numbers are
denoted by N, W, I, Q, R respectively.
Methods of Writing Sets
There are two methods of writing a set:
a.
Listing method or Roster form
b.
Rule method or Set builder form
a.
Listing method or Roster form
In this method:
i.
Elements of the set are enclosed within
curly brackets.
ii.
Each element is written only once.
iii. Elements are separated by commas.
iv. The order of writing the elements in a
set is not important.
Example:
A = {a, b, c, d, e} or A = {b, d, a, c, e} are
same or equal sets that represent first five
letters of the English alphabet.
Few examples of writing a set by listing
method are:
i.
L is a set of letters of the word “fatal”.

L = {f, a, t, l}
ii.
M is a set of integers less than 5.

M = {… , 3, 2, 1, 0, 1, 2, 3, 4}
iii. O is a set of even natural numbers from
1 to 100.

O = {2, 4, 6, 8, … , 100}
b.
Rule method or Set builder form
In this method, elements of the set are
described by specifying the property or rule
that uniquely determines the elements of a set.
Example:
i.
Y = {x|x is a vowel in the English
alphabet}
In the above notation, curly brackets
denotes ‘set of’, vertical line (|) denotes
‘such that’.
Set Y is read as:
“Y is a set of all ‘x’ such that ‘x’ is a
vowel in the English alphabet”.
Chapter 01: Sets
ii.
B = {x|x W, x < 10}
Set B is read as:
“B is a set of all ‘x’ such that ‘x’ is a
whole number less than 10”.
Solution:
i.
F = {x|x = 5n, n  N, n  4}
ii.
G = {x|x = n2, n  N, 3  n < 10}
iii.
H = {x|x = 5n, n  N, n  4}
iv.
X = {x| square of x is 64}
or X = {x|x is a square root of 64}
Which of the following collections are sets?
i.
The collection of prime numbers.
ii.
The collection of easy sub topics in
this chapter.
iii. The collection of good teachers in
your school.
iv. The collection of girls in your class.
v.
The collection of odd natural numbers.
Solution:
i.
It is a set.
ii.
Meaning of ‘easy sub topics’ may vary from
person to person, as it is a relative term.
Therefore, it is not a set.
iii. Choice of good teachers varies from student to
student as ‘good’ is a relative term. Therefore,
it is not a set.
iv. It is a set.
v.
It is a set.
v.
Y = {x|x =
2.
Examples:
A
Note:
Instead of ‘|’ sometimes two vertical dots ‘:’ are also
used.
Exercise 1.1
1.
Write the following sets in the roster form:
i.
A = {x|x is a month of the Gregarian
year not having 30 days}
ii.
B = {y|y is a colour in the rainbow}
iii. C = {x|x is an integer and 4 < x < 4}
iv. D = {x|x  I, 3 < x  3}
v.
E = {x|x = (n  1)3, n < 3, n  W}
Solution:
i.
A = {January, February, March, May, July,
August, October, December}
ii.
B = {violet, indigo, blue, green, yellow, orange,
red}
iii. C = {3, 2, 1, 0, 1, 2, 3}
iv. D = {2, 1, 0, 1, 2, 3}
v.
Putting n = 0, 1, 2, we have, E = {1, 0, 1}
3.
Write the following sets in the set builder
form:
i.
F = {5, 10, 15, 20}
ii.
G = {9, 16, 25, 36, … , 81}
iii. H = {5, 52, 53, 54}
iv. X = {8, 8}
v.
1 
 1 1 1
Y = 1, , , ,

 8 27 64 125 
1
n3
, n  N, n  5}
4.
Write the set of first five positive integers
whose square is odd.
Solution:
P = {1, 3, 5, 7, 9}
1.3
Venn Diagrams
L. Euler, a great Mathematician, introduced the idea
of diagrammatic representation of sets. Later, British
logician, John-Venn (1834-1923) used and
developed the idea of the above concept to study
sets. Such representations are called Venn Diagrams.
A set is represented by a ‘closed’ figure in a Venn
Diagram, where the elements of the set are
represented by points in the closed figure. Some of
the closed figures used to represent Venn Diagrams
are: rectangle, circle, triangle, etc.
.a
.e .i
.o .u
C
.
0
B
.
2
.
1
.a
.
.
.b .c
2
4
.d
.
8
.
.e
D
6
1.4
Types of Sets
i.
Singleton set: A set containing exactly one
element is called as a singleton set.
Example:
a.
A = {5}
b.
B = {x|x + 3 = 0}
Set B having only one element i.e., 3
ii.
Empty set: A set which does not contain any
element is called as an empty or a null set. It is
represented as {} or  (phi).
3
Std. IX : Algebra iii.

iv.
Example:
a.
A = {a|a is a natural number, 5 < a < 6}

A = { } or A = 
iii.
p3 = 8

p3 = (2)3
b.


p = 2

C = {2}

It is a singleton set.
iv.
(q  4)2 = 0

q4=0

q=4

D = {4}

It is a singleton set.
v.
1 + 2x = 3x

1 = 3x  2x

1=x

x=1

E = {1}

It is a singleton set.
B = {x|x is a natural number, x < 1}
B=
Finite set: If counting of elements in a set
terminates at a certain stage, the set is called
as finite set.
Example:
A = {1, 2, 3, 4, 5, 6, 7}
B = {x|x is days in a week}
The above sets A and B have finite elements.
Set A and set B are finite sets.
Infinite set: If counting of elements in a set
does not terminate at any stage, the set is
called as infinite set.
Example:
P = {1, 2, 3, 4, 5, 6, …}
W = {x|x is a whole number}
The above sets P and W have elements that
cannot be counted. They are sets that do not
terminate at any stage. Therefore, P and W are
infinite sets.
Note:
i.
X = {0} is not a null set as ‘0’ is an element of
set X.
ii.
An empty set is a finite set.
iii. Sets of Natural numbers, Whole numbers,
Integers, Rational numbers and Real numbers
are all infinite sets.
Exercise 1.2
1.
State which of the following sets are
singleton sets:
i.
A= x


x  16
ii.
B = {y|y2 = 36}
iii. C = {p|p  I, p3 = 8}
iv. D = {q|(q  4)2 = 0}
v.
E = {x|1 + 2x = 3x, x  W}
Solution:
x = 16
i.

x = 256

A = {256}

It is a singleton set.
y2 = 36
y = 6
B = {–6, +6}
It is not a singleton set.
ii.



4
2.
Which of the following sets are empty?
i.
A set of all even prime numbers
ii.
B = {x|x is a capital of India}
iii. F = {y|y is a point of intersection of two
parallel lines}
iv. G = {z|z  N, 3 < z < 4}
v.
H = {t|t is a triangle having four sides}
Solution:
i.
A = {2}

It is not an empty set.
ii.

B = {Delhi}
It is not an empty set.
iii.


Parallel lines do not intersect each other.
F={}
It is an empty set.
iv.


z is a natural number. There is no natural
number between 3 and 4.
G={}
It is an empty set.
v.


A triangle is a three-sided figure.
H={}
It is an empty set.
Chapter 01: Sets
3.
Classify the following sets into finite or
infinite:
i.
A = {1, 3, 5, 7, …}
ii.
B = {101, 102, 103, … , 1000}
iii. C = {x|x  Q, 3 < x < 5}
iv. D = {y|y = 3n, n  N}
Solution:
i.
Here, counting of elements do not terminate at
any stage.

A is an infinite set.
ii.

Here, counting of elements terminate at 1000.
B is a finite set.
iii.
There is infinite number of rational elements
between 3 and 5.
C is an infinite set.

iv.

Here, counting of elements do not terminate at
any stage.
D is an infinite set.
4.
Let G = {x|x is a boy of your class} and
H = {y|y is a girl of your class}. What type
of sets G and H are?
Solution:
Set G and set H are finite sets.
1.5
Subset
If every element of set Y is an element of set X, then
Y is said to be subset of set X.
Symbolically, it is represented as Y  X
If we have say ‘a’, an element which belongs to set
Y, we can say that, it (‘a’) also belongs to set X.
But if a  Y and a  X then it is said that set Y is
not a subset of X or Y  X.
Example:
If Y = {b, z} and X = {b, l, z} then we say that
Y  X.
If Y is a subset of X and set X contains atleast one
element which is not in set Y, then set Y is the
proper subset of set X. It is denoted as Y  X.
Set X is said to be the ‘super set’ of set Y and is
denoted as X  Y.
If X = {a, b} and Y = {b, a}, then set X is a subset of
set Y and Y is also subset of set X.
In this case set X is the improper subset of the set Y.
It is denoted as X  Y and it is read as “X is an
improper subset of Y.”
Also set Y is the improper subset of the set X.
It is denoted as Y  X and it is read as “Y is an
improper subset of X.”
Note:
i.
Every set is a subset of itself i.e. Y  Y.
ii.
Empty set is a subset of every set i.e.,   X.
1.6
Universal Set
A suitably chosen non-empty set of which all the
sets under consideration are the subsets of that set is
called the Universal set.
It is denoted by ‘U’.
Example:
A = {x|x is Physics laboratory in your school}
B = {y|y is Chemistry laboratory in your school}
C = {z|z is Biology laboratory in your school}
U = {l|l is laboratories in your school}
It can be seen that A  U, B  U, C  U.

Set U is the universal set of sets A, B and C.
Note: Universal set is a set that cannot be changed
once fixed for a particular solution.
In Venn diagram, generally universal set is
represented by a rectangle.
Exercise 1.3
1.
Observe the following sets and answer the
questions given below:
A = The set of all residents in Mumbai
B = The set of all residents in Bhopal
C = The set of all residents in Maharashtra
D = The set of all residents in India
E = The set of all residents in Madhya Pradesh
i.
Write the subset relation between the
sets A and C.
ii.
Write the subset relation between the
sets E and D.
iii. Which set can be chosen suitably as
the universal set?
Solution:
i.
All residents of Mumbai are residents of
Maharashtra.

AC
ii.

iii.

All residents of Madhya Pradesh are residents
of India.
ED
Mumbai, Maharashtra, Bhopal, Madhya
Pradesh are parts of India.
Set D can be chosen as the universal set.
5
Std. IX : Algebra 2.
Let A = {a, b, c}, B = {a}, C = {a, b}, then
i.
Which sets given above are the proper
subsets of the set A?
ii.
Which set is the super set of set C?
Solution:
i.
Elements of set B and set C are the elements
of set A. Also, there exists an element viz. c
which is not an element of set B and set C but
is in set A.

Set B and set C are the proper subsets of
set A.
ii.
Set A is the super set of set C i.e. A  C.
3.
Draw a Venn diagram, showing sub set
relations of the following sets:
A = {2, 4}
B = {x|x = 2n, n < 5, n  N}
C = {x|x is an even natural number  16}
Solution:
A = {2, 4}
B = {2, 4, 8, 16}
C = {2, 4, 6, 8, 10, 12, 14, 16}

ABC
C
B
A
.12
.10
.8 .2 .16
.4
.6
.14
Prove that, if A  B and B  C, then A  C.
(Hint: Start with an arbitrary element
x  A and show that x  C)
Solution:
Let us assume that x  A
….(i)
But, A  B

xB

BC
4.


xC
From (i) and (ii),
AC
….(ii)
5.
If X = {1, 2, 3}, write all possible subsets of X.
Solution:
All possible subsets of X are as follows:
i.
{ } or  .…[a null set is a subset of every set]
ii.
{1}
iii. {2}
iv. {3}
6
v.
vi.
vii.
viii.
{1, 2}
{1, 3}
{2, 3}
{1, 2, 3}
1.7
Operations on Sets
.…[every set is a subset of itself]
a.
Equality:
If A is a subset of B and B is a subset of A,
then A and B are said to be equal sets and are
denoted by A = B.
Both the sets A and B contain exactly the
same elements.
If the elements of A and B are not same, then
we write A  B.
Note: To prove that sets A and B are equal, it is
always necessary to prove that A  B and
B  A.
i.
Let A = {x|x = 2n, n  N and x < 10}
and B = {2, 4, 6, 8}

A = {2, 4, 6, 8}

A  B and B  A

A=B
ii.
Let P = {x|x is an odd natural number,
x < 8}
and Q = {y|y is an even natural number,
y <10}
In roster form,
P = {1, 3, 5, 7}
Q = {2, 4, 6, 8}

P  Q and Q  P

PQ
b.

Intersection of Sets:
If A and B are two sets then a set of common
elements in A and B is called intersection of
set A and B. It is represented as A  B and is
read as ‘A intersection B’.
Example:
Let A = {1, 3, 5, 7, 9}
B = {3, 9, 12}
A  B = {3, 9}
or A  B = {x|x  A and x  B}
.1
.3
.5
.7 .9
.12
B
A
AB
Shaded part in the Venn diagram represents
intersection of sets A and B.
Chapter 01: Sets
Properties of Intersection of Sets:
i.
ii.
iii.
iv.
v.
vi.
c.

AB=BA
[Commutative property]
A  (B  C) = (A  B)  C
[Associative property]
A  B  A; A  B  B
A  P; B  P then A  B  P
If A  B then A  B = A
If B  A then A  B = B
A   =  and A  A = A
Disjoint Sets:
Let A = {2, 4, 6, 8} and B = {1, 3, 5, 7}
AB=
If there are no common elements in two sets,
then such sets are called disjoint sets.
.2
.4
.1
.3
.7 .5
.6 .8

d.

A
B
The Venn diagram represents two disjoint sets
A and B.
AB=
For disjoint sets
i.
AB=
ii.
A  B
iii. B  A
Union of Sets:
If A and B are two sets then a set containing
all the elements of A and B together is called
union of sets A and B.
Union of two sets A and B is denoted as
‘A  B’ and is read as ‘A union B’.
Let A = {1, 2, 3, 4, 5} and B = {3, 5, 7, 9}
A  B = {1, 2, 3, 4, 5, 7, 9}
or A  B = {x|x  A or x  B}
A
.1 .2 .3 .7
.5
.9
.4
B
AB
The shaded portion in the Venn diagram
represents A  B.
Properties of Union of sets:
i.
AB=BA
[Commutative property]
ii.
A  (B  C) = (A  B)  C
[Associative property]
iii. A  A  B and B  A  B
iv.
v.
vi.
If A  B, then A  B = B and if B  A, then
AB=A
A=A
AA=A
Distributive Property:
i.
A  (B  C) = (A  B)  (A  C)
ii.
A  (B  C) = (A  B)  (A  C)
e.
Complement of a Set:
If U is a universal set and set A is a subset of
the universal set, then set of all elements in U
which are not in set A is called the
complement of set A.
It is denoted by A or Ac.
Let U = {x|x is a natural number, x  9}
and A = {1, 3, 5, 7}
In roster form,
U = {1, 2, 3, 4, 5, 6, 7, 8, 9}
A = { 2, 4, 6, 8, 9}
or A = {x|x  U and x  A}
In Venn diagram, complement of set A is
given as:
U
A
.6
.2
.3 .4
.1 A .8
.5 .7
.9
Note:
i.
A  A = 
ii.
A  A = U
Properties of Complement of a Set:
i.
(A) = A
ii.
 = U
iii. U = 
iv. If A  B, then B  A
v.
A  A = 
vi. A  A = U
If A and B are any two sets, then
i.
(A  B) = A  B and
ii.
(A  B) = A  B
Exercise 1.4
1.
Let P = {x|x is a letter in the word
‘CATARACT’} and Q = {y|y is a letter in
the word ‘TRAC’}. Show that P = Q.
Solution:
Roster form of set P and set Q is as follows:
P = {C, A, T, R}
Q = {T, R, A, C}
Set P and set Q are subset of each other.
Also, the elements of set P and set Q are same.

P=Q
7
Std. IX : Algebra 2.
Find the union of each of the following
pairs of sets:
i.
A = {2, 3, 5, 6, 7}, B = {4, 5, 7, 8}
ii.
C = {a, e, i, o, u}, D = {a, b, c, d}
iii. E = {x|x  N and x is a divisor of 12}
F = {y|y  N and y is a divisor of 18}
Solution:
i.
A = {2, 3, 5, 6, 7}, B = {4, 5, 7, 8}

A  B = {2, 3, 4, 5, 6, 7, 8}
i.
ii.
iii.

iv.

A = {40, 41, 42, 43} = {1, 4, 16, 64}
B = {80, 81, 82} = {1, 8, 64}
A = {2, 8, 32, 128}
B = {2, 4, 16, 32, 128}
A  B = {1, 4, 8, 16, 64}
(A  B) = {2, 32, 128}
(A  B) = {1, 64}
(A  B) = {2, 4, 8, 16, 32, 128}
5.
Find the intersection of each of the
following pairs of sets:
i.
A = {1, 2, 4, 5, 7}, B = {2, 3, 4, 8}
ii.
C = {x|x  N, 5 < x  10}
D = {y|y  W, 5  y < 10}
iii. E = {x|x  I, x < 0}, F = {y|y  I, y > 0}
Solution:
i.
A = {1, 2, 4, 5, 7}, B = {2, 3, 4, 8}

A  B = {2, 4}
Let A = {a|a is a letter in the word ‘college’}
and B = {b|b is a letter in the word ‘luggage’}
and U = {a, b, c, d, e, f, g, l, o, u}. Verify:
i.
(A  B) = A  B
ii.
(A  B) = A  B
Proof:
i.
In roster form, set A and set B can be written
as:
A = {c, o, l, e, g}
B = {l, u, g, a, e}
U = {a, b, c, d, e, f, g, l, o, u}

A = {a, b, d, f, u}

B = {b, c, d, f, o}

A  B = {a, c, e, g, l, u, o}
L.H.S. = (A  B) = {b, d, f}
.… (i)
R.H.S. = A  B = {b, d, f}
.… (ii)
From (i) and (ii), we get
L.H.S. = R.H.S.
(A  B) = A  B
ii.
ii.
A  B = {l, g, e}
L.H.S. = (A  B) = {a, b, c, d, f, o, u} .… (iii)
R.H.S. = A  B = {a, b, c, d, f, o, u} …. (iv)
From (iii) and (iv), we get
L.H.S. = R.H.S.
(A  B) = A  B
1.8
Number of Elements in the Set
ii.

C = {a, e, i, o, u}, D = {a, b, c, d}
C  D = {a, b, c, d, e, i, o, u}
iii.
The Roster form of set E and set F is as
follows:
E = {1, 2, 3, 4, 6, 12}
F = {1, 2, 3, 6, 9, 18}
E  F = {1, 2, 3, 4, 6, 9, 12, 18}

3.
The Roster form of set C and set D is as
follows:
C = {6, 7, 8, 9, 10}
D = {5, 6, 7, 8, 9}
C  D = {6, 7, 8, 9}

iii.

The Roster form of set E and set F is as
follows:
E = {… , 4, 3, 2, 1}
F = {1, 2, 3, 4, …}
E  F = { } or 
Let U = {x|x = 2n, n  W, n < 8} be the
universal set. A = {y|y = 4n, n  W, n < 4};
B = {z|z = 8n, n  W, n  2}. Then find:
i.
A
ii.
B
iii. (A  B)
iv. (A  B)
Solution:
The Roster form of set U, set A and set B is as
follows:
U = {20, 21, 22, 23, 24, 25, 26, 27}
= {1, 2, 4, 8, 16, 32, 64, 128}
4.
8
If A is any set then the number of elements in set A
is denoted by n (A).
Illustrations:
i.
Let A = {x|x  N, 7 < x  12}

A = {8, 9, 10, 11, 12}

n(A) = 5
ii.
For an empty set,
n() = 0
iii. n(A  B) = n(A) + n(B)  n(A  B)
To verify this identity,
let us consider the following example,
A = {2, 3, 4} and B = {3, 4, 5, 6}

A  B = {2, 3, 4, 5, 6} and
A  B = {3, 4}
Chapter 01: Sets


n(A) = 3, n(B) = 4, n(A  B) = 5 and
n(A  B) = 2
L.H.S. = n(A  B) = 5
.... (i)
R.H.S. = n(A) + n(B)  n(A  B)
=3+42
=5
.... (ii)
n(A  B) = n(A) + n(B)  n(A  B)
.... [From (i) and (ii)]




n(T  C) = 80
There are 80 students who either drink tea or
coffee or both. But there are 100 students in
the hostel.
Number of students who neither drink tea nor
coffee = n(U)  n(T  C) = 100  80 = 20
Students who do not drink tea or coffee is 20.
4.
Exercise 1.5
1.
Let A = {1, 3, 5, 6, 7}, B = {4, 6, 7, 9}, then
verify the following:
n (A  B) = n(A) + n(B)  n(A  B)
Proof:
A = {1, 3, 5, 6, 7} and
B = {4, 6, 7, 9}

A  B = {1, 3, 4, 5, 6, 7, 9} and A B = {6, 7}

n(A) = 5, n(B) = 4, n(A  B) = 7 and
n(A  B) = 2
L.H.S. = n(A  B) = 7
....(i)
R.H.S. = n(A) + n(B)  n(A  B)
=5+42=7
....(ii)

L.H.S. = R.H.S.
....[From (i) and (ii)]

n(A  B) = n(A) + n(B)  n(A  B)
2.
Let A and B be two sets such that n(A) = 5,
n(A  B) = 9, n(A  B) = 2. Find n(B).
Solution:
Given, n(A) = 5, n(A  B) = 9, n(A  B) = 2
n(B) = ?
By using identity,
n(A  B) = n(A) + n(B)  n(A  B)

9 = 5 + n(B)  2

9 – 5 + 2 = n(B)

n(B) = 6
3.
In a school hostel, there are 100 students,
out of which 60 drink tea, 50 drink coffee
and 30 drink both tea and coffee. Find the
number of students who do not drink tea or
coffee.
Solution:
Let U be the universal set of students in hostel,
T be the set of students who drink tea and
C be the set of students who drink coffee.

n(U) = 100, n(T) = 60, n(C) = 50, n(T  C) = 30
By using the identity,
n(T  C) = n(T) + n(C)  n(T  C)
= 60 + 50  30
= 110  30
110 children choose their favourite colour
from blue and pink. Every student has to
choose at least one of the colours. 60
children choose blue colour, while 70
children choose pink colour. How many
children choose both the colours as their
favourite colour?
Solution:
Let the number of children who choose blue
colour be n(B) and number of children who
choose pink colour be n(P).

n(B) = 60 and n(P) = 70

Number of children who choose their
favourite colour from blue or pink.

n(B  P) = 110
By using the identity,
n(B  P) = n(B) + n(P)  n(B P)

110 = 60 + 70  n(B P)

n(B P) = 60 + 70  110

n(B P) = 20

The number of students who choose both
the colours as their favourite colours is 20.
5.
Observe the figure and verify the following
equation:
A
.1
.5
.4
.2
.3
.9
B
.8
.7
.6
C
n(A  B  C) = n(A) + n(B) + n(C)  n(A  B)
 n(B  C)  n(C  A) + n(A  B  C)
Proof:
L.H.S. = n(A  B  C)
A  B  C = {1, 2, 3, 4, 5, 6, 7, 8, 9}

n(A  B  C) = 9
…. (i)
Now, A = {1, 2, 3, 4, 5}

n(A) = 5
B = {2, 3, 6, 7, 8}

n(B) = 5
C = {3, 4, 6, 9}

n(C) = 4
A B = {2, 3}

n(A B) = 2
9
Std. IX : Algebra 

B C = {3, 6}

n(B C) = 2
C A = {3, 4}

n(C A) = 2
A B C = {3}

n(A B C) = 1
R.H.S.
= n(A) + n(B) + n(C)  n(A  B)  n(B  C)
 n(C  A) + n(A  B  C)
=5+5+4222+1
=9
…. (ii)
L.H.S. = R.H.S.
....[From (i) and (ii)]
n(A  B  C) = n(A) + n(B) + n(C)
 n(A  B)  n(B  C)  n(C  A)
+ n(A  B  C)
Solution:
i.
A = {… , 3, 2, 1}
ii.
B = {25, 45, 52, 54, 56, 58, 65, 85}
iii. C = {2, 3, 5}
iv. D = {3, 2, 1, 0, 1, 2, 3}
v.
Since, 2  n  4

n = 2, 3, 4
n
2
2

for n = 2, 2
=
=
2
n  1 (2)  1 3
for n = 3,
n
3
3
=
=
2
n  1 (3)  1 8
for n = 4,
n
4
4
=
=
2
n  1 (4)  1 15
2
2
Problem Set - 1
1.
Which of the following collections are sets?
i.
The collection of rich people in your
district.
ii.
The collection of natural numbers less
than 50.
iii. The collection of most talented
persons of India.
iv. The collection of first ten prime integers.
v.
The collection of all days in a week
starting with the letter ‘T’.
vi. The collection of some months in a year.
vii. The collection of all books in your
school library.
viii. The collection of smart boys in your
class.
ix. The collection of multiples of 7.
x.
The collection of students in your
class who got a lot of marks in the
first unit test.
Solution:
From the given collections (ii), (iv), (v), (vii) and
(ix) are sets. Remaining collections are not
considered as sets as they have relative terms and
their meaning may vary from person to person.
2.
10
E=  , ,
3.
Write the following sets in the set builder
form:
i.
F = {I, N, D, A}
ii.
G = {1, 1}
iii. H = {3, 9, 27, 81, 243}
iv. J = {15, 24, 33, 42, 51, 60}

E = x x 


, 2  n  4,n  N 
n 1

n
2
1 2 3 4 5 
, , 
 2 5 10 17 26 
v.
K = , ,
Solution:
i.
F = {x|x is a letter in the word ‘INDIA’}
ii.
G = {y| square of y is 1}
or G = {y|y is a square root of 1}
iii. H = {a|a = 3n, n  N, n  5}
iv. J = {b|b is a two digit number whose sum of
digits is 6}
v.
Write the following sets in roster form:
i.
A = {x|x  I, x  W}
ii.
B = {x|x is two digit number such that
the product of its digits is a
multiple of ten}
iii. C = {x|x is a prime divisor of 120}
iv. D = {x|x  I and x2 < 10}
v.
2 3 4 

 3 8 15 

When n = 1, c =
1
(1)  1 2
When n = 2, c =
2
2

2
(2)  1 5
3
(3)  1 10
When n = 4, c =
4
4

2
(4)  1 17

n

n 2 +1
K = c c =
3

When n = 3, c =
When n = 5, c =

1
2
2
5

5
(5)  1 26
2


, n  N, n  5

Chapter 01: Sets
4.
Classify the following sets as ‘singleton’ or
‘empty’:
i.
A = {x|x is a negative natural number}
ii.
B = {y|y is an odd prime number < 4}
iii. C = {z|z is a natural number, 5 < z < 7}
iv. D = {d|d  N, d2  0}
Solution:
i.
Each natural number is positive.
A={}
It is an empty set.

ii.

B = {3}
It is a singleton set.
iii.

C = {6}
It is a singleton set.
iv.
There is no natural number whose square is
less than or equal to zero.
D={}
It is an empty set.


5.
Classify the following sets as ‘finite’ or
‘infinite’:
i.
A = {x|x is a multiple of 3}
ii.
B = {y|y is a factor of 13}
iii. C = {…, 3, 2, 1, 0}
iv. D = {x|x = 2n, n  N}
Solution:
i.
A = {3, 6, 9, 12, …}
It is an infinite set.

ii.

B = {1, 13}
It is a finite set.
iii.
C is an infinite set.
iv.

D = {20, 21, 22, 23, 24,...} = {2, 4, 8, 16, 32, …}
D is an infinite set.
6.
State which of the following sets are equal.
i.
N = {1, 2, 3, 4, …}
ii.
W = {0, 1, 2, 3, …}
iii. A = {x|x = 2n, n  W}
iv. B = W  {0}
Solution:
i.
N ={1, 2, 3, 4, …}
ii.
W = {0, 1, 2, 3, …}
iii. A = {20, 21, 22, 23, …} = {1, 2, 4, 8, …}
iv. B = W  {0} = {1, 2, 3, 4, …}
Here, set N and set B are subset of each other.
In set N and set B, the elements are the same.

N=B
7.
Let A = {7, 5, 2} and B =

3

125, 4, 49 .
Are the sets A and B equal? Justify your
answer.
Solution:
A = {7, 5, 2},
B = 3 125, 4, 49





B = {5, 2, 2, 7, 7}
Here, A is a subset of B, but B is not a subset
of A.
Elements of set A and set B are not equal.
AB
8.
If A = {1, 2, 3, 4}, B = {2, 4, 6, 8},
C = {3, 4, 5, 6} and U = {x|x  N, x < 10}.
Verify the following properties:
i.
A  (B  C) = (A  B)  C
ii.
A  (B  C) = (A  B)  (A  C)
iii. A  (C) = (A  B)  (A  C)
iv. (A  B) = A  B
v.
(A  B) = A  B
vi. (A) = A
Solution:
Roster form of set U is as follows:
U = {1, 2, 3, 4, 5, 6, 7, 8, 9}
A = {1, 2, 3, 4}
B = {2, 4, 6, 8}
C = {3, 4, 5, 6}
i.
B  C = {2, 3, 4, 5, 6, 8}
A  B = {1, 2, 3, 4, 6, 8}
L.H.S. = A  (B  C)
= {1, 2, 3, 4, 5, 6, 8}
…. (i)
R.H.S. = (A  B)  C
= {1, 2, 3, 4, 5, 6, 8}
…. (ii)

From (i) and (ii), we get
L.H.S. = R.H.S.
A  (B  C) = (A  B)  C

ii.


iii.
B  C = {4, 6}
A  B = {1, 2, 3, 4, 6, 8}
A  C = {1, 2, 3, 4, 5, 6}
L.H.S. = A  (B  C)
= {1, 2, 3, 4, 6}
…. (i)
R.H.S. = (A  B)  (A  C)
= {1, 2, 3, 4, 6}
…. (ii)
From (i) and (ii), we get
L.H.S. = R.H.S.
A  (B  C) = (A  B)  (A  C)
B  C = {2, 3, 4, 5, 6, 8}
L.H.S. = A  (B  C)
= {2, 3, 4}
…. (i)
11
Std. IX : Algebra 

iv.



v.


vi.


9.
A  B = {2, 4}
A  C = {3, 4}
R.H.S. = (A  B)  (A  C)
= {2, 3, 4}
.… (ii)
From (i) and (ii), we get
L.H.S. = R.H.S.
A  (C) = (A  B)  (A  C)
A  B = {1, 2, 3, 4, 6, 8}
(A  B) = {5, 7, 9}
A = {1, 2, 3, 4}
 A = {5, 6, 7, 8, 9}
B = {2, 4, 6, 8}
 B = {1, 3, 5, 7, 9}
L.H.S. = (A  B)
= {5, 7, 9}
…. (i)
R.H.S. = A  B
= {5, 7, 9}
…. (ii)
From (i) and (ii), we get
L.H.S. = R.H.S.
(A  B) = A  B
A  B = {2, 4}
(A  B) = {1, 3, 5, 6, 7, 8, 9}
A = {5, 6, 7, 8, 9}
B = {1, 3, 5, 7, 9}
L.H.S. = (A  B)
= {1, 3, 5, 6, 7, 8, 9}
R.H.S. = A  B
= {1, 3, 5, 6, 7, 8, 9}
From (i) and (ii), we get
L.H.S. = R.H.S.
(A  B) = A  B
A = {5, 6, 7, 8, 9}
L.H.S. = (A)
= {1, 2, 3, 4}
R.H.S. = A
= {1, 2, 3, 4}
From (i) and (ii), we get
L.H.S. = R.H.S.
(A) = A
A = {0}
Set A is not a null set.
ii.
Here, 2b is an even number and 1 is an odd
number.
Since, addition of even and odd number
always give odd number,
value of 2b + 1, for any value of b  N
is an odd number.
Set B is a null set.


iii.


Square of an odd number is always an odd
number.
c2 cannot be even.
Set C is a null set.
10.
Give an example of the set which can be
written in set builder form but cannot be
written in roster form.
Solution:
Consider the set of rational numbers ‘Q’.
In set builder form, it is
a
b


Q =  a  I, b  I and b  0 
But same set Q cannot be written in roster form.
…. (i)
…. (ii)
…. (i)
11.
Write down all possible subsets of each of
the following sets:
i.

ii.
A = {1}
iii. B = {1, 2}
iv. C = {a, b, c, d}
Solution:
i.
Subset of a null set is only one i.e. 
ii.
Subsets of set A are empty set { } and set A
itself.
i.e. and {1}
iii.
All possible subsets of set B are
, {1}, {2}, {1, 2}.
iv.
All possible subsets of set C are
{a}, {b}, {c}, {d}, {a, b}, {a, c}, {a, d},
{b, c}, {b, d}, {c, d}, {a, b, c}, {b, c, d},
{a, c, d}, {a, b, d}, {a, b, c, d}
…. (ii)
For each of the following sets, state with
reasons, whether it is a null set or not:
i.
A = {x|x  I, x2 is not positive}
ii.
B = {b|b  N, 2b + 1 is even}
iii. C = {c|c  N, c is odd and c2 is even}
Solution:
i.
Whether a number is positive or negative its
square is always a positive.

Square of an integer cannot be negative,
except zero, whose square is neither positive
nor negative.
12


12.
Write proper subsets of the following sets:
i.
A = {a, b}
ii.
B = {a, b, c}
Solution:
i.
Proper subsets of A are {a} and {b}.
ii.
Proper subsets of B are {a}, {b}, {c}, {a, b},
{b, c}, {c, a}.
Chapter 01: Sets
13.
Write the sets A and B such that A is finite,
B is finite, A and B are disjoint sets.
Solution:
Let B = {1, 3, 5, 7}, A = {2, 4, 6, 8} be the
finite sets.
But their intersection is a null set,
i.e. A  B = 

A and B are disjoint sets.
14.
Let A = {a, b, c, d}, B = {a, b, c},
C = {b, d, e}, then find the sets D and E
satisfying the following conditions:
i.
D  A, D  B
ii.
C  E, B E = 
Solution:
i.
Since, D  A and D  B

D = {a, b}/{a, c}/{a, d}/{a, b, d}/{b, c, d}
/{a, c, d}
ii.
Since, C  E and B  E = 

E must not contain any element of set B. C is
superset of set E.

E = {d}/{e}/{d, e}
Let U = {x|x  N, x < 10},
A = {a|a is even, a  U},
B = {b|b is a factor of 6, b  U}. Verify that:
n(A) + n(B) = n(A  B) + n(A  B).
Solution:
Roster form of set U and set A is as follows:
U = {1, 2, 3, 4, 5, 6, 7, 8, 9}
A = {2, 4, 6, 8}
….(Even numbers within
the universal set)

n(A) = 4
Roster form of set B is as follows:
B = {1, 2, 3, 6}
….( b  U)
15.





n(B) = 4
A  B = {1, 2, 3, 4, 6, 8}
n(A  B) = 6
A  B = {2, 6}
n(A  B) = 2
L.H.S. = n(A) + n(B)
=4+4
=8
....(i)
R.H.S. = n(A  B) + n(A  B)
=6+2
=8
....(ii)
From (i) and (ii), we get
L.H.S. = R.H.S.
n(A) + n(B) = n(A  B) + n(A  B)
16.
In a group of students, 50 students passed
in English, 60 students passed in
Mathematics and 40 students passed in
both. Find the number of students who
passed either in English or in Mathematics.
Solution:
Let the number of students who passed in
English be denoted by set E.

n(E) = 50
Let the number of students who passed in
Mathematics be denoted by set M.

n(M) = 60

The number of students who passed in English
and Mathematics = n(M  E) = 40
By using identity,
n(E  M) = n(E) + n(M)  n(E  M)
= 50 + 60 – 40
= 110  40
= 70

The number of students who passed either
in English or in Mathematics is 70.
17.
A T.V. survey says 136 students watch only
programme P1, 107 watch only programme
P2, 27 watch only programme P3. 25
students watch P1 and P2 but not P3. 37
watch P2 and P3 but not P1. 53 students
watch P1 and P3 but not P2. 40 students
watch all three programmes and 80
students do not watch any programme.
Find, with the help of Venn diagram.
i.
Number of P1 viewers.
ii.
Number of P2 or P3 viewers.
iii. Total number of viewers surveyed.
Solution:
Given data represented by Venn diagram is as
follows:
U
P2
P1
136
107
25
53
40
37
27
80
P3
i.

From the Venn diagram,
Number of P1 viewers = n(P1)
= 136 + 25 + 53 + 40
= 254
Number of P1 viewers is 254.
13
Std. IX : Algebra ii.

iii.

Number of P2 or P3 viewers = n(P2  P3)
= 107 + 25 + 40 + 37 + 53 + 27 = 289
Number of P2 or P3 viewers is 289.
Total number of viewers surveyed
= Number of only P1 viewers + Number of
P2 or P3 viewers + 80
= 136 + 289 + 80 = 505
Total number of viewers surveyed is 505.
18.
Show that, it is impossible to have sets A
and B such that set A has 32 elements, set B
has 42 elements, A  B has 12 elements and
A  B has 64 elements.
Solution:
Set A has 32 elements.

n(A) = 32
Set B has 42 elements.

n(B) = 42
Given that n(A  B) = 12 and n(A  B) = 64
By using the identity,
n(A  B) = n(A) + n(B)  n(A  B),
L.H.S. = n(A  B) = 64
....(i)
R.H.S. = n(A) + n(B)  n(A  B)
= 32 + 42  12 = 62 ....(ii)
From (i) and (ii), we get
L.H.S.  R.H.S.

Sets A and B are impossible.
19.
Let the universal set U be a set of all
students of your school. A is the set of boys,
B is the set of girls and C is the set of
students participating in sports. Describe
the following sets in words and represent
them by a Venn diagram:
i.
BC
ii.
A  (B  C)
Solution:
i.
B  C represents the set of girls participating
in sports. A
B
C
Represent sets A, B, C such that A  B,
A  C =  and B  C   by Venn diagram
and shade the portion representing
A  (B  C).
Solution:
i.
AB

set A is proper subset of set B.
i.e. set A is inside set B.
ii.
A  C = 
 set A and set C do not intersect.
iii. B  C  
 set B and set C intersect.

B C
B
C

A


20.
A  (B C)
Let A, B, C be sets such that A  B  ,
B  C   and A  C  . Do you claim that
A  B  C  ? Justify your answer.
Solution:
There are two possibilities:
i.
If A = {a, b}
B = {b, c}
C = {c, a}
then, it is observed that A  B  ,
B  C  , C  A  ,
but, A  B  C = 
ii.
If A = {a, b}
B = {a, c}
C = {a, b, c}
then, it is observed that A  B  ,
B  C  , C  A  ,
but, A  B  C = {a}  
We cannot claim that A  B  C  .

21.
22.
ii.
BC
A  (B C) represents the set of all boys or
set of girls that participate in sports.
BC
A
C
B
A  (B  C)
14
With the help of suitable example, verify
the following statements:
If A  B, B  C, then A  C.
Solution:
Let A = {x, y, z}, B = {a, x, y}, C = {y, w}
Since, each element of set A does not exist in
set B.

A  B
Each element of set B does not exist in set C.

B  C
Each element of set A does not exist in set C.

A  C
Chapter 01: Sets
23.
If A and B are any two sets, then prove that
i.
(A  B) = A  B
ii.
(A  B) = A  B
[Hint: Show (A  B)  A  B and vice
versa]
Solution:
Let U = {1, 2, 3, 4, 5, 6, 7, 8}
A = {1, 2, 3, 4, 5}
B = {3, 4, 5, 6, 7}
i.
A  B = {1, 2, 3, 4, 5, 6, 7}
L.H.S. = (A  B) = {8}
…. (i)
A = {6, 7, 8}
B = {1, 2, 8}
R.H.S. = A  B = {8}
…. (ii)
From (i) and (ii), we get
L.H.S. = R.H.S.

(A  B) = A  B
ii.


Now,
A  B = {3, 4, 5}
L.H.S. = (A  B) = {1, 2, 6, 7, 8}
R.H.S. = A  B = {1, 2, 6, 7, 8}
From (i) and (ii), we get
L.H.S. = R.H.S.
(A  B) = A  B
…. (i)
…. (ii)
One-Mark Questions
1.
Write the following set in set builder form.
A = {2, 3, 5, 7, 11, 13, 17}
Solution:
The set builder r form of set B is:
A = {x|x is a prime number, x < 18}
2.
Write the following set in roster form.
B = {x|x is a natural number and 4  x < 10
Solution:
The Roster form of set B is:
B = {4, 5, 6, 7, 8, 9}
3.
If A = {1, 2, 3, 4, 5, 6} and B = {1, 3, 5, 7}
then draw Venn diagram for A  B.
Solution:
U
B
A
2
4
6
1
3
5
AB
7
4.
If U = {1, 2, 3, 4, 5, 6, 7, 8, 9} is the universal
set and C = {5, 6, 7, 8} then find C.
Solution:
Given, U = {1, 2, 3, 4, 5, 6, 7, 8, 9} and
C = {5, 6, 7, 8}
C = {1, 2, 3, 4, 9}

5.
If A = {9, 11, 13, 15} and B = {1, 3, 5, 7}
then find A  B.
Solution:
Given, A = {9, 11, 13, 15} and B = {1, 3, 5, 7}

A  B = { } or 
6.
If A and B are two sets such that n(B) = 8,
n(A  B) = 11, n(A  B) = 6, find n(A).
Solution:
By using identify,
n(A  B) = n(A) + n(B)  n(A  B)

11 = n(A) + 8  6

n(A) = 11  2
n(A) = 9

7.
State which of the following sets are equal:
A = {x|x  W, x < 6}
B = {1, 2, 3, 4, 5, 6}
C = {0, 1, 2, 3, 4, 5}
Solution:
Here, A = {0, 1, 2, 3, 4, 5} and
C = {0, 1, 2, 3, 4, 5}
A=C

8.
Classify the following sets as singleton or
empty.
i.
A = {x|x is a natural number, x  5
and x  7}
ii.
B = {x|x is an even prime number}
Solution:
i.
There is no common number for x < 5
and x > 7

A = {}

Set A is empty set.
ii
B = {2}

Set B is singleton set.
9.
If A = {2, 3, 4, 5} and B = {1, 2, 5, 6}, then
find A  B.
Solution:
Given, A = {2, 3, 4, 5} and B = {1, 2, 5, 6}

A  B = {1, 2, 3, 4, 5, 6}
15
Std. IX : Algebra 10. If A = {3}, write all possible subsets of set A.
Solution:
 and {3}
11. If U = {1, 2, 3, 4} and X = {2, 4}, then find X.
Solution:
Given, U = {1, 2, 3, 4} and X = {2, 4}

X = {1, 3}
2.
Draw a Venn diagram showing subset
relations of the following sets:
A = {2 , 8}
B = {x|x = 2n, n  4 and n  N}
C = {x|x is an even natural number  20}
3.
If A = {x, y}, write all possible subsets of A.
4.
State true or false:
i.
 is a subset of itself.
ii.
If A  B and B  A, then A = B.
iii. The empty set is a subset of all sets.
Additional Problems for Practice
Based on Exercise 1.1 1.
2.
Write the following sets in the roster form:
i.
A = {x|x is a prime number which is a
divisor of 30}
ii.
B = {x|x is an even natural number}
iii. C = {x|x is an integer and x2 < 5}
iv. F = {x|x is a letter in the word ‘LITTLE’}
v.
E = {x|x  W, x  N}
vi. D = {x|x is a square root of 81}
Based on Exercise 1.4 1.
Find the union of each of the following pairs
of sets:
i.
A = {5, 15, 25}, B = {10, 20, 30}
ii.
H = {3, 6, 9, 12, 15} , F = {3, 4, 5, 6}
iii. M = {x|x  N and x is a divisor of 12}
N = {x|x  N and x is a prime divisor of
12}
Write the following sets in the set builder form:
i.
A = {2, 4, 6, 8, 10, 12, 14}
ii.
B = {5, 10, 15, 20, ….}
iii. C = {7, 72, 73, 74}
iv. D = {51, 53, 55, 57, 59}
v.
E = {2, 3, 5, 7, 11, 13, 17, 19}
2.
Find the Intersection of the following pairs of
sets:
i.
A = {5, 6, 7},
B = {8, 9, 10}
ii.
M = {10, 20, 30, 40, 50},
N = {20, 40, 60}
iii. N is a set of natural numbers and W is a
set of whole numbers.
iv. P = {a, b, p, d, q}, R = {q, r, s, p}
3.
If U = {x|x is a natural number less than 15} is
a universal set
A = {1, 3, 4, 5, 9}, B = {3, 5, 7, 9, 12}
Verify that (A  B) = A  B
4.
U = {x|x  I and 3  x  3},
A = {2, 0, 2}, B = {0, 1, 2, 3}
Find i.
A
ii.
B
iii. (A  B) iv. A  B
Based on Exercise 1.2 1.
2.
State which of the following sets are singleton
or empty sets:
i.
A = {x|x  5 = 0}
ii.
B = {y|y is an even prime number greater
than 2}
iii. D = {x|x  N and 3x  1 = 0}
iv. E = {x|x  I, x is neither a positive nor a
negative number}
v.
C = {x|x  N and x < 7 and x > 11}
Classify the following sets into finite or infinite:
i.
A = {x|x is a multiple of 1}
ii.
C = {x|x is a point on a line}
iii. D = {1, 2, 3, 4, …., 100}
iv. E = {x|x  N and x is an odd number}
Based on Exercise 1.3 1.
16
Write the subset relations among the following
sets:
P = set of all residents in Nagpur
X = set of all residents in Vadodara
Y = set of all residents in Maharashtra
T = set of all residents in Gujarat
Based on Exercise 1.5 1.
With the help of following figure, write the
following sets:
U
B
A
1
4
3
i.
iv.
9
5
10
11
A
AB
7
2
8
12
6
ii.
v.
B
AB
iii.
U
Chapter 01: Sets
2.
Let A and B be two sets such that n(A) = 17,
n(B) = 23, n(A  B) = 38. Find n(A  B)
7.
If A  P, B  P, then (A  B)  ________.
(A) A
(B) B
(C) P
(D) 
3.
240 students in a school were interviewed and
their hobbies were noted. 150 students were
interested in stamp collection. 80 took delight
in reading books, 40 of them do not like either.
What is the number of students who liked both
stamp collection and reading books?
8.
If A = {3, 4, 7, 8, 9} and B = {7, 8, 10, 11}
then A  B = ?
(A) {3, 4}
(B) {7, 8}
(C) {10, 11}
(D) {10}
9.
For any two sets A and B, A  B = ?
(A) {x|x  B or x  A}
(B) {x|x  A or x  B}
(C) {x|x  A and x  B}
(D) {x|x  A and x  B}
10.
If U = {1, 2, 3, 4, ….} and
A = {2, 4, 6, 8, ….} then A = ?
(A) {2, 4, 6, ….}
(B) {1, 3, 5, 7, ….}
(C) {0, 1, 3, 5, ….}
(D) {0, 2, 4, 6, 8, ….}
11.
If U = {4, 5, 6, 7, 8, 9}, P = {5, 6, 7, 8},
Q = {4, 6, 8, 9} then, P  Q = ?
(A) {4, 5, 7, 8, 9}
(B) {4, 5, 7, 9}
(C) {6, 7, 8}
(D) {4, 6, 7, 8, 9}
12.
In the following Venn
n(P  Q) = 70, then x = ?
4.
5.
In a class of 50 students, 35 like Physics, 30
like Mathematics and 3 like neither. How
many like both the subjects and how many
like Physics only?
U
A
3
6
15 12
9
B
18
24
From the above diagram find:
i.
AB
ii.
n (A  B)
iii. (A  B)
iv. n (A  B)
v.
A  B
Multiple Choice Questions
1.
If B = {x|x is a vowel in English alphabet},
then Roster form of B is
(A) {a, e, i, u}
(B) {a, e, p, o}
(C) {a, e, c, d}
(D) {a, e, i, o, u}
2.
Which of the following is not an Infinite set
(A) N
(B) W
(C) I
(D) None of these
3.
A = {z|z + 6 = 0} is a _______.
(A) empty set
(B) singleton set
(C) infinite set
(D) finite set
4.
Empty set is a _______ of every set.
(A) subset
(B) proper subset
(C) super set
(D) universal set
5.
If A = {x|x is worker in department  I of your
company}
B = {y|y is worker in department  II of your
company}
and C = {z|z is worker of your company) then
(A) C  A
(B) A  B
(C) A  C
(D) C  B
6.
If A  B and B  A, then set A and B are
_______ sets.
(A) equal
(B) disjoint
(C) super
(D) universal
P
13.
5
If
Q
40  x
(A)
diagram.
(B)
3
x
35  x
(C)
6
(D)
8
If n(A) = 10, n(B) = 25 and n(A  B) = 15,
then n(A  B) = ?
(A) 20 (B) 0
(C) 10 (D) 5
Answers to additional problems for practice
Based on Exercise 1.1 1.
i.
ii.
iii.
iv.
v.
vi.
A = {2, 3, 5}
B = {2, 4, 6, 8, ….}
C = {2, 1, 0, 1, 2}
F = {L, I, T, E}
E = {0}
D = {9, 9}
2.
i.
ii.
iii.
iv.
A = {x|x = 2n, n  N and n < 8}
B = {x|x = 5n and n  N}
C = {x|x = 7n, 1  n  4}
D = {x|x  N, x is an odd integer and
50 < x < 60}
E = {x|x is a prime number and 1 < x < 20}
v.
17
Std. IX : Algebra 5.
Based on Exercise 1.2
1.
Singleton sets are A, E
Empty sets are B, D, C
2.
Finite set is D.
Infinite sets are A, C, E.
1.
P  Y, X  T
2.
A = {2, 8}, B = {2, 4, 8, 16}
C = {2, 4, 6, 8, 10, 12, 14, 16, 18, 20}
C
B
6
A
12
4
2
8
16
18
20
14
3.
, {x}, {y}, {x, y}
4.
i.
True
ii.
True
iii.
True
Based on Exercise 1.4
1.
i.
ii.
iii.
A  B = {5, 10, 15, 20, 25, 30}
H  F = {3, 4, 5, 6, 9, 12, 15}
M  N = {1, 2, 3, 4, 6, 12}
2.
i.
ii.
iii.
iv.
AB=
M  N = {20, 40}
N  W = {1, 2, 3, ….}
P  R = {p, q}
4.
i.
ii.
iii.
iv.
A = {3, 1, 1, 3}
B = {3, 2, 1}
(A  B) = {3, 1}
A  B = {3, 1}
Based on Exercise 1.5
1.
i.
ii.
iii.
iv.
v.
2.
2
3.
30
4.
18 students like both subjects and 17 students
like only Physics.
18
A  B = {6, 12}
n(A  B) = 7
(A  B) = {3, 9, 15, 18, 24}
n(A  B) = 2
A  B = {3, 9, 15, 18, 24}
Answers to Multiple Choice Questions
Based on Exercise 1.3
10
i.
ii.
iii.
iv.
v.
A = {2, 3, 6, 7, 8, 11, 12}
B = {1, 3, 4, 6, 9, 11, 12}
U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}
A  B = {1, 2, 4, 5, 7, 8, 9, 10}
A  B = {5, 10}
1.
5.
9.
13.
(D)
(C)
(B)
(A)
2. (D)
6. (A)
10. (B)
3. (B)
7. (C)
11. (B)
4. (A)
8. (B)
12. (A)