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 IndiaRead More

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