数学专业英语[1]－The Real Number System

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数学专业英语－The Real Number System z1 x1 o0 d5 u* e' L5 ` 2 t" Y" B; O2 ?8 ^! _2 x& L

The Real-Number System $ a2 O; E$ Z+ b# ]. ~

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The real-number system is collection of mathematical objects, called real number, which acquire mathematical life by virtue fundamental principles, or rules, that we adopt. The situation is somewhat similar to a game, like chess, for example. The chess system, or game, is a collection of objects, called chess pieces, which acquire life by virtue of the rules of the game, that is, the principles that are adopted to define allowable moves for the pieces and the way in which they may interact.

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Our working experience with numbers has provided us all with some familiarity with the principles that govern the real-number system. However, to establish a common ground of understanding and avoid certain errors that have become very common, we shall explicitly state and illustrate many of these principles.

The real-number system includes such numbers as –27,-2,2/3,… It is worthy of note that positive numbers, 1/2, 1, for examples, are sometimes expressed as +(1/2), +1. The plus sign, “+”, used here does not express the operation of addition, but is rather part of the symbolism for the numbers themselves. Similarly, the minus sign, “-“, used in expressing such numbers as -(1/2), -1, is part of the symbolism for these numbers.

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Within the real number system, numbers of various kinds are identified and named. The numbers 1, 2, 3, 4,… which are used in the counting process, are called natural numbers. The natural numbers, together with–1,-2,-3,-4,…and zero, are called integers. Since 1,2,3,4,…are greater than 0, they are also called positive integers; -1,-2,-3,-4,…are less than 0, and for this reason are called negative integers. A real number is said to be a rational number if it can be expressed as the ratio of two integers, where the denominator is not zero. The integers are included among the rational numbers since any integer can be expressed as the ratio of the integer itself and one. A real number that cannot be expressed as the ratio of two integers is said to be an irrational number.

One of the basic properties of the real-number system is that any two real numbers can be compared for size. If a and b are real numbers, we write a<b to signify that a is less than b. Another way of saying the same thing is to write b>a, which is read “b is greater than a “.

Geometrically, real numbers are identified with points on a straight line. We choose a straight line, and an initial point f reference called the origin. To the origin we assign the number zero. By marking off the unit of length in both directions from the origin, we assign positive integers to marked-off points in one direction (by convention, to the right of the origin ) and negative integers to marked-off point in the other direction. By following through in terms of the chosen unit of length, a real number is attached to one point on the number line, and each point on the number line has attached to it one number.

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Geometrically, in terms of our number line, to say that a<b is to say that a is to the left of b; b>a means that b is to the right of a.

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Properties of Addition and Multiplication

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Addition and multiplication are primary operations on real numbers. Most, if not all, of the basic properties of these operations are familiar to us from experience.

(a) Closure property of addition and multiplication.

Whenever two real numbers are added or multiplied, we obtain a real number as the result. That is, performing the operations of addition and multiplication leaves us within the real-number system.

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(b) Commutative property of addition and multiplication.

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The order in which two real numbers are added or multiplied does not affect the result obtained. That is, if a and b are any two real numbers, then we have (i) a+ b=b+ a and (ii) ab = ba. Such a property is called a commutative property. Thus, addition and multiplication of real numbers are commutative operations.

(c) Associative property of addition and multiplication.

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Parentheses, brackets, and the like, we recall, are used in algebra to group together whatever terms are within them. Thus 2+(3+4) means that 2 is to be added to the sum of 3 and 4 yielding 2+7 =9 whereas (2+3)+4 means the sum of 2 and 3 is to be added to 4 yielding also 9. Similarly, 2•(3•4) yields 2•(12)=24 whereas (2•3) •4 yields the same end result by the route 6•4=24 . That such is the case in general is the content of the associative property of addition and multiplication of real numbers.

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(d) Distributive property of multiplication over addition.

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We know that 2•(3•4)=2•7=14 and that 2•3+ 2•4=14 ,thus 2•(3+4)=2•3+ 2•4. That such is the case in general for all real numbers is the content of the distributive property of multiplication over addition, more simply called the distributive property.

Substraction and Division

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The numbers zero and one. The following are the basic properties of the numbers zero and one.

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(a) There is a unique real number, called zero and denoted by 0, with the property that a+0=0+a, where a is any real number.

There is a unique real number, different from zero, called one and denoted by 1, with the property that a•1=1•a=a, where a is any real number.

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(b) If a is any real number, then there is a unique real number x, called the additive inverse of a , or negative of a, with the property that a+ x = x+ a .If a is any nonzero real number, then there is a unique real number y, called the multiplicative inverse of a, or reciprocal of a, with the property that ay = ya = 1

The concept of the negative of a number should not be confused with the concept of a negative number; they are not the same. ”Negative of“ means additive inverse of “. On the other hand, a “negative number” is a number that is less than zero.

The multiplicative inverse of a is often represented by the symbol 1/a or a^-1. Note that since the product of any number y and 0 is 0, 0 cannot have a multiplicative inverse. Thus 1/0 does not exist.

Now substraction is defined in terms of addition in the following way. & a& M0 N3 T$ L4 }& i6 j! ?

If a and b are any two real numbers, then the difference a-b is defined by a- b= c where c is such that b+ c=a or c= a+(-b). That is, to substract b from a means to add the negative of b (additive inverse of b) to a.

Division is defined in terms of multiplication in the following way.

If a and b are any real numbers, where b≠0, then a+ b is defined by a +b= a•(1/b) =a•b^-1. That is, to divide a by b means to multiply a by the multiplicative inverse ( reciprocal)of b. The quotient a +b is also expressed by the fraction symbol a/b.

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Vocabulary : I% T/ [5 k% J9 ]& T2 Q6 G

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real number 实数

negative 负的

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the real number system 实数系

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rational number 有理数

collection 集体,总体

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ratio 比,比率

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object 对象,目的

denominator 分母

principle 原理,规则

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numerator 分子

adopt 采用

irrational number 无理数

define 定义(动词)

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signify 表示

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definition 定义(名词)

geometrical 几何的

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establish 建立

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straight line 直线

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explicit 清晰的,明显的

initial point 初始点

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illustrate 说明

point of reference 参考点

positive 正的

origin 原点

express 表达

assign 指定

plus 加

unit 单位

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sign 记号,符号,正负号

property 性质

operation 运算,操作

closure property 封闭性质

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addition 加法

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commutative 交换的

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multiplication 乘法

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associative 结合的

substraction 减法

parentheses 圆括号

division 除法

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brackets 括号

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sum 和,总数

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algebra 代数

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procuct 乘积

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yield 产生

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difference 差,差分

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term 术语,项

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quotient 商

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distributive 分配的

symbolism 符号系统

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unique 唯一的

minus 减

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additive inverse 加法逆运算

identify 使同一

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multiplicative inverse 乘法逆运算

count 计数

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reciprocal 倒数,互逆

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natural number 自然数

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concept 概念

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zero 零

fraction 分数

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integer 整数

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arithmetic 算术的

greater than 大于

solution 解,解法

less than 小于

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even 偶的

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be equal to 等于

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odd 奇的

arbitrary 任意的

square 平方

absolute value 绝对值

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square root 平方根

cube 立方

induction 归纳法

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Note

1. Our working experience with numbers has provided us all with some familiarity with the principles that govern the real-number system.

意思是:我们对数的实际工作经验使我们大家对支配着实数系的各原则早已有了某些熟悉,这里working作”实际工作的”解,govern作”支配”解.

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2.The plus sign,”+”, used here not express the operation of addition, but is rather part of the symbolism for the numbers themselves.

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意思是:这里的正符号”+”不是表示加法运算,而是数本身的符号系统的一部分.

3. A real number is said to be a rational number if it can be expressed as the ratio of two integers, where the denominator is not zero.

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这是定义数学术语的一种形式.下面是另一种定义数学术语的形式.

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A matrix is called a square matrix if the number of its rows equals the number of its columns.

这里is called与is said to be 可以互用,注意is called后面一般不加to be而is said后面一般要加.

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4. A real number that cannot be expressed as the ratio of two integers is said to be an irrational number.

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与注3比较,这里用定语从句界定术语.

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5. There is a unique real number, called zero and denoted by 0, with the property that a+0=0+a, where a is any real number.

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意思是:存在唯一的一个实数,叫做零并记为0,具有性质a+0=0+a,这里(其中)a是任一实数.

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1) 这里called和denoted都是过去分词,与后面的词组成分词短语,修饰number.

2) with the property是前置短语,修饰number.

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3) 注意本句和注3.中where的用法,一般当需要附加说明句子中某一对象时可用此结构.

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Exercise

I. Turn the following arithmetic expressions into English:

i) 3+(-2)=1 ii) 2+3(-4)=-10

iii) = -5 iv) =3

v)2/5-1/6=7/30

II. Fill in each blank the missing mathematical term to mark the following sentences complete.

i) The of two real numbers of unlike signs is negative.

ii) An integer n is called if n=2m for some integer m.

iii) An solution to the equation xⁿ=c is called the n is of c.

iv) If x is a real number, then the of x is a nonnegative real number denoted by |x| and defined as follows

x, if x 0

|x|= -x, if x <0

III. Translate the following exercises into Chinese:

i) If x is an arbitrary real number, prove that there is exactly one integer n such that x<n<x+1.

ii) Prove that there is no rational number whose square in 2.

iii) Given positive real numbers a1,a2,a3,…such that an<ca_n-1 for all n>2, where c is a fixed positive number, use induction to prove that an<c^n-1a₁, for all n>1.

iv) Determine all positive integers n for which 2ⁿ<n!

Ⅳ Translate the following passage into Chinese:

There are many ways to introduce the real number system. One popular method is to begin with the positive integers 1,2,3,…and use them as building blocks to construct a more comprehensive system having the properties desired. Briefly, the idea of this method is to take the positive integers as undefined concepts, state some axioms concerning them, and them use the positive integers to build a larger system consisting of the positive rational numbers. The positive irrational numbers, in turn, may then be used as basis for constructing the positive irrational numbers. The final step is the introduction of the negative numbers and zero. The most difficult part of the whole process is the transition from the rational numbers to the irrational num

Ⅴ. Translate the following theorems into English:

1. 定理A: 每一非负数有唯一一个非负平方根.

2. 定理B: 若x>0, y是任意一实数,则存在一正整数n使得nx > y.

Ⅵ. 1. Try to show the structure of the set of real numbers graphically.

2. List and state the laws that operations of addition and multiplication of real numbers obey.

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

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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 19^th century, has had a profound influence on the development of mathematics in the 20^th 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.

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.

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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

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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:

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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.

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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.

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DEFINITION OF A SUBSET.A set A is said to be a subset of a set B, and we write

A B

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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 B does not rule out the possibility that B A. In fact, we may have both A B and B A, but this happens only if A and B have the same elements. In other words, A=B if and only if A B and B A .

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This theorem is an immediate consequence of the foregoing definitions of equality and inclusion. If A B but A≠B, then we say that A is a proper subset of B: we indicate this by writing A 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

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{x∣x>0}={y∣y>0}={t∣t>0}

and so on .

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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

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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.

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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.

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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 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.

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Vocabulary

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

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Notes + E; p$ \1 m. v0 j5 h

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.

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Exercise

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ⅰ. Turn the following mathematical expressions in English:

ⅰ)x∈A∪B ⅱ)A∩B=φ

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ⅲ)A={Φ} ⅳ)A={X: a<x<b}

ⅱ.Let A ={2,5,8,11,14} B={2,8,14} C={2,8}

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D={5,11} E={2,8,11}

ⅰ)B,C,D and E are ____________of A.

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ⅱ)C is the ______________of B and E.

ⅲ)A is the ______________of B and D.

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ⅳ)The intersection of B and D is ____________

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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:

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1.A two_ element set.

2.The difference set of A and B, where A and B are sets.

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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.

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b) Set A does not intersect set B.

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c) The intersection of A and B is zero.

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d) Set A and set B are B are disjoint.

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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.

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ⅵ Translate the following sentences into English:

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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。

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2． 集A的补（余）集的补集是A 。

[此贴子已经被作者于2004-11-27 12:29:10编辑过]

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数学专业英语－Continuous Functions of One Real Variable . P% z" k6 I: J' G, Y

This lesson deals with the concept of continuity, one of the most important and also one of the most fascinating ideas in all of mathematics. Before we give a preeise technical definition of continuity, we shall briefly discuss the concept in an informal and intuitive way to give the reader a feeling for its meaning.

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Roughly speaking the situation is this: Suppose a function f has the value f ( p ) at a certain point p. Then f is said to be continuous at p if at every nearby point x the function value f ( x ) is close to f ( p ). Another way of putting it is as follows: If we let x move toward p, we want the corresponding function value f ( x ) to become arbitrarily close to f ( p ), regardless of the manner in which x approaches p. We do not want sudden jumps in the values of a continuous function.

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Consider the graph of the function f defined by the equation f ( x ) = x –[ x ], where [ x ] denotes the greatest integer < x . At each integer we have what is known ad a jump discontinuity. For example, f ( 2 ) = 0 ,but as x approaches 2 from the left, f ( x ) approaches the value 1, which is not equal to f ( 2 ).Therefore we have a discontinuity at 2. Note that f ( x ) does approach f ( 2 ) if we let x approach 2 from the right, but this by itself is not enough to establish continuity at 2. In case like this, the function is called continuous from the right at 2 and discontinuous from the left at 2. Continuity at a point requires both continuity from the left and from the right.

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In the early development of calculus almost all functions that were dealt with were continuous and there was no real need at that time for a penetrating look into the exact meaning of continuity. It was not until late in the 18^th century that discontinuous functions began appearing in connection with various kinds of physical problems. In particular, the work of J.B.J. Fourier(1758-1830) on the theory of heat forced mathematicians the early 19^th century to examine more carefully the exact meaning of the word “continuity”.

A satisfactory mathematical definition of continuity, expressed entirely in terms of properties of the real number system, was first formulated in 1821 by the French mathematician, Augustin-Louis Cauchy (1789-1857). His definition, which is still used today, is most easily explained in terms of the limit concept to which we turn now.

The definition of the limit of a function.

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Let f be a function defined in some open interval containing a point p, although we do not insist that f be defined at the point p itself. Let A be a real number.

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The equation

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f ( x ) = A - p4 }$ F. m8 ]( ^6 S 5 r- D2 X% f+ k, s+ M4 M. ]# p

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is read “The limit of f ( x ) , as x approached p, is equal to A”, or “f ( x ) approached A as x approached p.” It is also written without the limit symbol, as follows:

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f ( x )→ A as x → p

This symbolism is intended to convey the idea that we can make f ( x ) as close to A as we please, provided we choose x sufficiently close to p.

Our first task is to explain the meaning of these symbols entirely in terms of real numbers. We shall do this in two stages. First we introduce the concept of a neighborhood of a point, the we define limits in terms of neighborhoods.

Definition of neighborhood of a point.

Any open interval containing a point p as its midpoint is called a neighborhood of p.

NOTATION. We denote neighborhoods by N ( p ), N₁ ( p ), N₂ ( p ) etc. Since a neighborhood N ( p ) is an open interval symmetric about p, it consists of all real x satisfying p-r < x < p+r for some r > 0. The positive number r is called the radius of the neighborhood. We designate N ( p ) by N ( p; r ) if we wish to specify its radius. The inequalities p-r < x < p+r are equivalent to –r<x-p<r, and to ∣x-p∣< r. Thus N ( p; r ) consists of all points x whose distance from p is less than r.

In the next definition, we assume that A is a real number and that f is a function defined on some neighborhood of a point p (except possibly at p ) . The function may also be defined at p but this is irrelevant in the definition.

Definition of limit of a function.

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The symbolism

f ( x ) = A or [ f ( x ) → A as x→ p ] 3 j5 i: c8 r: x \4 Y0 p5 Y: a

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means that for every neighborhood N₁ ( A ) there is some neighborhood N₂ ( p) such that

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f ( x ) ∈ N₁ ( A ) whenever x ∈ N₂ ( p ) and x ≠ p (*) - p' l* B6 `. g* @3 f

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The first thing to note about this definition is that it involves two neighborhoods, N₁ ( A) and

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N₂ ( p) . The neighborhood N₁ ( A) is specified first; it tells us how close we wish f ( x ) to be to the limit A. The second neighborhood, N₂ ( p ), tells us how close x should be to p so that f ( x ) will be within the first neighborhood N₁ ( A). The essential part of the definition is that, for every N₁ ( A), no matter how small, there is some neighborhood N₂ (p) to satisfy (*). In general, the neighborhood N₂ ( p) will depend on the choice of N₁ ( A). A neighborhood N₂ ( p ) that works for one particular N₁ ( A) will also work, of course, for every larger N₁ ( A), but it may not be suitable for any smaller N₁ ( A).

The definition of limit can also be formulated in terms of the radii of the neighborhoods

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N₁ ( A) and N₂ ( p ). It is customary to denote the radius of N₁ ( A) byεand the radius of N₂ ( p) by δ.The statement f ( x ) ∈ N₁ ( A ) is equivalent to the inequality ∣f ( x ) – A∣<ε,and the statement x ∈ N₁ ( A) ,x ≠ p ,is equivalent to the inequalities 0 <∣ x-p∣<δ. Therefore, the definition of limit can also be expressed as follows:

The symbol f ( x ) = A means that for everyε> 0, there is aδ> 0 such that

∣f ( x ) – A∣<ε whenever 0 <∣x – p∣<δ

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“One-sided” limits may be defined in a similar way. For example, if f ( x ) →A as x→ p through values greater than p, we say that A is right-hand limit of f at p, and we indicate this by writing

f ( x ) = A ' z! t/ u6 C0 O! u+ N

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In neighborhood terminology this means that for every neighborhood N₁ ( A) ,there is some neighborhood N₂( p) such that

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f ( x ) ∈ N₁ ( A) whenever x ∈ N₁ ( A) and x > p

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Left-hand limits, denoted by writing x→ p^-, are similarly defined by restricting x to values less than p.

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If f has a limit A at p, then it also has a right-hand limit and a left-hand limit at p, both of these being equal to A. But a function can have a right-hand limit at p different from the left-hand limit.

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The definition of continuity of a function.

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In the definition of limit we made no assertion about the behaviour of f at the point p itself. Moreover, even if f is defined at p, its value there need not be equal to the limit A. However, if it happens that f is defined at p and if it also happens that f ( p ) = A, then we say the function f is continuous at p. In other words, we have the following definition.

Definition of continuity of a function at a point.

A function f is said to be continuous at a point p if

( a ) f is defined at p, and ( b ) f ( x ) = f ( p )

This definition can also be formulated in term of neighborhoods. A function f is continuous at p if for every neighborhood N₁ ( f(p)) there is a neighborhood N₂ (p) such that

f ( x ) ∈ N₁ ( f (p)) whenever x ∈ N₂ ( p).

In theε-δterminology , where we specify the radii of the neighborhoods, the definition of continuity can be restated ad follows:

Function f is continuous at p if for every ε> 0 ,there is aδ> 0 such that

∣f ( x ) – f ( p )∣< ε whenever ∣x – p∣< δ

In the rest of this lesson we shall list certain special properties of continuous functions that are used quite frequently. Most of these properties appear obvious when interpreted geometrically ; consequently many people are inclined to accept them ad self-evident. However, it is important to realize that these statements are no more self-evident than the definition of continuity itself, and therefore they require proof if they are to be used with any degree of generality. The proofs of most of these properties make use of the least-upper bound axiom for the real number system.

THEOREM 1. (Bolzano’s theorem) Let f be continuous at each point of a closed interval [a, b] and assume that f ( a ) an f ( b ) have opposite signs. Then there is at least one c in the open interval (a ,b) such that f ( c ) = 0.

THEOREM 2. Sign-preserving property of continuous functions. Let f be continuious at c and suppose that f ( c ) ≠ 0. Then there is an interval (c-δ,c +δ) about c in which f has the same sign as f ( c ).

THEOREM 3. Let f be continuous at each point of a closed interval [a, b]. Choose two arbitrary points x₁ < x₂ in [a, b] such that f ( x₁ ) ≠ f ( x₂ ) . Then f takes every value between f ( x₁ ) and f (x₂ ) somewhere in the interval ( x_1, x₂ ).

THEOREM 4. Boundedness theorem for continuous functions. Let f be continuous on a closed interval [a, b]. Then f is bounded on [a, b]. That is , there is a number M > 0, such that∣f ( x )∣≤ M for all x in [a, b].

THEOREM 5. (extreme value theorem) Assume f is continuous on a closed interval [a, b]. Then there exist points c and d in [a, b] such that f ( c ) = sup f and f ( d ) = inf f .

Note. This theorem shows that if f is continuous on [a, b], then sup f is its absolute maximum, and inf f is its absolute minimum