# Elementary Set Theory, Part I/II

KAM-TIM LEUNG
DORIS LAI-CHUE CHEN
FOREWORD BY YUNG-CHOW WONG
Pages: 148
https://www.jstor.org/stable/j.ctt2jc0g0

1. Front Matter
(pp. i-iv)
2. FOREWORD
(pp. v-vi)
Y. C. Wong

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 are now special cases of a single one; and, amid these far-reaching changes, there have emerged certain basic concepts, notations and terminologies that are of considerable importance and frequent use in a large portion of mathematics.

By 1959, I felt that the time was ripe for this University to introduce into its first-year mathematics course the most fundamental and the more elementary of these basic concepts,...

3. PREFACE
(pp. vii-viii)
K. T. Leung and Doris L. C. Chen
(pp. ix-x)
5. PART I
• CHAPTER 1 STATEMENT CALCULUS
(pp. 3-21)

By a statement (or a proposition or a declarative sentence) we understand a sentence of which it is meaningful to say that its content is true or false. Obviously, each of the following sentences is a statement:

Geography is a science.

Confucius was a soldier.

Cheung Sam is dead and Lee Sai is in prison.

2 is smaller than 3 and 3 is a prime number.

The steering gear was loose or the driver was drunk.

If John is here, then the book is not his.

Whereas none of the following sentences can be regarded as a statement in the...

• CHAPTER 2 SETS
(pp. 22-42)

A fundamental concept in mathematics is that of a set. This concept can be used as a foundation of all known mathematics. In this and the 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 objects to be simply the individual things of our intuition and our thoughts. In what follows, objects are referred to by their names, usually letters. Thus in saying ‘an object is denoted by \$ x \$ ’ or ‘ \$ x \$ is an object’, we mean that ‘ \$ x \$ ’ is a...

• CHAPTER 3 RELATIONS
(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 callėd an unordered pair.

Let us recall a well-known technique used in plane analytic geometry. With respect to a fixed rectangular coordinate system, each point \$ P \$ in the plane is uniquely represented by a pair \$ (x,y) \$ of...

• CHAPTER 4 MAPPINGS
(pp. 53-64)

Most readers 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 calculus, an expression \$ y=f(x) \$ is used to represent an act of associating with each argument \$ x \$ (a real number) a unique value \$ y \$ (also a real number). Within the framework of set theory, this situation can be conveniently formulated by means of relations.

Definition 4.1. \$ A \$ mapping from (or of, or on) \$ a \$ set...

6. PART II
• CHAPTER 5 FAMILIES
(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 \$ I \$ , the indexing of certain elements of \$ A \$ by elements (indices) taken from \$ I \$ naturally involves the concept of mapping; after such a process is carried out, the indexed elements of \$ A \$ , together with their indices, will receive more attention than the process itself. To handle this kind of situation efficiently, we...

• CHAPTER 6 NATURAL NUMBERS
(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 called sets, we shall define natural numbers by means of certain sets.

Let us take, for instance, the natural number two. We all have an intuitive idea of twoness. For example, we say that each of the unordered pairs \$ \{a,b\} \$ , and \$ \{c,d\} \$ , where...

• CHAPTER 7 FINITE AND INFINITE SETS
(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 means that our main tool for comparison has so far been identity mappings. We now propose to compare sets on a broader base and in fact by means of bijective mappings. As a result of this comparison, sets may be grouped...

• CHAPTER 8 ORDERED SETS
(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 length the usual order relation of natural numbers. In this chapter we shall develop the general theory of order relations within the framework of set theory. The familiar results of the usual order relation of natural numbers may now serve as examples to illustrate the more abstract concepts of this chapter.

Definition 8.1. Let \$ A \$ be \$ a \$ set. Then a relation \$ R \$ defined in \$ A \$ is said to be an order relation...

• CHAPTER 9 ORDINAL NUMBERS AND CARDINAL NUMBERS
(pp. 117-128)

In Chapter 7 we have seen that each finite 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 regard natural numbers as standard finite well-ordered sets in the sense that

(NA) every finite well-ordered set is isomorphic to a unique natural number.

This means, therefore that (i) every finite well-ordered set is isomorphic to a standard one, and...

7. SPECIAL SYMBOLS AND ABBREVIATIONS
(pp. 129-131)
8. LIST OF AXIOMS
(pp. 131-132)
9. INDEX
(pp. 133-135)