中的元素是

x

。特别要注意的是空集和集合是不 同的。

In fact, the empty set contains no elements, whereas the set has one element.

Sets consisting of exactly one element are sometimes called one- element sets.

事实上，空集不含有任何元素，而有一个元素。由一个元 素构成的集合有时被

称为单元素集。

11

Diagrams often help us visualize relations between sets. For example, we may think

of a set S as a region in the plane and each of its elements as a point. Subsets of S may

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then be thought of the collections of points within S. For example, in Figure 2-3-1 the

shaded portion is a subset of A and also a subset of B. < /p>

（

35

页第五段）

图解有助于我们将集合之间的关系形象化。例如，可以把集合

S

看作平面内的

一个区域，其中的每一个元素即是一个点。< /p>

那么

S

的子集 就是

S

内某些点的全

体。例如，在图< /p>

2-3-1

中阴影部分是

A

的子集，同时也是

B

的子集。

12

Visual aids of this type, called Venn diagrams, are useful for testing the validity of

theorems in set theory or for suggesting methods to prove them. Of course, the proofs

themselves must rely only on the definitions of the concepts and not on the diagrams.

这种图解方法，叫做文氏图，在集合论中常用于检验定理 的有效性或者为证明

定理提供一些潜在的方法。当然证明本身必须依赖于概念的定义而不 是图解。

New Words & Expressions:

conversely

反之

geometric interpretation

几何意义

correspond

对应

induction

归纳法

deducible

可推导的

proof by induction

归纳证明

difference

差

inductive set

归纳集

distinguished

著名的

inequality

不等式

entirely complete

完整的

integer

整数

Euclid

欧几里得

interchangeably

可互相交换的

Euclidean

欧式的

intuitive

直观的

the field axiom

域公理

irrational

无理的

2.4

整数、有理数与实数

Integers, Rational Numbers and Real Numbers

1

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New Words & Expressions:

irrational number

无理数

rational

有理的

the order axiom

序公理

rational number

有理数

ordered

有序的

reasoning

推理

product

积

scale

尺度，刻度

quotient

商

sum

和

2

There exist certain subsets of R which are distinguished because they have special

properties not shared by all real numbers. In this section we shall discuss such subsets,

the integers and the rational numbers.

4

－

A Integers and rational numbers

有一些

R

的子集很著名，因为他们具有实数所不具备的特殊性质。在本节我们

将 讨论这样的子集，整数集和有理数集。

3

To introduce the positive integers we begin with the number 1, whose existence is

guaranteed by Axiom 4. The number 1+1 is denoted by 2, the number 2+1 by 3, and

so on.

The numbers 1,2,3,…, obtained in this way by repeated addition of 1 are all

positive, and they are called the positive integers.

我们从数字

1

开始介绍正整数，公理

4

保证了

1

的存在性。

1+1

用

2

表示，

2+1

用

3

表示，以此类推，由

1

重复累加的方式得到的数字

1,2,3

，…都是正的 ，它

们被叫做正整数。

4

Strictly speaking, this description of the positive integers is not entirely complete

because we have not explained in detail what we mean by the expressions “and so on”,

or “repeated addition of 1”.

严格地说，这种关于正整数的描述是不完整的，因为我们没有 详细解释“等等”

或者“

1

的重复累加 ”的含义。

5

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Although the intuitive meaning of expressions may seem clear, in careful treatment of

the real-number system it is necessary to give a more precise definition of the positive

integers. There are many ways to do this. One convenient method is to introduce first

the notion of an inductive set.

虽然这些说法的直观意思似乎是清楚的， 但是在认真处理实数系统时必须给出

一个更准确的关于正整数的定义。

< br>

有很多种方式来给出这个定义，一个简便的

方法是先引 进归纳集的概念。

6

DEFINITION OF AN INDUCTIVE SET. A set of real numbers is called an inductive

set if it has the following two properties:

The number 1 is in the set.

For every x in the set, the number x+1 is also in the set.

For example, R is an inductive set. So is the set . Now we shall define the positive

integers to be those real numbers which belong to every inductive set.

现在我们来定义正整数，就是属于每一个归纳集的实数。

7

Let P denote the set of all positive integers. Then P is itself an inductive set because (a)

it contains 1, and (b) it contains x+1 whenever it contains x. Since the members of P

belong to every inductive set, we refer to P as the smallest inductive set.

用

P

表示所有正整数的集合。那么

P

本身是一个归纳集，因为其中含

1

，满足

(a)

；只要包含

x

就包含

x+1,

满足

(b)

。由于

P

中的元素属于每一个归纳集，因此

P

是最小的归纳集。

8

This property of P forms the logical basis for a type of reasoning that mathematicians

call proof by induction, a detailed discussion of which is given in Part 4 of this

introduction.

P

的 这种性质形成了一种推理的逻辑基础，数学家称之为归纳证明，在介绍的

第四部分将给出 这种方法的详细论述。

9

The negatives of the positive integers are called the negative integers. The positive

integers, together with the negative integers and 0 (zero), form a set Z which we call

simply the set of integers.

正整数的相反数被叫做负整数。正整数，负整数和零构成了一个集合

Z

< br>，简称

为整数集。

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10

In a thorough treatment of the real- number system, it would be necessary at this stage

to prove certain theorems about integers. For example, the sum, difference, or product

of two integers is an integer, but the quotient of two integers need not to ne an integer.

However, we shall not enter into the details of such proofs.

在实数系统中 ，为了周密性，此时有必要证明一些整数的定理。例如，两个整

数的和、差和积仍是整数 ，但是商不一定是整数。然而还不能给出证明的细

节。

11

Quotients of integers a/b (where b

≠

0) are called rational numbers. The set of rational

numbers, denoted by Q, contains Z as a subset. The reader should realize that all the

field axioms and the order axioms are satisfied by Q. For this reason, we say that the

set of rational numbers is an ordered field. Real numbers that are not in Q are called

irrational.

整数

a

与

b

的商被叫做有理数，有理数集用

Q

表示，

Z

是

Q

的子集。读者应该

认识到

< br>Q

满足所有的域公理和序公理。因此说有理数集是一个有序的域。不是

12

The reader is undoubtedly familiar with the geometric interpretation of real numbers

by means of points on a straight line. A point is selected to represent 0 and another, to

the right of 0, to represent 1, as illustrated in Figure 2-4-1. This choice determines the

scale.

4

－

B Geometric interpretation of real numbers as points on a line

毫无疑问，读者都熟悉通过在直线上描点的方式表示实数的几何意义。如图

2-

4-1

所示，选择一个点表示

< br>0

，在

0

右边的另一个点表示< /p>

1

。这种做法决定了刻

度。

13

If one adopts an appropriate set of axioms for Euclidean geometry, then each real

number corresponds to exactly one point on this line and, conversely, each point on

the line corresponds to one and only one real number.

如果采用欧式几何 公理中一个恰当的集合，那么每一个实数刚好对应直线上的

一个点，反之，直线上的每一 个点也对应且只对应一个实数。

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14

For this reason the line is often called the real line or the real axis, and it is customary

to use the words real number and point interchangeably. Thus we often speak of the

point x rather than the point corresponding to the real number.

为此直线通 常被叫做实直线或者实轴，习惯上使用“实数”这个单词，而不是

“点”。因此我们经常 说点

x

不是指与实数对应的那个点。

15

This device for representing real numbers geometrically is a very worthwhile aid that

helps us to discover and understand better certain properties of real numbers.

However, the reader should realize that all properties of real numbers that are to be

accepted as theorems must be deducible from the axioms without any references to

geometry.

这种几何 化的表示实数的方法是非常值得推崇的，它有助于帮助我们发现和理

解实数的某些性质。 然而，读者应该认识到，拟被采用作为定理的所有关于实

数的性质都必须不借助于几何就 能从公理推出。

16

This does not mean that one should not make use of geometry in studying properties

of real numbers. On the contrary, the geometry often suggests the method of proof of

a particular theorem, and sometimes a geometric argument is more illuminating than a

purely analytic proof (one depending entirely on the axioms for the real numbers).

这并不意味着研究实数的性质时不会应用到几何。相反，几何经常会为证明一

< br>些定理提供思路，有时几何讨论比纯分析式的证明更清楚。

17

In this book, geometric arguments are used to a large extent to help motivate or clarity

a particular discuss. Nevertheless, the proofs of all the important theorems are

presented in analytic form.

在本书中，几何在很大程度上被用于激发或者阐明一些 特殊的讨论。不过，所

有重要定理的证明必须以分析的形式给出。

3

New Words & Expressions:

polygonal

多边形的

circular regions

圆域

parabolic

抛物线的

coordinate axis

坐标轴

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the unit distance

单位长度

the origin

坐标原点

horizontal

水平的

coordinate system

坐标系

perpendicular

互相垂直的，垂线

vertical

竖直的

an ordered pair

一个有序对

abscissa

横坐标

quadrant

象限

ordinate

纵坐标

intersect

相交

the theorem of Pythagoras

勾股定理

2.5 basic concepts of Cartesian geometry

4

As mentioned earlier, one of the applications of the integral is the calculation of area.

Ordinarily , we do not talk about area by itself ,instead, we talk about the area of

something . This means that we have certain objects (polygonal regions, circular

regions, parabolic segments etc.) whose areas we wish to measure. If we hope to

arrive at a treatment of area that will enable us to deal with many different kinds of

objects, we must first find an effective way to describe these objects.

就像前面提到的，积分的一个应用就是面积的计算，通常我们 不讨论面积本

身，相反，是讨论某事物的面积。这意味着我们有些想测量的面积的对象（ 多

边形区域，圆域，抛物线弓形等），如果我们希望获得面积的计算方法以便能

够用它来处理各种不同类型的图形，我们就必须首先找出表述这些对象的有效

方法。

5-A the coordinate system of Cartesian geometry

5

The most primitive way of doing this is by drawing figures, as was done by the

ancient Greeks. A much better way was suggested by Rene Descartes, who introduced

the subject of analytic geometry (also known as Cartesian geometry). Descartes’ idea

was to represent geometric points by numbers. The procedure for points in a plane is

this

：

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描述对象最基本的方法是画图，就像古希腊人做的那样。

R < /p>

笛卡儿提出了一种

比较好的方法，并建立了解析几何（也称为笛卡 儿几何）这门学科。笛卡儿的

思想就是用数来表示几何点，在平面上找点的过程如下：< /p>

5-A the coordinate system of Cartesian geometry

6

Two perpendicular reference lines (called coordinate axes) are chosen, one horizontal

(called the “x

-

axis”), the other vertical (the “y

-

axis”). Their point of intersection

denoted by O, is called the origin. On the x-axis a convenient point is chosen to the

right of O and its distance from O is called the unit distance. Vertical distances along

the Y-axis are usually measured with the same unit distance ,although sometimes it is

convenient to use a different scale on the y-axis. Now each point in the plane

(sometimes called the xy-plane) is assigned a pair of numbers, called its coordinates.

These numbers tell us how to locate the points.

选两条互相 垂直的参考线（称为坐标轴），一条水平（称为

x

轴），另一条 竖

直（称为

y

轴）。他们的交点记为< /p>

O,

称为原点。在

x

< br>轴上，原点的右侧选择一

个合适的点，该点与原点之间的距离称为单位长度，沿着

y

轴的垂直距离通常

用同样的单位长度 来测量，虽然有时候采用不同的尺度比较方便。现在平面上

的每一个点都分配了一对数， 称为坐标。这些数告诉我们如何定义一个点。

5-A the coordinate system of Cartesian geometry

7

A geometric figure, such as a curve in the plane , is a collection of points satisfying

one or more special conditions. By translating these conditions into expressions,,

involving the coordinates x and y, we obtain one or more equations which

characterize the figure in question , for example, consider a circle of radius r with its

center at the origin, as show in figure 2-5-2. let P be an arbitrary point on this circle,

and suppose P has coordinates (x, y).

一个几何图形是满足一个或多个特殊条件的 点集，比如平面上的曲线。通过把

这些条件转化成含有坐标

x< /p>

和

y

的表达式，我们就得到了一个或多个 能刻画该

图形特征的方程。例如，如图

2-5-2

所示的中心在原点，半径为

r

的圆，令

P

是

原上任意一点，假设

P

的坐标为

(x, y).

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5-B Geometric figure

9

New Words & Expressions:

prime

素数

displacement

位移

edge

棱，边

domain

定义域，区域

real variable

实变量

schematic representation

图解表示

tabulation

作表，表

mass

质量，许多，群众

absolute-value function

绝对值函数

2.6 function concept and function idea

10

Seldom has a single concept played so important a role in mathematics as has the

concept of function. It is desirable to know how the concept has developed.

在数学中，很少有个概念象函数的概念那样，起那么重要的作 用的。因此，需

要知道这个概念是如何发展起来的。

6-C The concept of function

11

This concept, like many others ,originates in physics. The physical quantities were the

forerunners of mathematical variables. And relation among them was called a

function relation in the later 16th century.

这个概念像许 多其他概念一样，起源于物理学。物理的量是数学的变量的先

驱，他们之间的关系在

16

世纪后期称为函数关系。

6-C The concept of function

12

For example , the formula s=16t2 for the number of feet s a body falls in any number

of seconds t is a function relation between s and t. it describes the way s varies with t.

the study of such relations led people in the 18th century to think of a function

relation as nothing but a formula.

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例如，代表一物体在若干秒

t

中下落若干英尺

s

< br>的公式

s=16t2

就是

s< /p>

和

t

之间的

函数 关系。它描述了

s

随

t

变化的公式，对这种关系的研究导致了

18

世纪的人

们认为函数关系只不过是一个公式罢了。

6-C The concept of function

13

Only after the rise of modern analysis in the early 19th century could the concept of

function be extended. In the extended sense , a function may be defined as follows: if

a variable y depends on another variable x in such a way that to each value of x

corresponds a definite value of y, then y is a function of x. this definition serves many

a practical purpose even today.

只有在

19