Lecture 4 - Sets and Functions | Algebra- Engineering Maths - Engineering Mathematics PDF Download

 Page 1


Sets and Functions 
Institute of Lifelong Learning, University of Delhi                                                     pg. 1 
 
 
 
 
 
 
 
 
 
Subject: Maths, Algebra-I 
Discipline Courses-1 
Semester-1 
Lesson : Sets and Functions 
Lesson Developer: Gurudatt Rao Ambedkar 
College/Department : A.N.D. College, Delhi University 
 
 
 
 
 
 
 
 
 
 
Page 2


Sets and Functions 
Institute of Lifelong Learning, University of Delhi                                                     pg. 1 
 
 
 
 
 
 
 
 
 
Subject: Maths, Algebra-I 
Discipline Courses-1 
Semester-1 
Lesson : Sets and Functions 
Lesson Developer: Gurudatt Rao Ambedkar 
College/Department : A.N.D. College, Delhi University 
 
 
 
 
 
 
 
 
 
 
Sets and Functions 
Institute of Lifelong Learning, University of Delhi                                                     pg. 2 
 
Table of Contents 
 Chapter : Set and Function 
? 1: Learning Outcomes 
? 2: Introduction Quantities 
? 3: Set 
? 3.1: Representation of sets 
? 3.2: Description of Sets 
? 4: Types of Sets 
o 4.1: Null set 
o 4.2: Equal sets 
o 4.3: Singleton set 
o 4.4: Sub-set 
o 4.5: Proper set 
o 4.6: Power set 
o 4.7: Universal set 
? 5: Venn-Euler Diagram 
? 6: Operation on sets 
o 6.1: Intersection of sets 
o 6.2: Union of sets 
o 6.3: Disjoint sets 
o 6.4: Difference of two sets 
o 6.5: Complement of a set 
o 6.6: Cartesian Product of sets 
? 7: Some useful results 
? 8: Function 
o 8.1: Domain of a function 
o 8.2: Range of a function 
? 9: Types of function  
? Summary 
? Exercises 
? References 
 
 
 
 
Page 3


Sets and Functions 
Institute of Lifelong Learning, University of Delhi                                                     pg. 1 
 
 
 
 
 
 
 
 
 
Subject: Maths, Algebra-I 
Discipline Courses-1 
Semester-1 
Lesson : Sets and Functions 
Lesson Developer: Gurudatt Rao Ambedkar 
College/Department : A.N.D. College, Delhi University 
 
 
 
 
 
 
 
 
 
 
Sets and Functions 
Institute of Lifelong Learning, University of Delhi                                                     pg. 2 
 
Table of Contents 
 Chapter : Set and Function 
? 1: Learning Outcomes 
? 2: Introduction Quantities 
? 3: Set 
? 3.1: Representation of sets 
? 3.2: Description of Sets 
? 4: Types of Sets 
o 4.1: Null set 
o 4.2: Equal sets 
o 4.3: Singleton set 
o 4.4: Sub-set 
o 4.5: Proper set 
o 4.6: Power set 
o 4.7: Universal set 
? 5: Venn-Euler Diagram 
? 6: Operation on sets 
o 6.1: Intersection of sets 
o 6.2: Union of sets 
o 6.3: Disjoint sets 
o 6.4: Difference of two sets 
o 6.5: Complement of a set 
o 6.6: Cartesian Product of sets 
? 7: Some useful results 
? 8: Function 
o 8.1: Domain of a function 
o 8.2: Range of a function 
? 9: Types of function  
? Summary 
? Exercises 
? References 
 
 
 
 
Sets and Functions 
Institute of Lifelong Learning, University of Delhi                                                     pg. 3 
 
1. Learning outcomes: 
After studying this chapter you should be able to 
? Understand the meaning of the term ‘Set’ 
? How to represent the sets 
? Distinguish between the different types of sets 
? Find the union, intersection, difference and complement of sets 
? Meaning of the term ‘function’  
? Understand different type of function 
? Plot different type of plots 
 
2. Introduction: 
The theory of sets was developed at the end of 19
th
 century. George Cantor 
(1845-1918), a German mathematician introduced the theory of sets which 
is now being used in many concepts of mathematics like sequences, 
probability etc. In this chapter we are presenting a brief idea about the 
theory of sets. 
Quantities – There are two kinds of Quantities: 
a) Constants   b)  Variables  
 
a) Constants –If any quantity does not change in mathematical operation 
then it is called constant. There are two types of Constants-  
i) Arbitrary constants 
ii) Absolute constants 
The constants remain unchanged in particular problems is called arbitrary 
constants; these are represented by . ,..., , , k c b a
 
The value of absolute 
constant remains fixed in all conditions; for example ? , 2 , 5 , 6 , 3 ? etc. are 
absolute constants.  
b) Variables – Variable are those quantities which are capable of assuming 
unlike values in a particular argument. These variables are represented 
by  etc w v u z y x , , , , , . Variables are of two types- 
i)   Dependent variables. 
ii)   Independent variables. 
Page 4


Sets and Functions 
Institute of Lifelong Learning, University of Delhi                                                     pg. 1 
 
 
 
 
 
 
 
 
 
Subject: Maths, Algebra-I 
Discipline Courses-1 
Semester-1 
Lesson : Sets and Functions 
Lesson Developer: Gurudatt Rao Ambedkar 
College/Department : A.N.D. College, Delhi University 
 
 
 
 
 
 
 
 
 
 
Sets and Functions 
Institute of Lifelong Learning, University of Delhi                                                     pg. 2 
 
Table of Contents 
 Chapter : Set and Function 
? 1: Learning Outcomes 
? 2: Introduction Quantities 
? 3: Set 
? 3.1: Representation of sets 
? 3.2: Description of Sets 
? 4: Types of Sets 
o 4.1: Null set 
o 4.2: Equal sets 
o 4.3: Singleton set 
o 4.4: Sub-set 
o 4.5: Proper set 
o 4.6: Power set 
o 4.7: Universal set 
? 5: Venn-Euler Diagram 
? 6: Operation on sets 
o 6.1: Intersection of sets 
o 6.2: Union of sets 
o 6.3: Disjoint sets 
o 6.4: Difference of two sets 
o 6.5: Complement of a set 
o 6.6: Cartesian Product of sets 
? 7: Some useful results 
? 8: Function 
o 8.1: Domain of a function 
o 8.2: Range of a function 
? 9: Types of function  
? Summary 
? Exercises 
? References 
 
 
 
 
Sets and Functions 
Institute of Lifelong Learning, University of Delhi                                                     pg. 3 
 
1. Learning outcomes: 
After studying this chapter you should be able to 
? Understand the meaning of the term ‘Set’ 
? How to represent the sets 
? Distinguish between the different types of sets 
? Find the union, intersection, difference and complement of sets 
? Meaning of the term ‘function’  
? Understand different type of function 
? Plot different type of plots 
 
2. Introduction: 
The theory of sets was developed at the end of 19
th
 century. George Cantor 
(1845-1918), a German mathematician introduced the theory of sets which 
is now being used in many concepts of mathematics like sequences, 
probability etc. In this chapter we are presenting a brief idea about the 
theory of sets. 
Quantities – There are two kinds of Quantities: 
a) Constants   b)  Variables  
 
a) Constants –If any quantity does not change in mathematical operation 
then it is called constant. There are two types of Constants-  
i) Arbitrary constants 
ii) Absolute constants 
The constants remain unchanged in particular problems is called arbitrary 
constants; these are represented by . ,..., , , k c b a
 
The value of absolute 
constant remains fixed in all conditions; for example ? , 2 , 5 , 6 , 3 ? etc. are 
absolute constants.  
b) Variables – Variable are those quantities which are capable of assuming 
unlike values in a particular argument. These variables are represented 
by  etc w v u z y x , , , , , . Variables are of two types- 
i)   Dependent variables. 
ii)   Independent variables. 
Sets and Functions 
Institute of Lifelong Learning, University of Delhi                                                     pg. 4 
 
Independent variables are those variables whose value can be changed 
independently and the dependent variables are those variables which 
depend on independent variables. 
For Example: Diameter of circle is d = 2r. Here diameter ‘d’ depends on the 
radius ‘r’ so r is the independent variable and d is dependent variable. 
3. Sets: 
“A well defined collection of distinct objects/things is called a Set.” 
We regularly speak the words which describes a particular category of 
objects like class, team, rivers etc. The adjective ‘well defined’ is most 
important which tells us that the object must have some definition. It is 
necessary to decide whether object belongs to a group or not. Students, 
players, numbers, alphabets, cities etc. are few examples of sets. 
Examples of Sets: 
? The cities of Uttar Pradesh. 
? States of India. 
? Solutions of the equations x
2
 – 4=0 i.e. 2 and -2. 
? Natural numbers N. 
? Letters in the word ALLAHABAD. 
Examples which are not set: 
?  The collection of all intelligent boys. 
?  The collection of all rich persons. 
3.1. Representation of Sets: 
Generally, we represent a set with capital letter (X, Y, Z etc) and the 
elements of sets i.e. object are denoted by small letters (a, b, c, etc). 
If   X = {a, b, c, 1, 2, 3} then we say that a, b, c, 1, 2, 3 are the elements 
of set X or element a belongs to the set X.  In mathematics we use Greek 
letter called epsilon, ? , means ‘belongs to’ to tell an element of a set. 
Page 5


Sets and Functions 
Institute of Lifelong Learning, University of Delhi                                                     pg. 1 
 
 
 
 
 
 
 
 
 
Subject: Maths, Algebra-I 
Discipline Courses-1 
Semester-1 
Lesson : Sets and Functions 
Lesson Developer: Gurudatt Rao Ambedkar 
College/Department : A.N.D. College, Delhi University 
 
 
 
 
 
 
 
 
 
 
Sets and Functions 
Institute of Lifelong Learning, University of Delhi                                                     pg. 2 
 
Table of Contents 
 Chapter : Set and Function 
? 1: Learning Outcomes 
? 2: Introduction Quantities 
? 3: Set 
? 3.1: Representation of sets 
? 3.2: Description of Sets 
? 4: Types of Sets 
o 4.1: Null set 
o 4.2: Equal sets 
o 4.3: Singleton set 
o 4.4: Sub-set 
o 4.5: Proper set 
o 4.6: Power set 
o 4.7: Universal set 
? 5: Venn-Euler Diagram 
? 6: Operation on sets 
o 6.1: Intersection of sets 
o 6.2: Union of sets 
o 6.3: Disjoint sets 
o 6.4: Difference of two sets 
o 6.5: Complement of a set 
o 6.6: Cartesian Product of sets 
? 7: Some useful results 
? 8: Function 
o 8.1: Domain of a function 
o 8.2: Range of a function 
? 9: Types of function  
? Summary 
? Exercises 
? References 
 
 
 
 
Sets and Functions 
Institute of Lifelong Learning, University of Delhi                                                     pg. 3 
 
1. Learning outcomes: 
After studying this chapter you should be able to 
? Understand the meaning of the term ‘Set’ 
? How to represent the sets 
? Distinguish between the different types of sets 
? Find the union, intersection, difference and complement of sets 
? Meaning of the term ‘function’  
? Understand different type of function 
? Plot different type of plots 
 
2. Introduction: 
The theory of sets was developed at the end of 19
th
 century. George Cantor 
(1845-1918), a German mathematician introduced the theory of sets which 
is now being used in many concepts of mathematics like sequences, 
probability etc. In this chapter we are presenting a brief idea about the 
theory of sets. 
Quantities – There are two kinds of Quantities: 
a) Constants   b)  Variables  
 
a) Constants –If any quantity does not change in mathematical operation 
then it is called constant. There are two types of Constants-  
i) Arbitrary constants 
ii) Absolute constants 
The constants remain unchanged in particular problems is called arbitrary 
constants; these are represented by . ,..., , , k c b a
 
The value of absolute 
constant remains fixed in all conditions; for example ? , 2 , 5 , 6 , 3 ? etc. are 
absolute constants.  
b) Variables – Variable are those quantities which are capable of assuming 
unlike values in a particular argument. These variables are represented 
by  etc w v u z y x , , , , , . Variables are of two types- 
i)   Dependent variables. 
ii)   Independent variables. 
Sets and Functions 
Institute of Lifelong Learning, University of Delhi                                                     pg. 4 
 
Independent variables are those variables whose value can be changed 
independently and the dependent variables are those variables which 
depend on independent variables. 
For Example: Diameter of circle is d = 2r. Here diameter ‘d’ depends on the 
radius ‘r’ so r is the independent variable and d is dependent variable. 
3. Sets: 
“A well defined collection of distinct objects/things is called a Set.” 
We regularly speak the words which describes a particular category of 
objects like class, team, rivers etc. The adjective ‘well defined’ is most 
important which tells us that the object must have some definition. It is 
necessary to decide whether object belongs to a group or not. Students, 
players, numbers, alphabets, cities etc. are few examples of sets. 
Examples of Sets: 
? The cities of Uttar Pradesh. 
? States of India. 
? Solutions of the equations x
2
 – 4=0 i.e. 2 and -2. 
? Natural numbers N. 
? Letters in the word ALLAHABAD. 
Examples which are not set: 
?  The collection of all intelligent boys. 
?  The collection of all rich persons. 
3.1. Representation of Sets: 
Generally, we represent a set with capital letter (X, Y, Z etc) and the 
elements of sets i.e. object are denoted by small letters (a, b, c, etc). 
If   X = {a, b, c, 1, 2, 3} then we say that a, b, c, 1, 2, 3 are the elements 
of set X or element a belongs to the set X.  In mathematics we use Greek 
letter called epsilon, ? , means ‘belongs to’ to tell an element of a set. 
Sets and Functions 
Institute of Lifelong Learning, University of Delhi                                                     pg. 5 
 
For Example:  b ? X i.e. b is an element of the set X. 
           b ? X i.e. b is not an element of the set X. 
3.2. Description of Sets: 
There are two ways to describe or specify the elements of a set: 
a) Roster method/ Tabular method: We list all the members of a set 
separated by commas. The list of members should be enclosed in curly 
bracket. 
e.g., X = {1, 2, 3, 4, 5} 
         Y= {a, l, h, b, d} 
b) Set builder method or rule method: We use a rule or definition to 
describe all the members of a set. 
e.g., X is a set whose elements are the first five natural numbers or X= 
{x : x ? N and   x =5}.  In this notation, the colon (":") means "such 
that", 
    Y is a set whose elements are the letters used in ALLAHABAD 
 
Value addition:  Do you Know? 
? Note 1:   Two element of a set may not be identical,  Every element of a 
set must be unique;  
{a, b} = {b, a} = {b, a, a, b, a} 
? Note 2:  The enumeration of elements can be abbreviated for sets with 
many elements. For example the set of all positive integer may be 
specified by tabular method as: N = {1, 2, 3, _ _ _}. 
 
4. Types of Sets: 
 
4.1. Null Set: 
A set with no element is call null set or void set or empty set. It is denoted 
by standard notation Ø i.e. Ø = { } 
For Example:  Ø = The set of countries in India