1、New Words & Expressions:brace 大括號(hào) roster 名冊(cè)consequence 結(jié)論，推論 roster notation 枚舉法designate 標(biāo)記，指定 rule out 排除，否決diagram 圖形，圖解 subset 子集distinct 互不相同的 the underlying set 基礎(chǔ)集distinguish 區(qū)別，辨別 universal set 全集divisible 可被除盡的 validity 有效性dummy 啞的，啞變量 visual 可視的even integer 偶數(shù) visualize 可視化irrelevant 無關(guān)緊要的

2、 void set(empty set) 空集2.3 集合論的基本概念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集合論的概念已經(jīng)被廣泛使用，遍及現(xiàn)代數(shù)學(xué)，因此對(duì)大學(xué)生來說，理解它的概念是必要的。集合是數(shù)學(xué)家們用抽象的方式來表述一些事物的集體的工具。Sets usually are denoted by capital letters; elements are designated by lower-case letters.集合通常用大寫字母表示，元素用小寫字母表示。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 . 我們用專用記號(hào)來表示x是S的元素或者x屬于S。如果x不屬于S，我們記為。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，我們可以用在大括號(hào)中列出元素的方式來表示集合。例如，小于10的正偶數(shù)的集合表示為2,4,6,8，而所有正偶數(shù)的集合表示為2,4,6, 三個(gè)圓點(diǎn)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.集合相等的定義 如果兩個(gè)集合A和B確切包含同樣的元素,則稱二者相等，此時(shí)記為A=B。如果一個(gè)集合包含了另一個(gè)集合以外的元素，則稱二者不等，記為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.根據(jù)這個(gè)定義，兩個(gè)集合2,4,6,8和2,8,6,4是相等的，因?yàn)樗麄兌及怂膫€(gè)整數(shù)2,4,6,8。因此，當(dāng)我們用枚舉法來描述集合的時(shí)候，元素出現(xiàn)的次序是無關(guān)緊要的。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，雖然在第二個(gè)集合中，2和4都出現(xiàn)兩次。兩個(gè)集合都包含了四個(gè)元素2,4,6,8，沒有其他元素，因此，依據(jù)定義這兩個(gè)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.這個(gè)例子表明我們沒有強(qiáng)調(diào)在枚舉法中所列出的元素要互不相同。一個(gè)相似的例子是，在單詞Mississippi中字母的集合等價(jià)于集合M,i,s,p, 其中包含了四個(gè)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一個(gè)給定的集合S

11、可以產(chǎn)生新的集合，這些集合叫做S的子集。例如，由可被4除盡的并且小于10的正整數(shù)所組成的集合是小于10的所有偶數(shù)所組成集合的子集。一般來說，我們有如下定義。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頁(yè)第三段）一個(gè)集合中不包含任何元素，這種情況是有可能的。這個(gè)

12、集合被叫做空集，用符號(hào)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.一些人認(rèn)為這樣的比喻是有益的，集合類似于容器（如背包和盒子）裝有某些東西那樣，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