数学专业英语2-3
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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。 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。
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） 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. 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。 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。
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。
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）
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.
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。 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.
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. The first basic concept that relates one set to another is equality of sets: 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。 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。
3－A Notations for denoting sets
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. Sets usually are denoted by capital letters; elements are designated by lower-case letters. 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
3－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. 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。
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 2-3-1 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。
2.3 集合论的基本概念 Basic Concepts of the Theory of Sets New Words & Expressions:
brace 大括号 consequence 结论，推论 designate 标记，指定 diagram 图形，图解 distinct 互不相同的 distinguish 区别，辨别 divisible 可被除尽的 dummy 哑的，哑变量 even integer 偶数 irrelevant 无关紧要的 roster 名册 roster notation 枚举法 rule out 排除，否决 subset 子集 the underlying set 基础集 universal set 全集 validity 有效性 visual 可视的 visualize 可视化 void set(empty set) 空集
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。
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）
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A≠B。
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.
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。
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. 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.
方式来表述一些事物的集体的工具。
集Fra Baidu bibliotek通常用大写字母表示，元素用小写字母表示。
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 .
※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.
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.” 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。 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”。
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） 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. 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。 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。
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。
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）
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.
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。 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.
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. The first basic concept that relates one set to another is equality of sets: 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。 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。
3－A Notations for denoting sets
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. Sets usually are denoted by capital letters; elements are designated by lower-case letters. 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
3－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. 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。
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 2-3-1 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。
2.3 集合论的基本概念 Basic Concepts of the Theory of Sets New Words & Expressions:
brace 大括号 consequence 结论，推论 designate 标记，指定 diagram 图形，图解 distinct 互不相同的 distinguish 区别，辨别 divisible 可被除尽的 dummy 哑的，哑变量 even integer 偶数 irrelevant 无关紧要的 roster 名册 roster notation 枚举法 rule out 排除，否决 subset 子集 the underlying set 基础集 universal set 全集 validity 有效性 visual 可视的 visualize 可视化 void set(empty set) 空集
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。
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）
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A≠B。
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.
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。
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. 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.
方式来表述一些事物的集体的工具。
集Fra Baidu bibliotek通常用大写字母表示，元素用小写字母表示。
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 .
※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.
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.” 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。 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”。