Page 7 - DMTH401_REAL ANALYSIS

Sachin Kaushal, Lovely Professional University Unit 1: Sets and Numbers

Unit 1: Sets and Numbers Notes

CONTENTS
Objectives
Introduction
1.1 Sets and Functions
1.1.1 Sets

1.1.2 Functions
1.2 System of Real Numbers
1.2.1 Natural Numbers
1.2.2 Integers
1.2.3 Rational Numbers
1.3 The Real Line
1.4 Complex Numbers

1.5 Mathematical Induction
1.6 Summary
1.7 Keywords
1.8 Review Questions
1.9 Further Readings

Objectives

After studying this unit, you will be able to:

 Discuss the basic concepts of sets and functions
 Explain the system of real numbers
 Describe the representation of real numbers

Introduction

As we know that one of the main features of Mathematics is the identification of the subject
matter, its analysis and its presentation in a satisfactory manner. In other words, the language
should be a vehicle which carries ideas through the mind without affecting their meaning in any
way. Set Theory comes closest to being such a language. Introduced between 1873 and 1895 by a
famous German mathematician, George Cantor (1845-1918), Set Theory became the foundation
of almost all the branches of Mathematics. Besides its universal appeal, it is quite amazing in its
simplicity and elegance.
A rigorous presentation of Set Theory is not the purpose here because we believe that you are
already familiar with it. We shall briefly recall some of its basic concepts which are essential for
a systematic study of Real Analysis. Closely linked with the sets, is the notion of a function,
which also you have learnt in your previous studies. In this unit, we shall review this as well as
other related concepts which are necessary for our discussion.

LOVELY PROFESSIONAL UNIVERSITY 1