数学专业英语－Basic Concepts of the Theory of Sets

yunjiang · 发表于 2004-5-6 09:24:07

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>In discussing any branch of mathematics, be it analysis, algebra, or geometry, it is helpful to use the notation and terminology of set theory. This subject, which was developed by Boole and Cantor in the latter part of the 19th century, has had a profound influence on the development of mathematics in the 20th century. It has unified many seemingly disconnected ideas and has helped to reduce many mathematical concepts to their logical foundations in an elegant and systematic way. A thorough treatment of theory of sets would require a lengthy discussion which we regard as outside the scope of this book. Fortunately, the basic noticns are few in number, and it is possible to develop a working knowledge of the methods and ideas of set theory through an informal discussion . Actually, we shall discuss not so much a new theory as an agreement about the precise terminology that we wish to apply to more or less familiar ideas.
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>In mathematics, the word “set” is used to represent a collection of objects viewed as a single entity
The collections called to mind by such nouns as “flock”, “tribe”, ‘crowd”, “team’, are all examples of sets, The individual objects in the collection are called elements or members of the set, and they are said to belong to or to be contained in the set. The set in turn ,is said to contain or be composed of its elements.
 We shall be interested primarily in sets of mathematical objects: sets of numbers, sets of curves, sets of geometric figures, and so on. In many applications it is convenient to deal with sets in which nothing special is assumed about the nature of the individual objects in the collection. These are called abstract sets. Abstract set theory has been developed to deal with such collections of arbitrary objects, and from this generality the theory derives its power.
 NOTATIONS. Sets usually are denoted by capital letters: A,B,C,….X,Y,Z ; elements are designated by lower-case letters: a, b, c,….x, y, z. We use the special notation 
X∈S
To mean that “x is an element of S “or” x belongs to S”. If x does not belong to S, we write x∈S. When convenient ,we shall designate sets by displaying the elements in braces; for example ，the set of positive even integers less than 10 is denoted by the symbol{2,4,6,8}whereas the set of all positive even integers is displayed as {2,4,6,…},the dots taking the place of “and so on”.
 The first basic concept that relates one set to another is equality of sets:
 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 are unequal and we write A≠B.
 SUBSETS. 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.
 DEFINITION OF A SUBSET.A set A is said to be a subset of a set B, and we write 
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Whenever every element of A also belongs to B. We also say that A is contained in B or B contains A. The relation is referred to as set inclusion.
 The statement A<v:shape> <v:imagedata></v:imagedata></v:shape>B does not rule out the possibility that B<v:shape> <v:imagedata></v:imagedata></v:shape>A. In fact, we may have both A<v:shape> <v:imagedata></v:imagedata></v:shape>B and B<v:shape> <v:imagedata></v:imagedata></v:shape>A, but this happens only if A and B have the same elements. In other words, A=B if and only if A<v:shape> <v:imagedata></v:imagedata></v:shape>B and B<v:shape> <v:imagedata></v:imagedata></v:shape>A .
 This theorem is an immediate consequence of the foregoing definitions of equality and inclusion. If A<v:shape> <v:imagedata></v:imagedata></v:shape>B but A≠B, then we say that A is a proper subset of B: we indicate this by writing A<v:shape> <v:imagedata></v:imagedata></v:shape>B.
 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.
 The notation 
 {X∣X∈S. and X satisfies P}
will designate the set of all elements X in S which satisfy the property P. When the universal set to which we are referring id understood, we omit the reference to S and we simply write{X∣X satisfies P}.This is read “the set of all x such that x satisfies p.” Sets designated in this way are said to be described by a defining property For example, the set of all positive real numbers could be designated as {X∣X>0};the universal set S in this case is understood to be the set of all real numbers. Of course, the letter x is a dummy and may be replaced by any other convenient symbol. Thus we may write 
{x∣x>0}={y∣y>0}={t∣t>0}
and so on .
 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 symbolφ.We will consider φto be a subset of every set. 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.
 To avoid logical difficulties, we must distinguish between the element x and the set {x} whose only element is x ,(A box with a hat in it is conceptually distinct from the hat itself.)In particular, the empty setφis not the same as the set {φ}.In fact, the empty set φcontains no elements whereas the set {φ} has one element φ(A box which contains an empty box is not empty).Sets consisting of exactly one element are sometimes called one-element sets.
 UNIONS,INTERSECTIONS, COMPLEMENTS. From two given sets A and B, we can form a new set called the union of A and B. This new set is denoted by the symbol
A∪B(read: “A union B”)
And is defined as the set of those elements which are in A, in B, or in both. That is to say, A∪B is the set of all elements which belong to at least one of the sets A,B.
 Similarly, the intersection of A and B, denoted by
A∩B(read: “A intersection B”)
Is defined as the set of those elements common to both A and B. Two sets A and B are said to be disjoint if A∩B=φ.
 If A and B are sets, the difference A-B (also called the complement of B relative to A)is defined to be the set of all elements of A which are not in B. Thus, by definition,
A- B={X|X∈A and X<v:shape> <v:imagedata></v:imagedata></v:shape>B}
 The operations of union and intersection have many formal similarities with (as well as differences from) ordinary addition and multiplications of union and intersection, it follows that A∪B=B∪A and A∩B=B∩A. That is to say, union and intersection are commutative operations. The definitions are also phrased in such a way that the operations are associative:
(A∪B)∪C=A∪(B∪C)and(A∩B)∩C=A=∩(B∩C).
 The operations of union and intersection can be extended to finite or infinite collections of sets.

yunjiang · 发表于 2004-5-6 09:24:30

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0cm 0cm 0pt; TEXT-ALIGN: center" align=center><A>Vocabulary</A><

0cm 0cm 0pt"> Set 集合 proper subset 真子集Set theory 集合论 universal set 泛集Branch 分支 empty set空集 Analysis 分析 void set 空集Geometry 几何学 union 并，并集Notation 记号，记法 intersection交，交集Terminology 术语，名词表 complement余，余集Logic 逻辑 relative to相对于Logical 逻辑的 finite有限的Systematic 系统的 disjoint不相交Informal 非正式的 infinite无限的Formal正式的 cardinal number基数，纯数Entity 实在物 ordinal number序数Element 元素 generality一般性，通性Abstract set 抽象集 subset子集Designate 指定， divisible可除的Notion 概念 set inclusion 集的包含Braces 大括号 immediate consequence直接结果Identical 恒同的，恒等的

yunjiang · 发表于 2004-5-6 09:24:51

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0cm 0cm 0pt; TEXT-ALIGN: center" align=center>Notes<

0cm 0cm 0pt; TEXT-INDENT: 189pt"> 1. In discussing any branch of mathematics, be it analysis, algebra, or geometry, it is helpful to use the notation and terminology of set theory.意思是：在讨论数学的任何分支时，无论是分析，代数或分析，利用集合论的记号和术语是有帮助的。这一句中be it analysis, algebra, or geometry 是以be开头的状语从句，用倒装形式。类似的句子还有：people will use the tools in further investigations, be it in mathematic, hysics , or what have you .2. Actually, we shall discuss not so much a new theory as an agreement about the precise terminology that we wish to more or less familiar ideas.意思是：事实上，我恩将讨论的与其说是一种新理论，不如说是关于精确术语的一种约定，我们希望将它们应用到或多或少熟悉的思想上去。注意：not so much A as B 在这里解释为“与其说A不如说B。”类似的用法如：This is not so much a lecture as a friendly chat.(与其说这是演讲不如说是朋友间的交谈。)3．Two sets A and B are said to be equal if they consist of exactly the same elements, in which case we write A=B. 数学上常常在给定了定义后，就 用符号来表达。上面句子是常见句型。类似的表达法有： A set A is said to be a subset of a set B, and we write A=B whenever every element of A also belongs to B. This set is called the empty set or the void set, and will be denoted by the symbol Φ.

yunjiang · 发表于 2004-5-6 09:25:07

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0cm 0cm 0pt; TEXT-ALIGN: center" align=center>Exercise<

0cm 0cm 0pt">ⅰ. Turn the following mathematical expressions in English:<

0cm 0cm 0pt"> ⅰ)x∈A∪B ⅱ)A∩B=φ ⅲ)A={Φ} ⅳ)A={X: a<x<b}ⅱ.Let A ={2,5,8,11,14} B={2,8,14} C={2,8}D={5,11} E={2,8,11}ⅰ)B,C,D and E are ____________of A.ⅱ)C is the ______________of B and E.ⅲ)A is the ______________of B and D.ⅳ)The intersection of B and D is ____________Read the text carefully and then insert the insert the correct mathematical term in each of the blanks.ⅲ)Give the definition of each of the following:1.A two_ element set.2.The difference set of A and B, where A and B are sets.ⅳ.Four statements are given below. Among them, there is one and only one statement that cannot be used to express the meaning of A∩B=ф.Point it out and give your reason.a) The intersection of A and B is zero.b) Set A does not intersect set B.c) The intersection of A and B is zero.d) Set A and set B are B are disjoint.ⅴ.Translate the following passage into Chinese: It was G.. Cantor who introduced the concept the concept of the set as an object of mathematical study. Cantor stated: “A set is a collection of definite, well_ distinguished Objects of out intuition or thought. These objects are called the elements of the set. “cantor introduced the notions of cardinal and ordinal number and developed what is now known as Set Theory.ⅵ Translate the following sentences into English:1． 若集A 与集B均是集C的子集，则集A与集B的并集仍是集C的子集。2． 集A的补（余）集的补集是A 。

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