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 .Read More

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