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# Lecture 4 - Sets and Functions Engineering Mathematics Notes | EduRev

## Engineering Mathematics : Lecture 4 - Sets and Functions Engineering Mathematics Notes | EduRev

``` 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

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

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

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

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

? 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
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