数学专业英语(吴炯圻)(3)

这个例子表明我们没有强调在枚举法中所列出的元素要互不相同。一个相似的例子是，在单

词Mississippi中字母的集合等价于集合{M,i,s,p}, 其中包含了四个互不相同的字母M,i,s,和p.

7

From a given set S we may form new sets, called subsets of S. For example, the set consisting of

those positive integers less than 10 which are divisible by 4 (the set {4,8}) is a subset of the set of

all even integers less than 10. In general, we have the following definition.

3－B Subsets

一个给定的集合S可以产生新的集合，这些集合叫做S的子集。例如，由可被4除尽的并且

小于10的正整数所组成的集合是小于10的所有偶数所组成集合的子集。一般来说，我们有

如下定义。

8

In all our applications of set theory, we have a fixed set S given in advance, and we are concerned

only with subsets of this given set. The underlying set S may vary from one application to another;

it will be referred to as the universal set of each particular discourse. （35页第二段）

当我们应用集合论时，总是事先给定一个固定的集合S，而我们只关心这个给定集合的子集。

基础集可以随意改变，可以在每一段特定的论述中表示全集。

9

It is possible for a set to contain no elements whatever. This set is called the empty set or the void

set, and will be denoted by the symbol . We will consider to be a subset of every set.（35

页第三段）

一个集合中不包含任何元素，这种情况是有可能的。这个集合被叫做空集，用符号表示。空

集是任何集合的子集。

Some people find it helpful to think of a set as analogous to a container (such as a bag or a box)

containing certain objects, its elements. The empty set is then analogous to an empty container.

一些人认为这样的比喻是有益的，集合类似于容器（如背包和盒子）装有某些东西那样，包

含它的元素。

10

To avoid logical difficulties, we must distinguish between the elements x and the set {x} whose

only element is x. In particular, the empty set is not the same as the set . （35页第

四段）

为了避免遇到逻辑困难，我们必须区分元素x和集合{x}，集合 {x}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

第二版 课文翻译及课后习题

region in the plane and each of its elements as a point. Subsets of S may 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. （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。那么S的子集就是S内某些点的全体。例如，在图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

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

第二版 课文翻译及课后习题

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

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.

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