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1. AN INTRODUCTION TO SET THEORY

2. AN INTRODUCTION TO SET THEORY
By
VEENA V
(B.Sc , M.Sc Mathematics)

3. Preface
The book “ An Introduction to Set theory” is intended for the secondary students and teachers in
Kerala syllabus. In this book all the topic have been deal with in a simple and lucid manner. A
sufficiently large number of problems have been solved. By studying this book , the student is
expected to understand the concept of set and their representations, types of sets, operations on
set, practical situations. To do more problems involving the types of sets, operations of sets and
express set builder form to roster form and roster form to set builder form.
Suggestion for the further improvement of this book will be highly
appreciated.
Veena v.

4. CONTENTS
Title Page No:
Preface
Chapter 1. Set and their representations 1 – 4
Chapter 2. Types of sets 5 – 7
Chapter 3. Subsets 8 – 13
Chapter 4. Operations on sets 14 – 20
Chapter 5. Practical problems on union and
intersection of two sets 21 – 22
Summary 23
Reference

5. 1
CHAPTER 1
SETS AND THEIR REPRESENTATIONS
In these days of conflict between ancient and modern studies ; there must surely be something to
be said for a study which did not begin with Pythagoras and will not end with Einstein ; but is
the oldest and the youngest. - G . H . HARDY *
Introduction
The concept of set serves as a fundamental part of the present day of mathematics . Today this
concept is being used in almost every branch of mathematics . Sets are used to define the
concept of relations and functions. The study of geometry, sequences, probability ,etc. requires
the knowledge of sets.
The theory of sets was developed by German mathematician George Cantor ( 1845 – 1918 ). He
first encountered sets while working on ‘ problems on trigonometric series’.
In this chapter , we discuss some basic definitions and operations involving sets.
Definition 1
A set is a well – defined collection of objects. The following points may be noted :
I. Objects , elements and members of a set are synonymous terms.
II. Sets are usually denoted by capital letters A, B, C, X, Y, Z, etc.
III. The elements of a set are represented by small letters a, b, c, x, y, z, etc.

6. If a is an element of a set A , we say that ‘a belongs to A’ the Greek symbol ϵ ( epsilon ) is
used to denote the phrase ‘ belongs to’ .Thus we write a∈A . If b is not an element of a set A,
we write b∉A and read ‘ b does not belongs to A’.
2
There are two methods of representing a set :
I. Roster or tabular form.
II. Set – builder form.
i. In roster form, all the elements of a set are listed, the elements are being separated by
commas and are enclosed within braces { }.For example , the set of all even positive
integers less than 7 is described in roster form as { 2,4,6 }
ii. In set – builder form, all the elements of a set possess a single common property which is
not possessed by any elements outside the set.
For example, in the set { a,e,i,o,u,s } all the elements possess a common property , namely each
of them is a vowel in the English alphabet. Denoting this set by V , we write,
V = { x: x is a vowel in English alphabet }.
We described the element of the set by using a symbol x ( any other symbol like the letters y,z,
etc. could be used ) which is followed by a colon ‘ :’ . After the sign of colon , we write the
characteristics property possessed by the elements of the set and then enclose the whole
description within braces.
The above description of the set V is read as ‘ the set of all x such that x is a vowel of the English
alphabet’. The braces stands for ‘ the set of all’ , the colon stands for ‘ such that’.
NOTE : It may be noted that while writing the set in roster form an element is not generally
repeated , i.e; all the elements are taken as distinct .
Example 1 : Write the solution set of the equation x2
+ x- 2 = 0 in roster form
Solution : The given equation can be written as,

7. ( x + 2 ) (x – 1 ) = 0 ; i.e; x = 1, -2 .
The roster form as { 1, -2 }.
3
Example 2 : Write the set { x : x is a positive integer and x2
< 40 } in the roster form .
Solution : The required numbers are 1,2,3,4,5,6. So , the given set in the roster form is
{1,2,3,4,5,6 }.
Example 3 : Write the set A = { 1,4,9,16,25,…} in set builder form.
Solution : We may write the set A as ,
A = { x : x is the square of a natural number }
Exercise
1. Which of the following are sets ? Justify your answer.
a) The collection of all the months of a year beginning with the letter J .
b) The collection of ten most talented writers of India.
c) A team of eleven best cricket bats man of the world.
d) The collection of all boy’s in your class.
e) The collection of all even integers.
2. Let A = { 1,2,3,4,5,6 }. Insert the appropriate symbol ∈ or ∉ in the blank spaces :
a) 5……A
b) 8…....A
c) 0…....A

8. d) 4……A
e) 2……A
3. Write the following sets in roster form:
a) A = { x : x is an integer and -3<x<7 }
b) B = { x : x is a natural number less than 6}
c) C = { x : x is a prime number which is divisor of 60}
d) D = The set of all letters in the word TRIGNOMETRY
e) E = The set of all letters in the word BETTER
4
4. Write the following sets in the set builder form :
i. {3,6,9,12 }
ii. {2,4,8,16,32}
iii. {5,2,5,125,625}
iv. {2,4,6,….}
v. {1,4,9,…..100}
5. List all the elements of the following sets :

9. i. A = { x : x is an odd natural number }
ii. B = { x : x is an integer, x2
≤4 }
iii. C = { x : x is a letter in the word ‘LOYAL’ }
iv. D = { x : x is a month of a year not having 31 days}
6. Match the following :
i. {1,2,3,6} a. {x : x is a prime number and a divisor of 6}
ii. {2,3} b. {x : x is an odd natural number less than 10}
iii. {M,A,T,H,E,I,C,S} c. { x : x is a natural number and a divisor of 6}
iv. {1,3,5,7,9} d. { x : x is a letter of the word MATHEMATICS}
5
CHAPTER 2
TYPES OF SETS

10. Definition 1
A set having no elements is called empty set or void set or null set. It is usually denoted by { } or
ᶲ.
Example:
The set of all boys in a girls school is a null set.
Definition 2
A set having only one element is called a singleton set.
Example:
The set of all principals in a college is a singleton set.
NOTE :
{ }ᶲ is not an empty set but it is a singleton set.
Definition 3
A set which is empty or having finite number of elements is a finite set. Otherwise set is
infinite.
Examples:
i. S = { 2,4,6,8}is finite.
ii. Set of all students in a country is finite .
iii. Set of all points in a line is infinite.
iv. Set N of natural numbers is an infinite set
6
Order of a finite set : The number of elements of a finite set is called the order of that set.
Order of a set A is denoted by n(A).
NOTE :

11. order of a set is also known as the cardinal number of the set
Example :
1. Order of empty set is 0 and order of singleton set is 1.
2. If A = { x : x is a divisor of 20}
Then, A = {1,2,4,5,10,20}
n(A) = 6
Definition 4
Two sets A and B are said to be equal, if they contain same elements.
i.e ; A = B , if all elements of A are in B and all elements of B are in A.
Example :
I. The sets {-1,1} and {x : x2
– 1 = 0 } are equal.
II. A={x : xєR and x2
-3x + 2 = 0} are not equal sets since A = {1,2,-1,-2} and
B = {1,2}
III. The set of all letters in the word LAST and the set of all letters in the word SALT are
equal sets.
Definition 5
Two sets A and B are said to be equivalent if n(A) = n(B). i.e; they contain same number of
elements.
Example :
I. A = {1} and B = {2} are equivalent sets since n(A) = 1 and n(B) =1.
II. A = {1,2,3,4}, B = {a,b,c} are not equivalent.
NOTE :
All the equal sets are equivalent. But the converse is not true. i.e; all the equivalent sets need
not be equal sets.
7
Exercise

12. 1. Which of the following are examples of the null set.
a) Set of even prime number
b) {x : x is a natural number , x<5 and x>7}
c) Set of odd natural number divisible by 2}
d) {y : y is a point common to any two parallel line}
2. Which of the following sets are finite or infinite .
a) The set of months of a year
b) The set of prime number less than 99
c) {1,2,3,…99,100}
3. In the following statement 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 a positive even integer and x≤10}
8
CHAPTER 3

13. SUBSETS
Definition 1
The set A is called a subset of set B if every element of A is also an element of B.
We write it as A⊂B. If A is not a subset of B we write A⊄B .
If A ⊂ B, then B is called the superset of A. Symbolically we can write , A ⊂B if xєA ⇒
xєB
Example
1. A= {-1,2,5}; B= {3,-1,2,7,5}
Clearly A⊂B. But B⊄A.
2. The set of all odd natural numbers is a subset of the set of all natural numbers
3. Let A= { Vowels in the English alphabet}
B= {Letters in the English alphabet}
Then A is a subset of B and therefore B is the superset of A.
NOTE :
I. A⊂B and B⊂A if and only if A = B.
II. Every set is a subset of it self. i.e, A⊂A.
III. Empty set is a subset of every set. i.e, ᶲ⊂A
Definition 2
Let A and B be two sets such that A⊂B and A ≠ B.
9
Then A is called proper subset of B.

14. That is, there exists at least one element of B which is not in A.
If A is not a proper subset, it is called an improper subset.
NOTE :
I. Every set is an improper subset of itself
II. Every set except the null set will have minimum two subsets.
III. Empty set has only one subset.
IV. An element of a set cannot be a subset. That is, a⊂{a,b}is an incorrect statement.
*Subsets of set of Real numbers:
There are many important subsets of R.
The set of natural numbers,
N = {1,2,3,….}
The set of integers,
Z = {…,-3,-2,-1,0,1,2,3….}
The set of rational numbers,
Q = { x : x = p/q , p,qєZ and q ≠ 0}
The set of irrational numbers,
T = { x : xєR and x ∉Q}
Then , N⊂Z⊂Q, Q⊂R, T⊂R, N⊄T.
10
*Intervals as subsets of :

15. Let a,bєR and a< b.
We can define four intervals as follows.
1. Open interval (a,b) = { x : x єR, a <x <b}
2. Closed interval [a,b] = { x : x єR, a≤ x≤ b}
3. Closed but open interval [a,b) = { x : x єR, a≤ x<b}
4. Open but closed interval (a,b] = { x : x єR, a< x≤ b}
All these intervals are subset of R.
NOTE :
The set of real numbers can be written as an interval as (- ∞ ,∞ ).
The number (b - a) is called the length of any of the intervals (a,b) , [a,b] , [a,b) or (a,b].
Definition 3.
Power set of a set A is the set of all subsets of A. It is usually denoted by P(A).
Example :
• Let A = {1,2}
Then P(A) = { {1,2}, {1} ,{2} , ᶲ}, n(P(A)) = 4.
• If A = ᶲ ,P(A) = {ᶲ }. So n(P(A)) = 1.
NOTE:
• The elements of the power set are subsets.
• If n(A) = m, then the number of subsets of A =2m
.
In other words, if n(A) = m, then n(P(A)) = 2m
.
• If n(A) = m, then the number of proper subsets of A =2m
-1
11
Example

16. If A = {1,2,3,4}, then number of subsets of A = 24
= 16.
Number of proper subsets of A = 24
-1 = 15
n(P(A)) = 24
= 16
i.e, n(A) = m then n(P(A)) =2m
.
Definition 4
Universal set of given sets, is the superset of all sets under consideration. It is denoted
by U.
Example
• A = {x : x is an even natural number }
• B = {x : x is an odd natural number }
• C = {x : x is a prime number }
Then the universal set are considered as ,
• U = {x : x is an even natural number }
Definition
Relationships between the sets can be represented by means of diagram known as Venn
diagrams.
The universal set is represented usually by a rectangle. The elements of the sets are
written in their respective circles.
Example
If U = set of natural numbers less than 10 is the universal set of which A = The set of all
prime numbers less than 10 is a subset. The corresponding Venn diagram is given
below,
12

17. Exercise
1. Examine whether the following statements are true or false.
• {a,b} ⊄{b,c,a}
• {1,2,3}⊂{1,3,5}
• {a} є{a,b,c}
• {a} ⊂{a,b,c}
2. Let A = {1,2 {3,4}, 5}. Which of the following statements are incorrect and why?
• {3,4} ⊂ A
• {3,4}є A
• {2,4,5}є A
• {1,2,3⊂A
3. Write down all the subsets of the following sets.
• {a}
• {a,b}
13

18. • {1,2,3}
• {5,6,7,8}
4. How many elements has P(A) if A = ?ᶲ
5. Write the following intervals in the set builder form;
• (-3,0)
• [6,12]
• (6,12]
• [-23,5]

19. 14
CHAPTER 4
OPERATIONS ON SETS
Definition 1
The union of two sets A and B is the set C which consists of all those elements which are either
in A or in B( including those which are in both).
In symbols, we write
A∪B = { x: x ∈A or x∈ B}
Example
I. A = {1,2,3,4,5}, B = {3,4,5,6}
∴ A∪ B = {1,2,3,4,5,6 }
II. A = {x : x is an even natural number}
B = {x : x is an odd natural number }
∴ A ∪B = {x : x is a natural number}
III. A = { 1,2,3,4}, B = {1,3}
∴ A ∪B = {1,2,3,4}
Note that here B⊂A and A∪ B =A
The Venn diagram of A∪ B is as follows:

20. 15
*Some properties of the operation of union:
a. A ∪ B = B ∪A ( Commutative law)
b. (A ∪B) ∪ C = A∪ (B∪ C) (Associative law )
c. A∪ ᶲ = A (Law of identity element ,ᶲ is the identity of U )
d. A∪ A = A (Idempotent law )
e. U ∪A = U (Law of U )
.
Definition 2
The intersection of sets A and B is the set of all elements which are common to both A and B.
The symbol ∩ is used to denote the intersection.
Symbolically, we write A∩ B = { x : x ∈ A and x ∈B}.
Examples:
1) A = {1,2,3,4} , B = {3,4,5,6}
A ∩B = {3,4}
2) A = {a,b,c}, B = {a,b,c,d,e}
A ∩B = {a,b,c} = A
Note that here A⊂ B
*Some properties of operation of intersection:
i. A ∩B = B ∩A (Commutative law)
ii. (A∩ B)∩ C = A∩( B ∩C) (Associative law)
iii. ᶲ∩ A = ᶲ , U ∩A = A (Law of ᶲ and U )
iv. A∩ A = A (Idempotent law )

21. 16
v. A∩( B∪ C )= (A∩ B)∪ (A∩ C) (Distributive law)
The venn diagram of A∩B ,
Definition 3
Two sets A and B are called disjoint sets, if there is no element common to them.
That is, A∩ B = ᶲ.
The venn diagram of disjoint set ,
Example
A = {x : x is an even natural number }
B = {x : x is an odd natural number }
∴A∩ B = ᶲ.
Here , we say A and B are disjoint set.

22. 17
Definition 4
The difference of two sets A and B in this order is the set of elements that are in A and not in B.
i.e, A-B = { x : x ∈ A and x∉ B }
B-A = { x : x ∈ B and x∉ A }
Example
A = {1,2,3,4,5} , B = {4,5,6,7} then ,
A-B = {1,2,3} , B-A = {6,7}
The venn diagram of A-B , B-A, are as shown below,
1. A-B 2. B-A
a. A∪ B = (A- B)∪ (A∩B)∪( B-A)
b. A-B = A- (A ∩B)
c. A-B , B-A , and A∩ B are disjoint sets.

23. 18
Exercise
1. Find the union of each of the following pairs of sets;
i. X = {1,3,5} , Y = {1,2,3}
ii. A = {x : x is a natural number and multiple of 3}
B = {x : x is a natural number and less than 6}
2. If A = {1,2,3,4} , B = {3,4,5,6} , C = {5,6,7,8} and D = {7,8,9,10} ,find
i. A∪ B
ii. A∪C
iii. B∪ C
iv. B∪ D
v. A ∪B∪ C
vi. A∪ B∪D
vii. B∪ C∪ D
3. If A = {3,5,7,9,11} , B = {7,9,11,13} ,C = {11,13,15} and D = {15,17}. Find
i. A∩ B
ii. A∩ C
iii. A∩ D
iv. B∩D
v. A∩C∩D
4. If X = {a,b,c,d } , Y = {f,b,d,g}. find X-Y, Y-X, X∩Y.

24. 19
Definition 5
If A is any set and U is the universal set, then complement of A denoted by A is the set of allꞌ
elements that are in U and not in A.
i.e,A = { x: xꞌ ∈ U and x∉ A }
Obviously A = U – A.ꞌ
NOTE:
The complement of A is also denoted by AC
.
Example:
i. U = {1,2,3,4,5,6,7}
A = {1,3,5,7}, B = {2,3,4,6} then,
A = {2,4,6}, B = {1,5,7}.ꞌ ꞌ
ii. U = {x : x is a natural number}
A = {x : x is an even natural number}
Then A = {x : x is an odd natural number}ꞌ
The venn diagram of A is as given below,ꞌ

25. 20
*Some properties of complement sets:
a) A∪A = U and A∩A =ꞌ ꞌ ᶲ.
b) (A ) = Aꞌ ꞌ
c) ᶲꞌ. =U and U =ꞌ ᶲ.
d) De morgan’s laws
If A and B are any two subset of the universal set U, then
1. (A∪ B) = A ∩Bꞌ ꞌ ꞌ
2. (A∩B) = Aꞌ ꞌ∪Bꞌ
These two results are stated in words as,
The complement of the union of two sets is the intersection of their complements and the
complements of the intersection of two sets is the union of their complements. These are
called De morgan’s laws.
Exercise
1. If U = {a,b,c,d,e,f,g,h} . find the complements of the following sets.
• A = {a,b,c}
• B = {d,e,f,g}
• C = {a,c,e,g}
2. If U = {1,2,3,4,5,6,7,8,9} , A = {2,4,6,8}, B = { 2,3,5,7} . Verify that (A∪ B ) =ꞌ
A ∩B and (A∩B) = Aꞌ ꞌ ꞌ ꞌ∪Bꞌ

26. 21
CHAPTER 5
PRACTICAL PROBLEMS ON UNION AND INTERSECTION OF
TWO SETS
The following results are very useful in doing practical problems;
• If A and B are two disjoint sets,
then n(A∪ B) = n(A) + n(B)
• If A and B are any finite sets,
then n(A∪ B) = n(A) + n(B) –n(A∩B)
• If A,B,C are three disjoint sets ,
then n(A∪B∪C) = n(A) +n(B)+n(C)
• If A,B,C are any three finite sets,
then n(A∪B∪C) = n(A) + n(B) +n(C) – n(A∩B) – n(A∩C) – n(B∩C) + n(A∩B∩C)
We have already seen that
A∪ B = (A – B )∪ (A∩B )∪(B-A)
and A-B , A∩ B and B-A are disjoint set.
So, n(A∪B) = n(A - B) + n(B - A) + n(A∩ B)
Example 1: In a school there are 20 teachers who teach mathematics or physics. Of these 12
teach maths and 4 teach both physics and maths. How many teach physics ?

27. Solution : Let M denote the set of teachers who teach maths and P denote the set of teachers
who teach physics.
22
We have, n(M∪P) = 20, n(M) = 12, n(M∩P) = 4.
n(M∪P) = n(M) + n(P) --n(M∩P).
20 = 12 + n(P) – 4
∴n(P) = 12
Hence 12 teachers teach physics.
Example 2: In a class of 35 students , 24 like to play cricket and 16 like to play football. Also ,
each student likes to play at least one of the two games. How many students like to play both
cricket and football?
Solution : Let X be the set of student who like to play cricket and Y be the set of
students who like to play football. Then X∪Y is the set of students who like to play at
least one game and X∩Y is the set of students who like to play both games.
Given ,n(X) = 24 , n(Y) = 16 , n(X∪Y) = 35.
n(X∪Y) = n(X) + n(Y) –n(X∩Y)
35 = 24 + 16 - n(X∩Y)
∴n(X∩Y) = 5
Hence 5 students likes to play both games.
Exercise
1. If X and Y are two sets such that n(X) = 17, n(Y) = 23, and n(X∪ Y) = 38. Find
n(X ∩Y) ?
2. In a group of 400 people 250 can speak Hindi and 200 speak English. How many
people can speak both Hindi and English ?

28. 3. In a committee 50 people speak French 20 speak Spanish and 10 speak both
French and Spanish. How many speak at least one of these two language ?
23
Summary
This chapter deals with some basic definitions and operations involving sets. These are
summarized below ;
1. A set is a well defined collection of objects.
2. A set which does not contain any elements is called empty set
3. A set which consist of a definite number of elements is called finite set. Other wise the
set is called infinite set.
4. Two sets A and B are said to be equal, if they have exactly the same elements.
5. A set A is said to be subset of a set B , if every element of A is also an element of B.
6. A power set of a set A is collection of all subsets of A.
7. The union of two sets A and B is the set of all those elements which are either in A or
in B.
8. The intersection of two sets A and B is the set of all elements which are common.
9. The complement of a subset A of universal set U is the set of all elements of U which
are not in A.
10. If A and B are finite sets such that A∩ B =ᶲ. ,then n(A∪B) =n(A) + n(B).
11. If A∩B ≠ᶲ , then n(A∪B) = n(A) + n(B) –n(A∩ B).

29. REFERENCE:
MATHEMATICS TEXT BOOK FOR CLASS PLUS ONE : C.P. Janardhanan pillay
$
V. G .Vijayalekshmy