Lecture 3 - Sets and Relations | Algebra- Engineering Maths - Engineering Mathematics PDF Download

 Page 1


Sets and Relations 
Institute of Lifelong Learning, University of Delhi      pg. 1 
 
 
 
 
 
 
 
 
Subject: Maths, Algebra-I 
Lesson: Sets and Relations 
Lesson Developer: B.K.Tyagi 
College/Department: A.R.S.D College, Delhi University 
 
 
 
 
 
 
 
 
 
 
Page 2


Sets and Relations 
Institute of Lifelong Learning, University of Delhi      pg. 1 
 
 
 
 
 
 
 
 
Subject: Maths, Algebra-I 
Lesson: Sets and Relations 
Lesson Developer: B.K.Tyagi 
College/Department: A.R.S.D College, Delhi University 
 
 
 
 
 
 
 
 
 
 
Sets and Relations 
Institute of Lifelong Learning, University of Delhi      pg. 2 
 
 
Table of Contents: 
 Chapter : Functions and Relations 
? 1: Learning Outcomes 
? 2: Introduction 
? 3: Sets 
o 3.1 Subset of a Set 
? 4: Union and Intersection of sets 
o 4.1: Arbitrary Unions and Intersections 
? 5: Relations 
o 5.1: Binary Relations 
o 5.2: Reflexive, Symmetric and Transitive Relations 
o 5.3: Equivalence Relations 
o 5.4 : Partitions of Sets 
? 6: Mappings 
o 6.1: Identical Mapping 
o 6.2: Composition of Mappings 
o 6.3: One-One or Injective Mapping 
o 6.4: Onto or Surjective Mapping 
o 6.5: Bijective Mapping 
o 6.6: Inverse Mapping 
? 7: Finite Sets and Infinite Sets 
o 7.1: Finite Sets 
o 7.2: Infinite Sets 
? Summary 
? Exercises 
? Glossary 
? References/ Bibliography/ Further Reading 
 
 
 
 
Page 3


Sets and Relations 
Institute of Lifelong Learning, University of Delhi      pg. 1 
 
 
 
 
 
 
 
 
Subject: Maths, Algebra-I 
Lesson: Sets and Relations 
Lesson Developer: B.K.Tyagi 
College/Department: A.R.S.D College, Delhi University 
 
 
 
 
 
 
 
 
 
 
Sets and Relations 
Institute of Lifelong Learning, University of Delhi      pg. 2 
 
 
Table of Contents: 
 Chapter : Functions and Relations 
? 1: Learning Outcomes 
? 2: Introduction 
? 3: Sets 
o 3.1 Subset of a Set 
? 4: Union and Intersection of sets 
o 4.1: Arbitrary Unions and Intersections 
? 5: Relations 
o 5.1: Binary Relations 
o 5.2: Reflexive, Symmetric and Transitive Relations 
o 5.3: Equivalence Relations 
o 5.4 : Partitions of Sets 
? 6: Mappings 
o 6.1: Identical Mapping 
o 6.2: Composition of Mappings 
o 6.3: One-One or Injective Mapping 
o 6.4: Onto or Surjective Mapping 
o 6.5: Bijective Mapping 
o 6.6: Inverse Mapping 
? 7: Finite Sets and Infinite Sets 
o 7.1: Finite Sets 
o 7.2: Infinite Sets 
? Summary 
? Exercises 
? Glossary 
? References/ Bibliography/ Further Reading 
 
 
 
 
Sets and Relations 
Institute of Lifelong Learning, University of Delhi      pg. 3 
 
 
1. Learning Outcomes: 
After you have read this chapter, you should be able to  
? define 
? understand 
? identify 
? differentiate 
? appreciate 
? apply the knowledge of  
set theory, functions and relations to any field of knowledge. 
2. Introduction: 
In this chapter, the algebra of sets is described. It consists of the 
operations of union, intersection, Cartesian product, power set etc. .These 
operations are fundamental to any branch of mathematics. Relations and 
functions are sets which are objects of investigation in every branch of 
knowledge. Various types of relations and functions are described.  
3. Sets: 
Relations and functions are sets. So in order to understand what is a 
function or a relation we need to understand the definition of a set. But 
this is not straightforward to introduce the definition of a set .There are 
many formal ways in which the definition of a set is introduced .The 
collections of objects which an undergraduate student of mathematics 
encounters in his daily life are accepted as sets by all definitions of a set. 
For examples, the collections of: numbers, the voters of a state, the 
undergraduate students in Delhi University in 2012 are all examples of  
sets. Some collections of sets are also accepted as sets, but the collection 
of all sets is not a set. 
Value Addition: Remark 
The definition of set: ”a set is a well-defined collection of objects” is not 
valid since the term well-defined in the above definition remains 
Page 4


Sets and Relations 
Institute of Lifelong Learning, University of Delhi      pg. 1 
 
 
 
 
 
 
 
 
Subject: Maths, Algebra-I 
Lesson: Sets and Relations 
Lesson Developer: B.K.Tyagi 
College/Department: A.R.S.D College, Delhi University 
 
 
 
 
 
 
 
 
 
 
Sets and Relations 
Institute of Lifelong Learning, University of Delhi      pg. 2 
 
 
Table of Contents: 
 Chapter : Functions and Relations 
? 1: Learning Outcomes 
? 2: Introduction 
? 3: Sets 
o 3.1 Subset of a Set 
? 4: Union and Intersection of sets 
o 4.1: Arbitrary Unions and Intersections 
? 5: Relations 
o 5.1: Binary Relations 
o 5.2: Reflexive, Symmetric and Transitive Relations 
o 5.3: Equivalence Relations 
o 5.4 : Partitions of Sets 
? 6: Mappings 
o 6.1: Identical Mapping 
o 6.2: Composition of Mappings 
o 6.3: One-One or Injective Mapping 
o 6.4: Onto or Surjective Mapping 
o 6.5: Bijective Mapping 
o 6.6: Inverse Mapping 
? 7: Finite Sets and Infinite Sets 
o 7.1: Finite Sets 
o 7.2: Infinite Sets 
? Summary 
? Exercises 
? Glossary 
? References/ Bibliography/ Further Reading 
 
 
 
 
Sets and Relations 
Institute of Lifelong Learning, University of Delhi      pg. 3 
 
 
1. Learning Outcomes: 
After you have read this chapter, you should be able to  
? define 
? understand 
? identify 
? differentiate 
? appreciate 
? apply the knowledge of  
set theory, functions and relations to any field of knowledge. 
2. Introduction: 
In this chapter, the algebra of sets is described. It consists of the 
operations of union, intersection, Cartesian product, power set etc. .These 
operations are fundamental to any branch of mathematics. Relations and 
functions are sets which are objects of investigation in every branch of 
knowledge. Various types of relations and functions are described.  
3. Sets: 
Relations and functions are sets. So in order to understand what is a 
function or a relation we need to understand the definition of a set. But 
this is not straightforward to introduce the definition of a set .There are 
many formal ways in which the definition of a set is introduced .The 
collections of objects which an undergraduate student of mathematics 
encounters in his daily life are accepted as sets by all definitions of a set. 
For examples, the collections of: numbers, the voters of a state, the 
undergraduate students in Delhi University in 2012 are all examples of  
sets. Some collections of sets are also accepted as sets, but the collection 
of all sets is not a set. 
Value Addition: Remark 
The definition of set: ”a set is a well-defined collection of objects” is not 
valid since the term well-defined in the above definition remains 
Sets and Relations 
Institute of Lifelong Learning, University of Delhi      pg. 4 
 
undefined. 
 
In this chapter, the capital letters A,B, …., X,Y,Z, possibly with suffixes, 
are used to denote sets of objects. If an element a is in the set A, we 
express this fact by the notation A a ? and read it as: "a belongs to A", "a 
is in A", or "a is member of A". An object in A is also called an element of 
A or a point in
1
A .  
3.1. Subset of a Set: 
A set A is said to be a subset of a set B , written as B A ? , if A x ? implies 
that B x ? . Two sets A and B are said to be equal, denoted B A ? , if B A ?
 
and A B ? . For example, if integers, of set ? A
 
and numbers real of set  the ? B , 
then B A ? , B is not a subset of A, and A and B are, of course, not equal, 
that is B A ? . A set A is said to be a proper subset of set B, denoted B A ?
or B A
?
? , if B A ?
 
and B A ? . 
Most often a set can be identified by a property that is common to all its 
elements. For example, 
 ? ? country a of capital a is :x x A ? 
 ? ? 4 by  divisible integer  posiitve a is :k k B ? . 
In the nineteenth century, a set was considered to be a collection of 
elements having some property P, and clearly is the basis of the notation :
? ? P x x property   the has : . Intuitively, it is difficult to see what possibly could go 
wrong with this. In 1901, however, Russel made the crucial observation 
that the above understanding of a set leads to a contradiction : Let A be 
the set of all sets x such that x is not a member of x, that is  
 ? ? x x x A ? ? : . 
From this we get the contradiction : A A ? if and only if A A ? . This 
contradiction is known as the Russel's paradox. Nevertheless, the notation 
? ? P x x property   the has : for a set is used extensively in mathematics . 
Page 5


Sets and Relations 
Institute of Lifelong Learning, University of Delhi      pg. 1 
 
 
 
 
 
 
 
 
Subject: Maths, Algebra-I 
Lesson: Sets and Relations 
Lesson Developer: B.K.Tyagi 
College/Department: A.R.S.D College, Delhi University 
 
 
 
 
 
 
 
 
 
 
Sets and Relations 
Institute of Lifelong Learning, University of Delhi      pg. 2 
 
 
Table of Contents: 
 Chapter : Functions and Relations 
? 1: Learning Outcomes 
? 2: Introduction 
? 3: Sets 
o 3.1 Subset of a Set 
? 4: Union and Intersection of sets 
o 4.1: Arbitrary Unions and Intersections 
? 5: Relations 
o 5.1: Binary Relations 
o 5.2: Reflexive, Symmetric and Transitive Relations 
o 5.3: Equivalence Relations 
o 5.4 : Partitions of Sets 
? 6: Mappings 
o 6.1: Identical Mapping 
o 6.2: Composition of Mappings 
o 6.3: One-One or Injective Mapping 
o 6.4: Onto or Surjective Mapping 
o 6.5: Bijective Mapping 
o 6.6: Inverse Mapping 
? 7: Finite Sets and Infinite Sets 
o 7.1: Finite Sets 
o 7.2: Infinite Sets 
? Summary 
? Exercises 
? Glossary 
? References/ Bibliography/ Further Reading 
 
 
 
 
Sets and Relations 
Institute of Lifelong Learning, University of Delhi      pg. 3 
 
 
1. Learning Outcomes: 
After you have read this chapter, you should be able to  
? define 
? understand 
? identify 
? differentiate 
? appreciate 
? apply the knowledge of  
set theory, functions and relations to any field of knowledge. 
2. Introduction: 
In this chapter, the algebra of sets is described. It consists of the 
operations of union, intersection, Cartesian product, power set etc. .These 
operations are fundamental to any branch of mathematics. Relations and 
functions are sets which are objects of investigation in every branch of 
knowledge. Various types of relations and functions are described.  
3. Sets: 
Relations and functions are sets. So in order to understand what is a 
function or a relation we need to understand the definition of a set. But 
this is not straightforward to introduce the definition of a set .There are 
many formal ways in which the definition of a set is introduced .The 
collections of objects which an undergraduate student of mathematics 
encounters in his daily life are accepted as sets by all definitions of a set. 
For examples, the collections of: numbers, the voters of a state, the 
undergraduate students in Delhi University in 2012 are all examples of  
sets. Some collections of sets are also accepted as sets, but the collection 
of all sets is not a set. 
Value Addition: Remark 
The definition of set: ”a set is a well-defined collection of objects” is not 
valid since the term well-defined in the above definition remains 
Sets and Relations 
Institute of Lifelong Learning, University of Delhi      pg. 4 
 
undefined. 
 
In this chapter, the capital letters A,B, …., X,Y,Z, possibly with suffixes, 
are used to denote sets of objects. If an element a is in the set A, we 
express this fact by the notation A a ? and read it as: "a belongs to A", "a 
is in A", or "a is member of A". An object in A is also called an element of 
A or a point in
1
A .  
3.1. Subset of a Set: 
A set A is said to be a subset of a set B , written as B A ? , if A x ? implies 
that B x ? . Two sets A and B are said to be equal, denoted B A ? , if B A ?
 
and A B ? . For example, if integers, of set ? A
 
and numbers real of set  the ? B , 
then B A ? , B is not a subset of A, and A and B are, of course, not equal, 
that is B A ? . A set A is said to be a proper subset of set B, denoted B A ?
or B A
?
? , if B A ?
 
and B A ? . 
Most often a set can be identified by a property that is common to all its 
elements. For example, 
 ? ? country a of capital a is :x x A ? 
 ? ? 4 by  divisible integer  posiitve a is :k k B ? . 
In the nineteenth century, a set was considered to be a collection of 
elements having some property P, and clearly is the basis of the notation :
? ? P x x property   the has : . Intuitively, it is difficult to see what possibly could go 
wrong with this. In 1901, however, Russel made the crucial observation 
that the above understanding of a set leads to a contradiction : Let A be 
the set of all sets x such that x is not a member of x, that is  
 ? ? x x x A ? ? : . 
From this we get the contradiction : A A ? if and only if A A ? . This 
contradiction is known as the Russel's paradox. Nevertheless, the notation 
? ? P x x property   the has : for a set is used extensively in mathematics . 
Sets and Relations 
Institute of Lifelong Learning, University of Delhi      pg. 5 
 
Consider the set 
 ? ? 4 an greater th and 3  than less is  that such integer  an is : x x x A ? . 
Since, there is no integer less than 3 and greater than 4, the set A does 
not contain any element. Such a set is called an empty set, and is usually 
denoted by F. Thus, an empty set F is a set which contains no element. 
An empty set is also called a null set. An empty set is unique, and 
therefore we speak of "the empty set". The empty set is a proper subset 
of every non-empty set. 
If a set contains only one element x, it is denoted by ? ? x and called a 
singleton.  
4. Union and Intersection of Sets: 
The sets are combined in several ways to obtain new sets. The operation 
of union " " ? applied to two sets 
1
A and 
2
A ,denoted 
2 1
A A ? , is defined as 
 ? ?
2 1 2 1
or : A x A x x A A ? ? ? ? . 
Similarly, if there are n sets
n
A A A ,..., ,
2 1
, their union, denoted 
?
n
i
i
A
1 ?
, is 
defined as 
 ? ? ? ? n i A x x A
i
n
i
i
,..., 2 , 1 some for  :
1
? ? ?
?
?
. 
The notation 
?
n
i
i
A
1 ?
 is an abbreviation for 
n
A A A ? ? ? ...
2 1
. The union of a 
finite number of sets is termed as "the finite union". 
The operation of intersection when applied to two sets produces the set of 
common elements in those two sets. Formally, if 
1
A and 
2
A are two sets, 
then the  intersection of 
1
A and 
2
A , denoted 
2 1
A A ? , is defined as  
 ? ?
2 1 2 1
 and : A x A x x A A ? ? ? ? . 
The symbol " " ? stands for the intersection. Similarly, if 
n
A A A ,..., ,
2 1
 are 
sets, their intersection, denoted 
?
n
i
i
A
1 ?
, is defined as