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Sets, Relations and Graphing Functions, Study Guides, Projects, Research of Mathematics

Bharathiar UniversityMathematics

Recalling the concepts of sets and relations, student understand graphing the functions

Typology: Study Guides, Projects, Research

2017/2018

Uploaded on 09/18/2018

veeraragavan-subrama
veeraragavan-subrama 🇮🇳

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2. SETS, RELATIONS AND FUNCTIONS
2.1 Introduction....................................................................................................................................2
2.2 Sets – Recalling............................................................................................................................. 2
2.3 Cartesian Product – Recalling........................................................................................................6
2.4 Constants and Variables, Intervals and Neigbourhoods...............................................................12
2.4.1 Constants and Variables...........................................................................................13
2.4.2 Intervals and Neigbourhoods...................................................................................13
2.5 Relations..................................................................................................................................... 15
2.5.1 Type of relations...................................................................................................... 21
2.6 Functions......................................................................................................................................30
2.6.1 Some elementary functions......................................................................................34
2.6.2 Types of Functions:..................................................................................................36
2.6.3 Operations on functions...........................................................................................45
2.6.4 Inverse of a function................................................................................................48
2.6.5 Algebra of Functions................................................................................................50
2.6.6 Some Special Functions...........................................................................................52
2.7 Graphing functions using transformations:..................................................................................56
Page 1 of 62
2. SETS , RELATIONS AND FUNCTIONS
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2. SETS, RELATIONS AND FUNCTIONS

  • 2.1 Introduction....................................................................................................................................
  • 2.2 Sets – Recalling.............................................................................................................................
  • 2.3 Cartesian Product – Recalling........................................................................................................
  • 2.4 Constants and Variables, Intervals and Neigbourhoods...............................................................
    • 2.4.1 Constants and Variables...........................................................................................
    • 2.4.2 Intervals and Neigbourhoods...................................................................................
  • 2.5 Relations.....................................................................................................................................
    • 2.5.1 Type of relations......................................................................................................
  • 2.6 Functions......................................................................................................................................
    • 2.6.1 Some elementary functions......................................................................................
    • 2.6.2 Types of Functions:..................................................................................................
    • 2.6.3 Operations on functions...........................................................................................
    • 2.6.4 Inverse of a function................................................................................................
    • 2.6.5 Algebra of Functions................................................................................................
    • 2.6.6 Some Special Functions...........................................................................................
  • 2.7 Graphing functions using transformations:.................................................................................. - Page 1 of

2. SETS, RELATIONS AND FUNCTIONS

(DRAFT MATERIAL)

2.1 Introduction

In the previous classes, students have studied and are well versed with the real numbers and their arithmetic operations. Even though students have learned about sets of real numbers, Venn diagrams, cartesian product of sets for the continuation of better understanding, we will recall more about sets and cartesian products of sets. Relation between these sets of real numbers and some of their applications in the form of “Relations” and “Functions” were dealt with already in previous standards. However we give a new facelift to the mathematical notion of “Relations” and “Functions”.

Nicole d’Orsene ( ~ 1320 – 1382), philosopher, mathematician, a counsellor of King Charles V of France, one of the most original thinkers of 14th^ century, records some general

ideas about independent and dependent variable quantities, the first ever record of relation or function found in early history. Notion of function implicit in trigonometric and logarithmic tables, but as a mathematical term “function” was coined in 1692 by Gottfried Von Leibnitz (1646 – 1716), a German mathematician and philosopher. The most prolific mathematician, Leonard Euler (1707 – 1783) was the first scientist to give the concept of a function. Johann Peter Gustav Lejeune Dirichlet (1805 – 1859) was a German mathematician credited with the modern “formal” definition of function with notation y = f(x). He was a student of Gauss. After Gauss, in 1855, he was appointed as Gauss’ successor at Gottingen. The idea evolved for close to 200 years in intimate connection with problems in Calculus and Analysis. The concept of function is one of the distinguishing features of modern against classical mathematics.

To know about the concept of functions, it is necessary to recall the concepts of sets, cartesian product of sets and relations.

Page 1 of 62

  1. SETS , RELATIONS AND FUNCTIONS

and B, if A ⊆ B and B ⊆ A, then the two sets are equal. For any set A, the empty set ∅ and the set A are subsets of A. These two subsets are called trivial subsets. Further, we say A is a proper subset of B if A is a subset of B and A ≠ B. That is, B contains at least one element which is not in A. As already noted, any set is a subset of itself. This subset is called an improper subset. In other words, for any set A, A is the improper subset of A. Note that, N⊂ W ⊂ Z ⊂ Q ⊂ R , where N denotes set of all Natural Numbers or positive integers, W denotes set of all non–negative integers, Z denotes set of all integers, Q denotes set of all rational numbers and R denotes set of all real numbers. Note that the set of all irrational numbers is a subset of R but not a subset of any other set denoted above.

Already we learned that the union of two sets A and B is denoted by A ∪ B and defined as A ∪ B = {x : x ∈ A or x ∈ B } and the intersection of two sets A and B is A ∩ B = {x : x ∈ A and x ∈ B}.