Lecture 1 - Real Numbers, Functions - Real Numbers, Functions and Sequences Notes | EduRev

Created by: Bhuvan Midha

: Lecture 1 - Real Numbers, Functions - Real Numbers, Functions and Sequences Notes | EduRev

 Page 1


Module 1 : Real Numbers, Functions and Sequences
Lecture 1 : Real Numbers, Functions    [ Section 1.1 : Real Numbers ]
   Objectives
   In this section you will learn the following
Axiomatic definition of real numbers.
Properties of real numbers.
 
1.1.1 The Real Numbers :
        Real Numbers are the elements of a set, denoted by , with the following properties:
    1) Algebraic properties of real numbers:
 
There are two binary operations defined on , one called addition, denoted by  , and the
other called multiplication, denoted by  , with the usual algebric properties: for all 
.
.
.
 There exist two distinct elements in , denoted by 0 and 1, with following properties:
 
0 +  =     for all      ;     1 =    for all  0   .
The elements 0, read as zero, is called the additive identity and 1, read as one, is called the multiplicative
identity.
For every   , there exists unique element     such that    + (- ) = 0 ; for   0 in ,
there exists unique element    such that  = 1.
 
     2)  Order properties of real numbers:
        There exists an order, denoted by <, between the elements of  with the following properties:
For  , one and only one of the following relations hold : .
Page 2


Module 1 : Real Numbers, Functions and Sequences
Lecture 1 : Real Numbers, Functions    [ Section 1.1 : Real Numbers ]
   Objectives
   In this section you will learn the following
Axiomatic definition of real numbers.
Properties of real numbers.
 
1.1.1 The Real Numbers :
        Real Numbers are the elements of a set, denoted by , with the following properties:
    1) Algebraic properties of real numbers:
 
There are two binary operations defined on , one called addition, denoted by  , and the
other called multiplication, denoted by  , with the usual algebric properties: for all 
.
.
.
 There exist two distinct elements in , denoted by 0 and 1, with following properties:
 
0 +  =     for all      ;     1 =    for all  0   .
The elements 0, read as zero, is called the additive identity and 1, read as one, is called the multiplicative
identity.
For every   , there exists unique element     such that    + (- ) = 0 ; for   0 in ,
there exists unique element    such that  = 1.
 
     2)  Order properties of real numbers:
        There exists an order, denoted by <, between the elements of  with the following properties:
For  , one and only one of the following relations hold : .
.
 There are two more properties that real numbers have which we shall describe later :
 
 
3)
Archimedean property
4) Completeness property
 
Geometrically, set of all points on a line represent the set of all real numbers. There are some special subsets
of  which are important. These are the familiar number systems.
1.1.2 , the set of Natural Numbers:
Recall that, there exist unique elements 0, 1  such that , for x   and , for all 0 
 x  .One can show that . The set  is the 'smallest' subset of  having the property : and
, whenever n . This is also called the Principle of Mathematical Induction. One can
show that such a subset of  exists, and is unique. Elements of  are called natural numbers. We
shall use the familiar notation,  = {1,2,....,}. Geometrically, we can select any arbitrary point O on the
real line and associate it with 0 . Equidistant points on the right of O can be labeled as     
                                                   
     The set  has the following properties, which we shall assume: 
 for all .
For every .
For every , there is no element m such that .
 
 
Archimedean property
 
For every , there exists such that .
  
1.1.3 Definition:
(i) A subset    is said to be bounded above if there exists  such that r    for all r  E. That is,
all the elements of E lie to the left of s, up to s at most.                                   
 
                                     
(ii) Similarly, we say  is bounded below if there exists t  such that t r  for all r .
 
                                     
(iii)
A set  is said to be bounded if it is both bounded above and below, i.e., there exist s, t  such that t  r 
 s  for every r . 
 
                                     
1.1.4 Example :
 
 is bounded below by 1. In fact, every    is bounded below. Archimedian property says that is not
 bounded  above.
  
Page 3


Module 1 : Real Numbers, Functions and Sequences
Lecture 1 : Real Numbers, Functions    [ Section 1.1 : Real Numbers ]
   Objectives
   In this section you will learn the following
Axiomatic definition of real numbers.
Properties of real numbers.
 
1.1.1 The Real Numbers :
        Real Numbers are the elements of a set, denoted by , with the following properties:
    1) Algebraic properties of real numbers:
 
There are two binary operations defined on , one called addition, denoted by  , and the
other called multiplication, denoted by  , with the usual algebric properties: for all 
.
.
.
 There exist two distinct elements in , denoted by 0 and 1, with following properties:
 
0 +  =     for all      ;     1 =    for all  0   .
The elements 0, read as zero, is called the additive identity and 1, read as one, is called the multiplicative
identity.
For every   , there exists unique element     such that    + (- ) = 0 ; for   0 in ,
there exists unique element    such that  = 1.
 
     2)  Order properties of real numbers:
        There exists an order, denoted by <, between the elements of  with the following properties:
For  , one and only one of the following relations hold : .
.
 There are two more properties that real numbers have which we shall describe later :
 
 
3)
Archimedean property
4) Completeness property
 
Geometrically, set of all points on a line represent the set of all real numbers. There are some special subsets
of  which are important. These are the familiar number systems.
1.1.2 , the set of Natural Numbers:
Recall that, there exist unique elements 0, 1  such that , for x   and , for all 0 
 x  .One can show that . The set  is the 'smallest' subset of  having the property : and
, whenever n . This is also called the Principle of Mathematical Induction. One can
show that such a subset of  exists, and is unique. Elements of  are called natural numbers. We
shall use the familiar notation,  = {1,2,....,}. Geometrically, we can select any arbitrary point O on the
real line and associate it with 0 . Equidistant points on the right of O can be labeled as     
                                                   
     The set  has the following properties, which we shall assume: 
 for all .
For every .
For every , there is no element m such that .
 
 
Archimedean property
 
For every , there exists such that .
  
1.1.3 Definition:
(i) A subset    is said to be bounded above if there exists  such that r    for all r  E. That is,
all the elements of E lie to the left of s, up to s at most.                                   
 
                                     
(ii) Similarly, we say  is bounded below if there exists t  such that t r  for all r .
 
                                     
(iii)
A set  is said to be bounded if it is both bounded above and below, i.e., there exist s, t  such that t  r 
 s  for every r . 
 
                                     
1.1.4 Example :
 
 is bounded below by 1. In fact, every    is bounded below. Archimedian property says that is not
 bounded  above.
  
1.1.5   , the set of Integers :
For every , let  be the unique element of  such that . Let 
                                    : = { ..., -2, -1, 0, 1, 2, ... } 
Elements of  are called integers. Clearly,  is neither bounded above nor bounded below.
1.1.6 , the set of rational numbers :
 
For every  , let   be such that  . The element  is also denoted by . Let 
                                       : =  
The set is called the set of rational numbers and the elements of the set \ are called the irrational
numbers.
Both, the rational and the irrational numbers have the following denseness property :
1.1.7 Denseness of rational and the irrational numbers:
 
For every real numbers x and y, with ,there exist a rational and an irrational such that 
and
.
1.1.8  Note: Why real numbers?
 
At this stage one can ask the following questions: What is the need to work with real numbers? Can one
not work always with rational numbers? How real numbers are different from rational numbers? You will
see answer to some of these questions in this course. Hopefully, you would have realized by now that
arithmetic is necessary for day-to-day life.
 
Also, you would have seen (in your school courses) that there does not exist any rational r such that  =
2. (This was discovered by the Greek mathematicians in 500 B.C.) This is one of the reasons why
mathematicians were forced to invent a set of numbers which is 'bigger' than that of rationals, and which
satisfy equations of the type  for all . The property of the real numbers that distinguishes
them from the rational numbers, is called the completeness property, which we shall discuss in section 1.6
of lecture 3. Geometrically, rational numbers when represented by points on the line, do not cover every
point of the line. For example, the point Q on the line such that OQ is equal to OA, the length of the
diagonal of a square with unit length, does not correspond to any rational number.
 
 
Thus, some gaps are left when rational numbers are represented as points on a horizontal line. Filling up
these gaps is the "completeness property" of the real numbers. We will make it mathematically precise in
the next section. These gaps are the irrational numbers.
1.1.9Intervals :
 
We describe next another important class of subsets of  , called intervals. For a, b  with , we
write
 ( a, b ) : =  { x    | a  < x  <  b }, [ a , b ]  : =  { x   | a   x  b },
 
 ( a, b ] : =  { x   | a < x  b }, [ a, b )  : =  { x   | a   x  <  b },                        
 ( a, + ) : =  { x   | a <  x }, [ a, + )  : =  { x   |  a    x },
 
  ( -  , a ] : =  { x   | a  x }, ( - , a )  : =  { x   |  a  >  x },
 
When a  = b, interval ( a, a ) : =  and [ a,  a ]  : =  { a }.  Intervals of the type (a ,b ] are called left-open
right-closed intervals.
 
Simililarly, intervals of the type [ a,  b )  are called left-closed right-open Intervals. And intervals of the type  
( a, b ), ( - , a ), ( a, + ) are called open intervals and of the type [ a,b ], ( - ,a ], ( a, + } are called
closed intervals. Note that  - ,  +  are just symbols, and not numbers. 
Page 4


Module 1 : Real Numbers, Functions and Sequences
Lecture 1 : Real Numbers, Functions    [ Section 1.1 : Real Numbers ]
   Objectives
   In this section you will learn the following
Axiomatic definition of real numbers.
Properties of real numbers.
 
1.1.1 The Real Numbers :
        Real Numbers are the elements of a set, denoted by , with the following properties:
    1) Algebraic properties of real numbers:
 
There are two binary operations defined on , one called addition, denoted by  , and the
other called multiplication, denoted by  , with the usual algebric properties: for all 
.
.
.
 There exist two distinct elements in , denoted by 0 and 1, with following properties:
 
0 +  =     for all      ;     1 =    for all  0   .
The elements 0, read as zero, is called the additive identity and 1, read as one, is called the multiplicative
identity.
For every   , there exists unique element     such that    + (- ) = 0 ; for   0 in ,
there exists unique element    such that  = 1.
 
     2)  Order properties of real numbers:
        There exists an order, denoted by <, between the elements of  with the following properties:
For  , one and only one of the following relations hold : .
.
 There are two more properties that real numbers have which we shall describe later :
 
 
3)
Archimedean property
4) Completeness property
 
Geometrically, set of all points on a line represent the set of all real numbers. There are some special subsets
of  which are important. These are the familiar number systems.
1.1.2 , the set of Natural Numbers:
Recall that, there exist unique elements 0, 1  such that , for x   and , for all 0 
 x  .One can show that . The set  is the 'smallest' subset of  having the property : and
, whenever n . This is also called the Principle of Mathematical Induction. One can
show that such a subset of  exists, and is unique. Elements of  are called natural numbers. We
shall use the familiar notation,  = {1,2,....,}. Geometrically, we can select any arbitrary point O on the
real line and associate it with 0 . Equidistant points on the right of O can be labeled as     
                                                   
     The set  has the following properties, which we shall assume: 
 for all .
For every .
For every , there is no element m such that .
 
 
Archimedean property
 
For every , there exists such that .
  
1.1.3 Definition:
(i) A subset    is said to be bounded above if there exists  such that r    for all r  E. That is,
all the elements of E lie to the left of s, up to s at most.                                   
 
                                     
(ii) Similarly, we say  is bounded below if there exists t  such that t r  for all r .
 
                                     
(iii)
A set  is said to be bounded if it is both bounded above and below, i.e., there exist s, t  such that t  r 
 s  for every r . 
 
                                     
1.1.4 Example :
 
 is bounded below by 1. In fact, every    is bounded below. Archimedian property says that is not
 bounded  above.
  
1.1.5   , the set of Integers :
For every , let  be the unique element of  such that . Let 
                                    : = { ..., -2, -1, 0, 1, 2, ... } 
Elements of  are called integers. Clearly,  is neither bounded above nor bounded below.
1.1.6 , the set of rational numbers :
 
For every  , let   be such that  . The element  is also denoted by . Let 
                                       : =  
The set is called the set of rational numbers and the elements of the set \ are called the irrational
numbers.
Both, the rational and the irrational numbers have the following denseness property :
1.1.7 Denseness of rational and the irrational numbers:
 
For every real numbers x and y, with ,there exist a rational and an irrational such that 
and
.
1.1.8  Note: Why real numbers?
 
At this stage one can ask the following questions: What is the need to work with real numbers? Can one
not work always with rational numbers? How real numbers are different from rational numbers? You will
see answer to some of these questions in this course. Hopefully, you would have realized by now that
arithmetic is necessary for day-to-day life.
 
Also, you would have seen (in your school courses) that there does not exist any rational r such that  =
2. (This was discovered by the Greek mathematicians in 500 B.C.) This is one of the reasons why
mathematicians were forced to invent a set of numbers which is 'bigger' than that of rationals, and which
satisfy equations of the type  for all . The property of the real numbers that distinguishes
them from the rational numbers, is called the completeness property, which we shall discuss in section 1.6
of lecture 3. Geometrically, rational numbers when represented by points on the line, do not cover every
point of the line. For example, the point Q on the line such that OQ is equal to OA, the length of the
diagonal of a square with unit length, does not correspond to any rational number.
 
 
Thus, some gaps are left when rational numbers are represented as points on a horizontal line. Filling up
these gaps is the "completeness property" of the real numbers. We will make it mathematically precise in
the next section. These gaps are the irrational numbers.
1.1.9Intervals :
 
We describe next another important class of subsets of  , called intervals. For a, b  with , we
write
 ( a, b ) : =  { x    | a  < x  <  b }, [ a , b ]  : =  { x   | a   x  b },
 
 ( a, b ] : =  { x   | a < x  b }, [ a, b )  : =  { x   | a   x  <  b },                        
 ( a, + ) : =  { x   | a <  x }, [ a, + )  : =  { x   |  a    x },
 
  ( -  , a ] : =  { x   | a  x }, ( - , a )  : =  { x   |  a  >  x },
 
When a  = b, interval ( a, a ) : =  and [ a,  a ]  : =  { a }.  Intervals of the type (a ,b ] are called left-open
right-closed intervals.
 
Simililarly, intervals of the type [ a,  b )  are called left-closed right-open Intervals. And intervals of the type  
( a, b ), ( - , a ), ( a, + ) are called open intervals and of the type [ a,b ], ( - ,a ], ( a, + } are called
closed intervals. Note that  - ,  +  are just symbols, and not numbers. 
 
As stated above, shall assume that the set of real numbers can be identified with points on the straight line.
If point O represent the number 0, then points on the left of O represent negative real numbers and points on
the right of O represent positive real numbers. Intervals are part of the line as shown:                    
   
 
 
This identification is useful in visualizing various properties of real numbers. An open interval of the type 
( ) is called an  -  neighborhood of a  .
                                                                                       
        For Quiz refer the WebSite
 
         Practice Exercises 1.1 :  Real Numbers
(1) Using the Principle of Mathematical Induction, prove the following for all n  :
(i) n  > 1  implies  n - 1   .
(ii)
For x     with  x  >  0,  if  x  +  n   ,  then  x   .
(iii)
 m  +  n,  m n     for  all   m,n  .
 
(2) Using the Principle of induction prove the following:
(i) If    then  and  .
(ii)
 If     and  ,  then
                                                for all  .
(3) Show that the product of any two even (odd) integers is also an even (odd) integer.
(4) Find an irrational number between 3 and 4.     
   Recap
   In this section you have learnt the following :
Axiomatic definition of real numbers and the construction of other number systems.
The algebraic and order properties of real numbers.
Page 5


Module 1 : Real Numbers, Functions and Sequences
Lecture 1 : Real Numbers, Functions    [ Section 1.1 : Real Numbers ]
   Objectives
   In this section you will learn the following
Axiomatic definition of real numbers.
Properties of real numbers.
 
1.1.1 The Real Numbers :
        Real Numbers are the elements of a set, denoted by , with the following properties:
    1) Algebraic properties of real numbers:
 
There are two binary operations defined on , one called addition, denoted by  , and the
other called multiplication, denoted by  , with the usual algebric properties: for all 
.
.
.
 There exist two distinct elements in , denoted by 0 and 1, with following properties:
 
0 +  =     for all      ;     1 =    for all  0   .
The elements 0, read as zero, is called the additive identity and 1, read as one, is called the multiplicative
identity.
For every   , there exists unique element     such that    + (- ) = 0 ; for   0 in ,
there exists unique element    such that  = 1.
 
     2)  Order properties of real numbers:
        There exists an order, denoted by <, between the elements of  with the following properties:
For  , one and only one of the following relations hold : .
.
 There are two more properties that real numbers have which we shall describe later :
 
 
3)
Archimedean property
4) Completeness property
 
Geometrically, set of all points on a line represent the set of all real numbers. There are some special subsets
of  which are important. These are the familiar number systems.
1.1.2 , the set of Natural Numbers:
Recall that, there exist unique elements 0, 1  such that , for x   and , for all 0 
 x  .One can show that . The set  is the 'smallest' subset of  having the property : and
, whenever n . This is also called the Principle of Mathematical Induction. One can
show that such a subset of  exists, and is unique. Elements of  are called natural numbers. We
shall use the familiar notation,  = {1,2,....,}. Geometrically, we can select any arbitrary point O on the
real line and associate it with 0 . Equidistant points on the right of O can be labeled as     
                                                   
     The set  has the following properties, which we shall assume: 
 for all .
For every .
For every , there is no element m such that .
 
 
Archimedean property
 
For every , there exists such that .
  
1.1.3 Definition:
(i) A subset    is said to be bounded above if there exists  such that r    for all r  E. That is,
all the elements of E lie to the left of s, up to s at most.                                   
 
                                     
(ii) Similarly, we say  is bounded below if there exists t  such that t r  for all r .
 
                                     
(iii)
A set  is said to be bounded if it is both bounded above and below, i.e., there exist s, t  such that t  r 
 s  for every r . 
 
                                     
1.1.4 Example :
 
 is bounded below by 1. In fact, every    is bounded below. Archimedian property says that is not
 bounded  above.
  
1.1.5   , the set of Integers :
For every , let  be the unique element of  such that . Let 
                                    : = { ..., -2, -1, 0, 1, 2, ... } 
Elements of  are called integers. Clearly,  is neither bounded above nor bounded below.
1.1.6 , the set of rational numbers :
 
For every  , let   be such that  . The element  is also denoted by . Let 
                                       : =  
The set is called the set of rational numbers and the elements of the set \ are called the irrational
numbers.
Both, the rational and the irrational numbers have the following denseness property :
1.1.7 Denseness of rational and the irrational numbers:
 
For every real numbers x and y, with ,there exist a rational and an irrational such that 
and
.
1.1.8  Note: Why real numbers?
 
At this stage one can ask the following questions: What is the need to work with real numbers? Can one
not work always with rational numbers? How real numbers are different from rational numbers? You will
see answer to some of these questions in this course. Hopefully, you would have realized by now that
arithmetic is necessary for day-to-day life.
 
Also, you would have seen (in your school courses) that there does not exist any rational r such that  =
2. (This was discovered by the Greek mathematicians in 500 B.C.) This is one of the reasons why
mathematicians were forced to invent a set of numbers which is 'bigger' than that of rationals, and which
satisfy equations of the type  for all . The property of the real numbers that distinguishes
them from the rational numbers, is called the completeness property, which we shall discuss in section 1.6
of lecture 3. Geometrically, rational numbers when represented by points on the line, do not cover every
point of the line. For example, the point Q on the line such that OQ is equal to OA, the length of the
diagonal of a square with unit length, does not correspond to any rational number.
 
 
Thus, some gaps are left when rational numbers are represented as points on a horizontal line. Filling up
these gaps is the "completeness property" of the real numbers. We will make it mathematically precise in
the next section. These gaps are the irrational numbers.
1.1.9Intervals :
 
We describe next another important class of subsets of  , called intervals. For a, b  with , we
write
 ( a, b ) : =  { x    | a  < x  <  b }, [ a , b ]  : =  { x   | a   x  b },
 
 ( a, b ] : =  { x   | a < x  b }, [ a, b )  : =  { x   | a   x  <  b },                        
 ( a, + ) : =  { x   | a <  x }, [ a, + )  : =  { x   |  a    x },
 
  ( -  , a ] : =  { x   | a  x }, ( - , a )  : =  { x   |  a  >  x },
 
When a  = b, interval ( a, a ) : =  and [ a,  a ]  : =  { a }.  Intervals of the type (a ,b ] are called left-open
right-closed intervals.
 
Simililarly, intervals of the type [ a,  b )  are called left-closed right-open Intervals. And intervals of the type  
( a, b ), ( - , a ), ( a, + ) are called open intervals and of the type [ a,b ], ( - ,a ], ( a, + } are called
closed intervals. Note that  - ,  +  are just symbols, and not numbers. 
 
As stated above, shall assume that the set of real numbers can be identified with points on the straight line.
If point O represent the number 0, then points on the left of O represent negative real numbers and points on
the right of O represent positive real numbers. Intervals are part of the line as shown:                    
   
 
 
This identification is useful in visualizing various properties of real numbers. An open interval of the type 
( ) is called an  -  neighborhood of a  .
                                                                                       
        For Quiz refer the WebSite
 
         Practice Exercises 1.1 :  Real Numbers
(1) Using the Principle of Mathematical Induction, prove the following for all n  :
(i) n  > 1  implies  n - 1   .
(ii)
For x     with  x  >  0,  if  x  +  n   ,  then  x   .
(iii)
 m  +  n,  m n     for  all   m,n  .
 
(2) Using the Principle of induction prove the following:
(i) If    then  and  .
(ii)
 If     and  ,  then
                                                for all  .
(3) Show that the product of any two even (odd) integers is also an even (odd) integer.
(4) Find an irrational number between 3 and 4.     
   Recap
   In this section you have learnt the following :
Axiomatic definition of real numbers and the construction of other number systems.
The algebraic and order properties of real numbers.
The Archemedian property of real numbers.
 
 
  [ Section 1.2 : Functions ]
   Objectives
   In this section you will learn the following
Concept of a function.
The absolute value function.
Concept of a bijective function.
 
 
1.2     FUNCTIONS :
 
You already have some familiarity with the concept of a function. Function is a kind relation
between various objects. For example, the volume  of a cube is a function of its side  ; in
physics velocity of a body at any time is a function of its initial velocity and time, and
acceleration; and so on. In mathematics, a function is defined as follows:
  
1.2.1  
Definition :
(i)
For sets    and   , a  function from    to    , denoted by  ,  is a correspondence which
assigns to every element     ,  a unique element  ( )   . The value of the function at an
element in  is denoted by  (  ), which is an element in . This is indicated by  . 
(ii)
For a function  , the set    is called the domain of     and the subset 
of , (set of images of  ) is called the range of  .  If    , then    is said to be real-valued. If
also , then the natural domain of    is the set of all   for which .
 
1.2.2  
Examples : 
(i) 
Let . Then,   has natural domain . Its range is also , because for any given , 
if   then we get  .
(ii)
Let . Then,   has natural domain , and its range is given by .
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