数学专业英语(吴炯圻-第2版)2-3课件

1、New Words & Expressions:brace 大括号 roster 名册consequence 结论，推论 roster notation 枚举法designate 标记，指定 rule out 排除，否决diagram 图形，图解 subset 子集distinct 互不相同的 the underlying set 基础集distinguish 区别，辨别 universal set 全集divisible 可被除尽的 validity 有效性dummy 哑的，哑变量 visual 可视的even integer 偶数 visualize 可视化irrelevant 无关紧要的

2、 void set(empty set) 空集2.3 集合论的基本概念Basic Concepts of the Theory of SetsThe 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 a

3、bstract way. 3A Notations for denoting sets集合论的概念已经被广泛使用，遍及现代数学，因此对大学生来说，理解它的概念是必要的。集合是数学家们用抽象的方式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。We use the special notation to mean that “x is an element of S” or “x belong

4、s to S”. If x does not belong to S, we write . 我们用专用记号来表示x是S的元素或者x属于S。如果x不属于S，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 whereas the set of all positive even integers is displ

5、ayed as 2,4,6, the three dots taking the place of “and so on.”如果方便，我们可以用在大括号中列出元素的方式来表示集合。例如，小于10的正偶数的集合表示为2,4,6,8，而所有正偶数的集合表示为2,4,6, 三个圆点表示 “等等”。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.

6、If one of the sets contains an element not in the other, we say the sets unequal and we write AB.集合相等的定义 如果两个集合A和B确切包含同样的元素,则称二者相等，此时记为A=B。如果一个集合包含了另一个集合以外的元素，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

7、2,4,6 and 8. Thus, when we use the roster notation to describe a set, the order in which the elements appear is irrelevant.根据这个定义，两个集合2,4,6,8和2,8,6,4是相等的，因为他们都包含了四个整数2,4,6,8。因此，当我们用枚举法来描述集合的时候，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 second set, each of

8、 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2. 集合2,4,6,8 和2,2,4,4,6,8也是相等的，虽然在第二个集合中，2和4都出现两次。两个集合都包含了四个元素2,4,6,8，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 th

9、at 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

10、,s,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 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.3B Subsets一个给定的集合S

11、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。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

12、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。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