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2.1 数学、方程与比例1A What is mathematics1B Equation2.2 几何与三角2A Why study geometry?2B Some geometrical terms2.3 集合论的基本概念3A Notations for denoting sets3B Subsets 子集2.4 整数、有理数与实数Integers, Rational Numbers and Real Numbers4A Integers and rational numbers整数和有理数4B Geometric interpretation of real numbers as points on a line在直线上描点的方式表示实数的几何意义2.5 basic concepts of Cartesian geometry解析几何的基本概念5-A the coordinate system of Cartesian geometry解析几何的坐标系5-B Geometric figure几何图形2.6 function concept and function idea函数的概念与函数思想6-C The concept of function2.7 Sequences and Their Limits序列及其极限7-A The definition of sequences7-B The limit of a sequence8-A 函数的导数2.9 微分方程简介differential equations2.10 线性空间中的相关与无关集Dependent and Independent Sets in a Linear Space2.11 数理逻辑入门Elementary Mathematical Logic11-A Predicates11-B Quantifiers2.12 概率论与数理统计Probability Theory and Mathematical StatisticsNew Words & Expressions:algebra 代数学geometrical 几何的 algebraic 代数的identity 恒等式arithmetic 算术, 算术的measure 测量，测度axiom 公理numerical 数值的, 数字的conception 概念，观点operation 运算constant 常数postulate 公设logical deduction 逻辑推理proposition 命题division 除，除法subtraction 减，减法 formula 公式term 项，术语trigonometry 三角学variable 变化的，变量2.1 数学、方程与比例Mathematics, Equation and Ratio1A What is mathematics4Mathematics comes from mans social practice, for example, industrial and agricultural production, commercial activities, military operations and scientific and technological researches.数学来源于人类的社会实践，比如工农业生产，商业活动， 军事行动和科学技术研究。And in turn, mathematics serves the practice and plays a great role in all fields. No modern scientific and technological branches could be regularly developed without the application of mathematics.反过来，数学服务于实践，并在各个领域中起着非常重要的作用。 没有应用数学，任何一个现在的科技的分支都不能正常发展。5From the early need of man came the concepts of numbers and forms. Then, geometry developed out of problems of measuring land , and trigonometry came from problems of surveying. To deal with some more complex practical problems, man established and then solved equation with unknown numbers , thus algebra occurred. 很早的时候，人类的需要产生了数和形的概念。接着，测量土地问题形成了几何学，测量问题产生了三角学。为了处理更复杂的实际问题，人类建立和解决了带未知数的方程，从而产生了代数学。Before 17th century, man confined himself to the elementary mathematics, i.e. , geometry, trigonometry and algebra, in which only the constants are considered.17世纪前，人类局限于只考虑常数的初等数学，即几何学，三角学和代数学。6The rapid development of industry in 17th century promoted the progress of economics and technology and required dealing with variable quantities. The leap from constants to variable quantities brought about two new branches of mathematics-analytic geometry and calculus, which belong to the higher mathematics.17世纪工业的快速发展推动了经济技术的进步， 从而遇到需要处理变量的问题。从常量到变量的跳跃产生了两个新的数学分支-解析几何和微积分，他们都属于高等数学。Now there are many branches in higher mathematics, among which are mathematical analysis, higher algebra, differential equations, function theory and so on.现在高等数学里面有很多分支，其中有数学分析，高等代数，微分方程，函数论等。7Mathematicians study conceptions and propositions, Axioms, postulates, definitions and theorems are all propositions. Notations are a special and powerful tool of mathematics and are used to express conceptions and propositions very often.数学家研究的是概念和命题，公理，公设，定义和定理都是命题。符号是数学中一个特殊而有用的工具，常用于表达概念和命题。Formulas ,figures and charts are full of different symbols. Some of the best known symbols of mathematics are the Arabic numerals 1,2,3,4,5,6,7,8,9,0 and the signs of addition “+”, subtraction “-” , multiplication “”, division “” and equality “=”.公式，图形和图表都是不同的符号.8The conclusions in mathematics are obtained mainly by logical deductions and computation. For a long period of the history of mathematics, the centric place of mathematics methods was occupied by the logical deductions.数学结论主要由逻辑推理和计算得到。在数学发展历史的很长时间内，逻辑推理一直占据着数学方法的中心地位。Now , since electronic computers are developed promptly and used widely, the role of computation becomes more and more important. In our times, computation is not only used to deal with a lot of information and data, but also to carry out some work that merely could be done earlier by logical deductions, for example, the proof of most of geometrical theorems.现在，由于电子计算机的迅速发展和广泛使用，计算机的地位越来越重要。现在计算机不仅用于处理大量的信息和数据，还可以完成一些之前只能由逻辑推理来做的工作，例如，证明大多数的几何定理。9回顾：1. 如果没有运用数学， 任何一个科学技术分支都不可能正常的发展 。2. 符号在数学中起着非常重要的作用，它常用于表示概念和命题。1B Equation10An equation is a statement of the equality between two equal numbers or number symbols.等式是关于两个数或者数的符号相等的一种描述。Equation are of two kinds- identities and equations of condition.An arithmetic or an algebraic identity is an equation. In such an equation either the two members are alike, or become alike on the performance of the indicated operation.等式有两种恒等式和条件等式。算术或者代数恒等式都是等式。这种等式的两端要么一样，要么经过执行指定的运算后变成一样。11An identity involving letters is true for any set of numerical values of the letters in it.含有字母的恒等式对其中字母的任一组数值都成立。An equation which is true only for certain values of a letter in it, or for certain sets of related values of two or more of its letters, is an equation of condition, or simply an equation. Thus 3x-5=7 is true for x=4 only; and 2x-y=10 is true for x=6 and y=2 and for many other pairs of values for x and y.一个等式若仅仅对其中一个字母的某些值成立，或对其中两个或者多个字母的若干组相关的值成立，则它是一个条件等式，简称方程。因此3x-5=7仅当x=4 时成立，而2x-y=0，当x=6,y=2时成立，且对x, y的其他许多对值也成立。12A root of an equation is any number or number symbol which satisfies the equation. To obtain the root or roots of an equation is called solving an equation.方程的根是满足方程的任意数或者数的符号。求方程根的过程被称为解方程。There are various kinds of equations. They are linear equation, quadratic equation, etc.方程有很多种，例如： 线性方程，二次方程等。13To solve an equation means to find the value of the unknown term. To do this , we must, of course, change the terms about until the unknown term stands alone on one side of the equation, thus making it equal to something on the other side. We then obtain the value of the unknown and the answer to the question.解方程意味着求未知项的值，为了求未知项的值，当然必须移项，直到未知项单独在方程的一边，令其等于方程的另一边，从而求得未知项的值，解决了问题。To solve the equation, therefore, means to move and change the terms about without making the equation untrue, until only the unknown quantity is left on one side ,no matter which side.因此解方程意味着进行一系列的移项和同解变形，直到未知量被单独留在方程的一边，无论那一边。14Equations are of very great use. We can use equations in many mathematical problems. We may notice that almost every problem gives us one or more statements that something is equal to something, this gives us equations, with which we may work if we need to.方程作用很大，可以用方程解决很多数学问题。注意到几乎每一个问题都给出一个或多个关于一个事情与另一个事情相等的陈述，这就给出了方程，利用该方程，如果我们需要的话，可以解方程。New Words & Expressions:numerical 数值的，数的position 位置，状态 cube n. 立方体sphere n. 球cylinder n. 柱体cone 圆锥geometrical 几何的triangle 三角形surface 面， 曲面pyramid 菱形plane 平面solid 立体，立体的straight line 直线line segment 直线段broken line 折线ray 射线equidistant 等距离的curve 曲线，弯曲2.2 几何与三角Geometry and Trigonology1New Words & Expressions:side 边angle 角radius（radii） 半径diameter 直径endpoint 端点circle 圆周，圆semicircle 半圆arc 弧minor arc 劣弧major arc 优弧acute angle 锐角right angle 直角hypotenuse 斜边adjacent side 邻边chord 弦circumference 周长2Many leading institutions of higher learning have recognized that positive benefits can be gained by all who study this branch of mathematics. 2A Why study geometry?许多居于领导地位的学术机构承认，所有学习这个数学分支的人都将得到确实的受益。This is evident from the fact that they require study of geometry as a prerequisite to matriculation in those schools.许多学校把几何的学习作为入学考试的先决条件，从这一点上可以证明。3Geometry had its origin long ago in the measurement by the Babylonians and Egyptians of their lands inundated by the floods of the Nile River. 几何学起源于很久以前巴比伦人和埃及人测量他们被尼罗河洪水淹没的土地。The greek word geometry is derived from geo, meaning “earth” and metron, meaning “measure” . 希腊语几何来源于geo ，意思是”土地“，和metron 意思是”测量“。4As early as 2000 B.C. we find the land surveyors of these people re-establishing vanishing landmarks and boundaries by utilizing the truths of geometry .公元前2000年之前，我们发现这些民族的土地测量者利用几何知识重新确定消失了的土地标志和边界。One of the most important objectives derived from a study of geometry is making the student be more critical in his listening, reading and thinking. In studying geometry he is led away from the practice of blind acceptance of statements and ideas and is taught to think clearly and critically before forming conclusions.几何的学习使学生在思考问题时更周密、审慎，他们将不会盲目接受任何结论.5A solid is a three-dimensional figure. Common examples of solids are cube, sphere, cylinder, cone and pyramid.2B Some geometrical terms立体是一个三维图形，立体常见的例子是立方体，球体，柱体，圆锥和棱锥。A cube has six faces which are smooth and flat. These faces are called plane surfaces or simply planes.立方体有6个面，都是光滑的和平的，这些面被称为平面曲面或者简称为平面。6A plane surface has two dimensions, length and width. The surface of a blackboard or of a tabletop is an example of a plane surface.平面曲面是二维的，有长度和宽度，黑板和桌子上面的面都是平面曲面的例子。A circle is a closed curve lying in one plane, all points of which are equidistant from a fixed point called the center.平面上的闭曲线当其中每点到一个固定点的距离均相当时叫做圆。固定点称为圆心。7A line segment drawn from the center of the circle to a point on the circle is a radius of the circle. The circumference is the length of a circle.经过圆心且其两个端点在圆周上的线段称为这个园的直径，这条曲线的长度叫做周长。One of the most important applications of trigonometry is the solution of triangles. Let us now take up the solution to right triangles. 三角形最重要的应用之一是解三角形，现在我们来解直角三角形。8A triangle is composed of six parts three sides and three angles. To solve a triangle is to find the parts not given. 一个三角形由6个部分组成，三条边和三只角。解一个三角形就是要求出未知的部分。A triangle may be solved if three parts (at least one of these is a side ) are given. A right triangle has one angle, the right angle, always given. Thus a right triangle can be solved when two sides, or one side and an acute angle, are given.如果三角形的三个部分（其中至少有一个为边）为已知，则此三角形就可以解出。直角三角形的一只角，即直角，总是已知的。因此，如果它的两边，或一边和一锐角为已知，则此直角三角形可解。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 void set(empty set) 空集2.3 集合论的基本概念Basic Concepts of the Theory of Sets1The 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. 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。3A Notations for denoting setsSets 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。2We 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 displayed 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”。3The 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.只有当省略的内容清楚时才能使用圆点。在大括号中列出集合元素的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:联系一个集合与另一个集合的第一个基本概念是集合相等。 4DEFINITION 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 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。5EXAMPLE 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.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。6EXAMPLE 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 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,s,和p.7From 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。8In 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的集合S，而我们只关心这个给定集合的子集。基础集可以随意改变，可以在每一段特定的论述中表示全集。9It 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。10To 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）为了避免遇到逻辑困难，我们必须区分元素x和集合x，集合 x中的元素是x。特别要注意的是空集和集合是不同的。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 sets.事实上，空集不含有任何元素，而有一个元素。由一个元素构成的集合有时被称为单元素集。11Diagrams 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 shaded portion is a subset of A and also a subset of B. （35页第五段）图解有助于我们将集合之间的关系形象化。例如，可以把集合S看作平面内的一个区域，其中的每一个元素即是一个点。 那么S的子集就是S内某些点的全体。例如，在图2-3-1中阴影部分是A的子集，同时也是B的子集。12Visual aids of this type, called Venn diagrams, 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，在集合论中常用于检验定理的有效性或者为证明定理提供一些潜在的方法。当然证明本身必须依赖于概念的定义而不是图解。New Words & Expressions:conversely 反之geometric interpretation 几何意义correspond 对应induction 归纳法deducible 可推导的proof by induction 归纳证明difference 差inductive set 归纳集distinguished 著名的inequality 不等式entirely complete 完整的integer 整数Euclid 欧几里得interchangeably 可互相交换的Euclidean 欧式的intuitive直观的the field axiom域公理irrational 无理的2.4 整数、有理数与实数Integers, Rational Numbers and Real Numbers1New Words & Expressions:irrational number 无理数rational 有理的the order axiom 序公理rational number 有理数ordered 有序的reasoning 推理product 积scale 尺度，刻度quotient商sum 和2There exist certain subsets of R which are distinguished because they have special properties not shared by all real numbers. In this section we shall discuss such subsets, the integers and the rational numbers.4A Integers and rational numbers有一些R的子集很著名，因为他们具有实数所不具备的特殊性质。在本节我们将讨论这样的子集，整数集和有理数集。3To introduce the positive integers we begin with the number 1, whose existence is guaranteed by Axiom 4. The number 1+1 is denoted by 2, the number 2+1 by 3, and so on. The numbers 1,2,3, obtained in this way by repeated addition of 1 are all positive, and they are called the positive integers.我们从数字1开始介绍正整数，公理4保证了1的存在性。1+1用2表示，2+1用3表示，以此类推，由1重复累加的方式得到的数字1,2,3，都是正的，它们被叫做正整数。4Strictly speaking, this description of the positive integers is not entirely complete because we have not explained in detail what we mean by the expressions “and so on”, or “repeated addition of 1”. 严格地说，这种关于正整数的描述是不完整的，因为我们没有详细解释“等等”或者“1的重复累加”的含义。 5Although the intuitive meaning of expressions may seem clear, in careful treatment of the real-number system it is necessary to give a more precise definition of the positive integers. There are many ways to do this. One convenient method is to introduce first the notion of an inductive set.虽然这些说法的直观意思似乎是清楚的，但是在认真处理实数系统时必须给出一个更准确的关于正整数的定义。 有很多种方式来给出这个定义，一个简便的方法是先引进归纳集的概念。6DEFINITION OF AN INDUCTIVE SET. A set of real numbers is called an inductive set if it has the following two properties:The number 1 is in the set.For every x in the set, the number x+1 is also in the set.For example, R is an inductive set. So is the set . Now we shall define the positive integers to be those real numbers which belong to every inductive set.现在我们来定义正整数，就是属于每一个归纳集的实数。7Let P denote the set of all positive integers. Then P is itself an inductive set because (a) it contains 1, and (b) it contains x+1 whenever it contains x. Since the members of P belong to every inductive set, we refer to P as the smallest inductive set.用P表示所有正整数的集合。那么P本身是一个归纳集，因为其中含1，满足(a)；只要包含x就包含x+1, 满足(b)。由于P中的元素属于每一个归纳集，因此P是最小的归纳集。8This property of P forms the logical basis for a type of reasoning that mathematicians call proof by induction, a detailed discussion of which is given in Part 4 of this introduction. P的这种性质形成了一种推理的逻辑基础，数学家称之为归纳证明，在介绍的第四部分将给出这种方法的详细论述。9The negatives of the positive integers are called the negative integers. The positive integers, together with the negative integers and 0 (zero), form a set Z which we call simply the set of integers.正整数的相反数被叫做负整数。正整数，负整数和零构成了一个集合Z，简称为整数集。10In a thorough treatment of the real-number system, it would be necessary at this stage to prove certain theorems about integers. For example, the sum, difference, or product of two integers is an integer, but the quotient of two integers need not to ne an integer. However, we shall not enter into the details of such proofs.在实数系统中，为了周密性，此时有必要证明一些整数的定理。例如，两个整数的和、差和积仍是整数，但是商不一定是整数。然而还不能给出证明的细节。11Quotients of integers a/b (where b0) are called rational numbers. The set of rational numbers, denoted by Q, contains Z as a subset. The reader should realize that all the field axioms and the order axioms are satisfied by Q. For this reason, we say that the set of rational numbers is an ordered field. Real numbers that are not in Q are called irrational.整数a与b的商被叫做有理数，有理数集用Q表示，Z是Q的子集。读者应该认识到Q满足所有的域公理和序公理。因此说有理数集是一个有序的域。不是有理数的实数被称为无理数。12The reader is undoubtedly familiar with
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