Elementary Set Theory, Part I/II

Cover

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Title Page, Copyright

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FOREWORD*

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pp. v-vi

The most striking characteristic of modern mathematics is its greater unity and generality. In modern mathematics, the boundaries between different areas have become obscured; very often, what used to be separate and unrelated disciplines...

PREFACE.

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pp. vii-

Elementary Set Theory is an cxtension of the lecture notes for the course ‘Fundamental Concepts of Mathematics' given each year to first-year undergraduate students of mathematics in the U niversity of Hong Kong since 1959...

CONTENTS

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pp. ix-x

PART 1

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pp. 1-

Chapter 1. Statement Calculus

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pp. 3-21

By a statement (or a proposition or a declarative sentence) we understand a sentcnce of which it is meaningful to say that its content is true or false. Obviously, each of the following sentcnces is a statement: Geography is a science. Confucius was a soldier. Cheung Sam...

Chapter 2. Sets

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pp. 22-42

A fundamental concept in rnathernatics is that of a set. This concept can be used as a foundation of all known rnathernatics. ln this and thc following chapters, we shall develop some of the basic properties of sets. In set theory, we shall be dealing with sets of objects. Here we take οbjects to be simply the...

Chapter 3. Relations

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pp. 43-52

We have seen in Section 2 E that, given any two objects x and y , there is a set {x,y} which has x and y as its only elements. Moreover, {x,y} = {y,x}; in other words, the order in which the objects x and y appear is immaterial to the construction of the set {x,y}. For this reason the set {x ,y} is called an unordered pair. Let us recall a well-known technique...

Chapter 4. Mappings

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pp. 53-64

Most readcrs are familiar with the graphical concept of functions. This involves in general a set A of objects called arguments, a set B of objects called values and an act of associating with each argument in A a unique value in B. In elementary...

PART II

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pp. 65-

Chapter 5. Families

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pp. 67-76

In mathematics, for the sake of convenient formulation and easy reference, we very often introduce subscripts, superscripts and the like to index the objects (e.g. points, lines, indeterminates, etc.) of our discussion. The indices are usually numbers or letters. More generally, given two sets A and...

Chapter 6. Natural Numbers

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pp. 77-90

What is a natural number? We are all familiar with the words ‘zero', ‘one', ‘two', etc., but do we know exactly what objects have these names? In this section, we shall try to answer these questions; in other words, we shall give a definition of natural numbers. As we already know something about some objects...

Chapter 7. Finite and Infinite Sets

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pp. 91-101

In this chapter we shall be mainly concerned with the problem of making comparisons between sets. So far our comparisons of any two sets have concerned whether or not one is a subset of the other; in other words, whether or not there exists an identity mapping of one set onto a subset of the other set. This also...

Chapter 8. Ordered Sets

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pp. 102-116

The concept of order in elementary mathematics and in daily life is so familiar to everybody that a motivation seems hardly to be necessary here. In fact we have discussed at some Iength the usual order relation of natural numbers. In this chapter we shall develop the general theory of order relations within...

Chapter 9. Ordinal Numbers and Cardinal Numbers

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pp. 117-128

In Chapter 7 we have seen that each infinite set A is equipotent to a unique natural number n. On the other hand the natural number n is a well-ordered set with respect to the usual order relation and no matter how the set A is well-ordered, A and n are isomorphic well-ordered sets. For this reason we may...

Special Symbols and Abbreviations

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pp. 129-130

List of Axioms

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pp. 131-132

Index

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pp. 133-135