数学专业英语翻译2-3省名师优质课赛课获奖课件市赛课一等奖课件.ppt

第二级,第三级,第四级,第五级,本资料仅供参考，不能作为科学依据。谢谢。本资料仅供参考，不能作为科学依据。谢谢,New Words&Expressions:,brace 大括号 roster 名册,consequence 结论，推论 roster notation 枚举法,designate 标识，指定 rule out 排除，否决,diagram 图形，图解 subset 子集,distinct 互不相同 the underlying set 基础集,distinguish 区分，区分 universal set 全集,divisible 可被除尽 validity 有效性,dummy 哑哑变量 visual 可视,even integer 偶数 visualize 可视化,irrelevant 无关紧要 void set(empty set)空集,2.3,集合论基本概念,Basic Concepts of the Theory of Sets,第1页,The 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,集合论概念已经被广泛使用，遍布当代数学，所以对大学生来说，了解它概念是必要。集合是数学家们用抽象方式来表述一些事物集体工具。,Sets usually are,denote,d by capital letters;,element,s are,designate,d by,lower-case,letters.,集合通惯用大写字母表示，元素用小写字母表示。,第2页,We use the special notation to mean that“,x,is an element of,S,”or“,x,belongs to,S,”.If,x,does not belong to,S,we write .,我们用专用记号来表示,x,是S元素或者x属于S。假如,x,不属于S，我们记为。,When convenient,we shall,designate,sets by displaying the elements in,brace,s;for example,the set of positive,even integer,s less than 10 is displayed as 2,4,6,8 whereas the set of all positive even integers is displayed as 2,4,6,the three dots taking the place of“and so on.”,假如方便，我们能够用在大括号中列出元素方式来表示集合。比如，小于10正偶数集合表示为2,4,6,8，而全部正偶数集合表示为2,4,6,三个圆点表示,“等等”。,第3页,The dots are used only when the meaning 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,.,只有当省略内容清楚时才能使用圆点。在大括号中列出集合元素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:,联络一个集合与另一个集合第一个基本概念是集合相等。,第4页,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 A,B.,集合相等定义 假如两个集合,A,和,B,确切包含一样元素,则称二者相等，此时记为,A=B。,假如一个集合包含了另一个集合以外元素，则称二者不等，记为,AB。,第5页,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 notation to describe a set,the,order,in which the elements appear is,irrelevant,.,依据这个定义，两个集合2,4,6,8和2,8,6,4是相等，因为他们都包含了四个整数2,4,6,8。所以，当我们用枚举法来描述集合时候，元素出现次序是无关紧要。,第6页,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 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,，,即使在第二个集合中，2和4都出现两次。两个集合都包含了四个元素2,4,6,8，没有其它元素，所以，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.,这个例子表明我们没有强调在枚举法中所列出元素要互不相同。一个相同例子是，在单词,Mississippi,中字母集合等价于集合M,i,s,p,其中包含了四个互不相同字母M,i,s,和p.,第7页,From a,given,set,S,we may form new sets,called,subset,s 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,一个给定集合,S,能够产生新集合，这些集合叫做,S,子集。比如，由可被4除尽而且小于10正整数所组成集合是小于10全部偶数所组成集合子集。普通来说，我们有以下定义。,第8页,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集合,S,，而我们只关心这个给定集合子集。基础集能够随意改变，能够在每一段特定叙述中表示全集。,第9页,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。这个集合被叫做空集，用符号表示。空集是任何集合子集。,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.,一些人认为这么比喻是有益，集合类似于容器（如背包和盒子）装有一些东西那样，包含它元素。,第10页,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）,为了防止碰到逻辑困难，我们必须区分元素,x,和集合,x,，,集合,x,中元素是,x。,尤其要注意是空集和集合是不一样。,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 set,s.,实际上，空集不含有任何元素，而有一个元素。由一个元素组成集合有时被称为单元素集。,第11页,Diagram,s 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,shade,d,portion,is a,subset,of,A,and also a subset of,B,.（35页第五段）,图解有利于我们将集合之间关系形象化。比如，能够把集合,S,看作平面内一个区域，其中每一个元素即是一个点。那么,S,子集就是,S,内一些点全体。比如，在图2-3-1中阴影部分是,A,子集，同时也是,B,子集。,第12页,Visual aids of this type,called Venn,diagram,s,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，在集合论中惯用于检验定理有效性或者为证实定理提供一些潜在方法。当然证实本身必须依赖于概念定义而不是图解。,第13页,作业：,P 37,2.汉译英(1),(2),第14页,谢 谢！,第15页,