2.3-A Basic Concepts of the Theory of Sets
合集下载
- 1、下载文档前请自行甄别文档内容的完整性，平台不提供额外的编辑、内容补充、找答案等附加服务。
- 2、"仅部分预览"的文档,不可在线预览部分如存在完整性等问题,可反馈申请退款(可完整预览的文档不适用该条件!)。
- 3、如文档侵犯您的权益，请联系客服反馈,我们会尽快为您处理(人工客服工作时间：9:00-18:30)。
第一个基本概念就是一个集合等于另一个集合
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.
还要了解单词： capital letters: 大写字母 lower-case letters:小写字母 Element：元素 Integer：整数
The concept of a set has been utilized so extensively throughout modern mathematics that an understanding of it is necessary for all college students. 集合的概念已经在经典的数学领域中被广泛使 用，大学生对集合概念的理解十分必要。
例2：{2,4,6,8} 和{2,2,4,4,6,8}这两个集合也 是相等的，即使在第二个集合中2和4都重复 出现。这两个集合中都只包含2,4,6,8这些元 素，所以，根据定义它们也是相等的
. This example shows that we do not insist that the objects listed in the roster notation be distinct.
集合相等的定义：如果A、B两个集合是由 相同的元素组成，那么就说这两个集合相 等，记为A=B。如果一个集合中有一个元 素不在另一个集合中，那么这两个集合不 相等，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.
elements are designated by lowercase letters:a,b,c,…,x,y,z.
集合（名）通常以大写字母表示： A,B,C,….X,Y,Z;
集合的元素通常以小写字母表示： a,b,c,…,x,y,z.
We use the special notation x S
2.3集合论的基本概念
（Basic Concepts of the Theory of Sets）
main content : this article describe a concept and characteristics of set and points out how to denote it.
to mean that “x is an element of S”or “x belong to S.”If x does not belong to S,we write .
xS
我们用特殊的符号来表示“x是S中的一个元素”或者 “x属于S。”如果x不属于S，我们用符号表达成xS
When convenient, we shall designate sets by displaying the elements in braces; 通常,我们应指定元素集通过在括号内显示。
总结：这篇文章讲述了集合的抽象概念，指出了集合 的表示方法：列举法和描述法；同时也介绍了集合的 特点：无序性和互异性。
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 are{2,4,6,…},
这三个点替代了“等等”，这些点只用于当意 思“等”是清楚的时候
The method of listing the members of a set within braces is sometimes referred to as the roster notation. 这种用括号列出集合中的元素的方法有时被称 为列举法 The first basic concept that relates one set to another is equality of sets :
另一个类似的例子，Mississippi这个字母集合 与集合{M,i,s,p}都包含了四个截然不同的字母 M,i,s, 和 p。所以这两个集合也是相等的。
summary ：From this article we know that Set are a means by which mathematicians talk of collections of things in an abstract way. We can describe it through the method of listing and description；Also, this article describe the characteristics of set.
例1：根据这个定义，由于{2,4,6,8}和 {2,8,6,4}这两个集合都包含2,4,6, 和 8四个 整数，所以说这两个集合是相等的。因此， 当我们使用符号来描述一集合时，元素出 现的顺序是无关紧要的。
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 .
摘要（1）集合 （2）集合的表示方法 （3）集合的性质
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
Set are a means by which mathematicians talk of collections of things in an abstract way.
集Байду номын сангаас是数学家们以抽象的方式来讨论某一 类对象的方法。
Set usually are denoted by capital letters:A,B,C,….X,Y,Z;
这个例子表明,我们不反对说在列出来的元 素中相同的符号是不同的。（集合中元素 的互异性）
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.
例如小于10的整型偶数集合用符号{2,4,6,8} 表示，而所有的偶数整型偶数集合表示为 {2,4,6,…},
the three dots taking the place of “and so on.” The dots are used only when the meaning of “and so on” is clear.
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
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.
还要了解单词： capital letters: 大写字母 lower-case letters:小写字母 Element：元素 Integer：整数
The concept of a set has been utilized so extensively throughout modern mathematics that an understanding of it is necessary for all college students. 集合的概念已经在经典的数学领域中被广泛使 用，大学生对集合概念的理解十分必要。
例2：{2,4,6,8} 和{2,2,4,4,6,8}这两个集合也 是相等的，即使在第二个集合中2和4都重复 出现。这两个集合中都只包含2,4,6,8这些元 素，所以，根据定义它们也是相等的
. This example shows that we do not insist that the objects listed in the roster notation be distinct.
集合相等的定义：如果A、B两个集合是由 相同的元素组成，那么就说这两个集合相 等，记为A=B。如果一个集合中有一个元 素不在另一个集合中，那么这两个集合不 相等，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.
elements are designated by lowercase letters:a,b,c,…,x,y,z.
集合（名）通常以大写字母表示： A,B,C,….X,Y,Z;
集合的元素通常以小写字母表示： a,b,c,…,x,y,z.
We use the special notation x S
2.3集合论的基本概念
（Basic Concepts of the Theory of Sets）
main content : this article describe a concept and characteristics of set and points out how to denote it.
to mean that “x is an element of S”or “x belong to S.”If x does not belong to S,we write .
xS
我们用特殊的符号来表示“x是S中的一个元素”或者 “x属于S。”如果x不属于S，我们用符号表达成xS
When convenient, we shall designate sets by displaying the elements in braces; 通常,我们应指定元素集通过在括号内显示。
总结：这篇文章讲述了集合的抽象概念，指出了集合 的表示方法：列举法和描述法；同时也介绍了集合的 特点：无序性和互异性。
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 are{2,4,6,…},
这三个点替代了“等等”，这些点只用于当意 思“等”是清楚的时候
The method of listing the members of a set within braces is sometimes referred to as the roster notation. 这种用括号列出集合中的元素的方法有时被称 为列举法 The first basic concept that relates one set to another is equality of sets :
另一个类似的例子，Mississippi这个字母集合 与集合{M,i,s,p}都包含了四个截然不同的字母 M,i,s, 和 p。所以这两个集合也是相等的。
summary ：From this article we know that Set are a means by which mathematicians talk of collections of things in an abstract way. We can describe it through the method of listing and description；Also, this article describe the characteristics of set.
例1：根据这个定义，由于{2,4,6,8}和 {2,8,6,4}这两个集合都包含2,4,6, 和 8四个 整数，所以说这两个集合是相等的。因此， 当我们使用符号来描述一集合时，元素出 现的顺序是无关紧要的。
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 .
摘要（1）集合 （2）集合的表示方法 （3）集合的性质
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
Set are a means by which mathematicians talk of collections of things in an abstract way.
集Байду номын сангаас是数学家们以抽象的方式来讨论某一 类对象的方法。
Set usually are denoted by capital letters:A,B,C,….X,Y,Z;
这个例子表明,我们不反对说在列出来的元 素中相同的符号是不同的。（集合中元素 的互异性）
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.
例如小于10的整型偶数集合用符号{2,4,6,8} 表示，而所有的偶数整型偶数集合表示为 {2,4,6,…},
the three dots taking the place of “and so on.” The dots are used only when the meaning of “and so on” is clear.
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