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1、数学专业英语翻译2-3The concept of a set has been utilized so extensively throughout modern mathematics that an understanding of it is necessary for all college students. Sets are a means by which mathematicians talk of collections of things in an abstract way. 3A Notations for denoting sets集集合合论论的的概概念念已已经经被
2、被广广泛泛使使用用，遍遍及及现现代代数数学学，因因此此对对大大学学生生来来说说，理理解解它它的的概概念念是是必必要要的的。集集合合是是数数学学家家们们用用抽抽象象的的方方式式来表述一些事物的集体的工具。来表述一些事物的集体的工具。Sets usually are denoted by capital letters; elements are designated by lower-case letters.集合通常用大写字母表示，元素用小写字母表示。集合通常用大写字母表示，元素用小写字母表示。We use the special notation to mean that “x is an
3、element of S” or “x belongs to S”. If x does not belong to S, we write . 我我们们用用专专用用记记号号来来表表示示x是是S的的元元素素或或者者x属属于于S。如如果果x不不属属于于S，我们记为。，我们记为。When convenient, we shall designate sets by displaying the elements in braces; for example, the set of positive even integers less than 10 is displayed as 2,4,6,8
4、 whereas the set of all positive even integers is displayed as 2,4,6, the three dots taking the place of “and so on.”如如果果方方便便，我我们们可可以以用用在在大大括括号号中中列列出出元元素素的的方方式式来来表表示示集集合合。例例如如，小小于于10的的正正偶偶数数的的集集合合表表示示为为2,4,6,8，而而所所有有正正偶偶数数的集合表示为的集合表示为2,4,6, 三个圆点表示三个圆点表示 “等等等等”。The dots are used only when the meaning
5、 of “and so on” is clear. The method of listing the members of a set within braces is sometimes referred to as the roster notation.只只有有当当省省略略的的内内容容清清楚楚时时才才能能使使用用圆圆点点。在在大大括括号号中中列列出出集集合元素的方法有时被归结为枚举法。合元素的方法有时被归结为枚举法。 The first basic concept that relates one set to another is equality of sets:联系一个集合与另一
6、个集合的第一个基本概念是集合相等。联系一个集合与另一个集合的第一个基本概念是集合相等。 DEFINITION OF SET EQUALITY Two sets A and B are said to be equal (or identical) if they consist of exactly the same elements, in which case we write A=B. If one of the sets contains an element not in the other, we say the sets unequal and we write AB.集集合合相
7、相等等的的定定义义 如如果果两两个个集集合合A和B确确切切包包含含同同样样的的元元素素,则则称称二二者者相相等等，此此时时记记为为A=B。如如果果一一个个集集合合包包含含了了另另一一个个集集合以外的元素，则称二者不等，记为合以外的元素，则称二者不等，记为AB。EXAMPLE 1. According to this definition, the two sets 2,4,6,8 and 2,8,6,4 are equal since they both consist of the four integers 2,4,6 and 8. Thus, when we use the roster
8、 notation to describe a set, the order in which the elements appear is irrelevant.根根据据这这个个定定义义，两两个个集集合合2,4,6,8和和2,8,6,4是是相相等等的的，因因为为他他们们都都包包含含了了四四个个整整数数2,4,6,8。因因此此，当当我我们们用用枚枚举举法法来来描描述集合的时候，元素出现的次序是无关紧要的。述集合的时候，元素出现的次序是无关紧要的。EXAMPLE 2. The sets 2,4,6,8 and 2,2,4,4,6,8 are equal even though, in the s
9、econd set, each of the elements 2 and 4 is listed twice. Both sets contain the four elements 2,4,6,8 and no others; therefore, the definition requires that we call these sets equal. 例例2. 集集合合2,4,6,8 和和2,2,4,4,6,8也也是是相相等等的的，虽虽然然在在第第二二个个集集合合中中，2和和4都都出出现现两两次次。两两个个集集合合都都包包含含了了四四个个元元素素2,4,6,8，没有其他元素，因此，依
10、据定义这两个集合相等。，没有其他元素，因此，依据定义这两个集合相等。This example shows that we do not insist that the objects listed in the roster notation be distinct. A similar example is the set of letters in the word Mississippi, which is equal to the set M,i,s,p, consisting of the four distinct letters M,i,s, and p.这这个个例例子子表表明明
11、我我们们没没有有强强调调在在枚枚举举法法中中所所列列出出的的元元素素要要互互不不相相同同。一一个个相相似似的的例例子子是是，在在单单词词Mississippi中中字字母母的的集集合合等等价于集合价于集合M,i,s,p, 其中包含了四个互不相同的字母其中包含了四个互不相同的字母M,i,s,和和p.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 divisibl
12、e 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.3B Subsets一一个个给给定定的的集集合合S可可以以产产生生新新的的集集合合，这这些些集集合合叫叫做做S的的子子集集。例例如如，由由可可被被4除除尽尽的的并并且且小小于于10的的正正整整数数所所组组成成的的集集合合是是小小于于10的的所所有有偶偶数数所所组组成成集集合合的的子子集集。一一般般来来说说，我我们们有有如如下定义。下定义。In all
13、 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）页第二段）当我们应用集合论时，总是事先给定一个固
14、定的集合当我们应用集合论时，总是事先给定一个固定的集合S，而，而我们只关心这个给定集合的子集。基础集可以随意改变，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。可以在每一段特定的论述中表示全集。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
15、 set.（35页第三段）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。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 emp
16、ty 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。和盒子）装有某些东西那样，包含它的元素。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）页第四段）为了避免遇到逻辑困难，
17、我们必须区分元素为了避免遇到逻辑困难，我们必须区分元素x和集合和集合x，集合集合 x中的元素是中的元素是x。特别要注意的是空集和集合是不同的。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
18、合有时被称为单元素集。成的集合有时被称为单元素集。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 then be thought of the collections of points within S. For example, in Figure 2-3-1 the shaded portion
19、 is a subset of A and also a subset of B. （35页第五段）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。看作平面内的一个区域，其中的每一个元素即是一个点。 那那么么S的子集就是的子集就是S内某些点的全体。例如，在图内某些点的全体。例如，在图2-3-1中阴影部中阴影部分是分是A的子集，同时也是的子集，同时也是B的子集。的子集。Visual aids of this type, called Venn diagrams, are us
20、eful 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。依赖于概念的定义而不是图解。作业：P 37 2. 汉译英 (1), (2)谢 谢！谢谢大家！
