Page 28 - DMTH401_REAL ANALYSIS

Real Analysis Sachin Kaushal, Lovely Professional University

Notes Unit 2: Algebraic Structure and Countability

CONTENTS
Objectives
Introduction

2.1 Order Relations in Real Numbers
2.1.1 Intervals
2.1.2 Extended Real Numbers

2.2 Algebraic Structure
2.2.1 Ordered Field
2.2.2 Complete Ordered Field
2.3 Countability
2.3.1 Countable Sets

2.3.2 Countability of Real Numbers
2.4 Summary
2.5 Keywords

2.6 Review Questions
2.7 Further Readings

Objectives

After studying this unit, you will be able to:
 Discuss the order relation – extended real number system
 Explain the field structure of the set of real numbers

 Describe the order-completeness
 Discuss countability to various infinite sets
Introduction

In Unit 1 we have discussed the construction of real numbers from the rational numbers which,
in turn, were constructed from integers. In this unit, we show that the set of real numbers has an
additional property which the set of rational numbers does not have, namely it is a complete
ordered field. The questions, therefore, that arise are: What is a field? What is an ordered field?
What does it mean for an ordered field to be complete? In order to answer these questions we
need a few concepts and definitions, e.g., those of order inequalities and intervals in R. We shall
discuss these concepts. Also in this unit, we shall explain the extended real number system.
You know that a given set is either finite or infinite. In fact a set is finite, if it contains just n
elements where n is some natural number. A set which is not finite is called an infinite set. The
elements of a finite set can be counted as one, two, three and so on, while those of an infinite set
can not be counted in this way. Can you count the elements of the set of natural numbers? Try it.
We shall show that this notion of counting can be extended in certain sense to even infinite sets.

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