《集合的基本概念》PPT课件.ppt

2.3 集合论的基本概念 （Basic Concepts of the Theory of Sets）,braces n.(复数)大括号 even integer=even number 偶数 positive number 正数 the underlying set n.基础集，底集 universal set 全集 void set=empty set 空集 dummy adj.哑的；n.哑标 rule out 排除，否决 diagram n.图形，图解 Venn diagrams 文氏图 be divisible by 4 可被4整除的 proper subset 真子集,be said to be 被称为(在下定义时用) the reference to 的注解 be referred to as 被称为 be contained in 被包含在 take the place of 代替,二、要求掌握的句子 1、The concept of a set has been utilized so extensively throughout modern mathematics that an understanding of it is necessary for all college students. 译文：在整个现代数学中集合的概念应用地如此广泛以至于对于所有的大学生来说有必要理解它。 注：这是一个sothat句型，表示“如此以至于”。the concept of 表示“的概念”。,2、Sets are a means by which mathematicians talk of collections of things in an abstract way. 译文：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. 3、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. 译文：常常用大写字母A, B, C, ,X, Y, Z来表示集合；而用小写字母a, b, c, x, y, z来表示(集合中的)元素。 注：be denoted by和be designated by都表示“用来表示”。,4、We use the special notation xS to mean that “x is an element of S” or “x belongs to S”. 译文：我们用专门的记号xS来表示“x是S中的一个元素”或者“x属于S”。 5、If x does not belong to S, we write . 译文：若x不属于S, 我们记 。,6、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 three dots taking the place of “and so on”. 译文：当方便时，我们应当用大括号中的元素来表示集合；例如，小于10的正偶数的集合用符号2, 4, 6, 8来表示，而所有正偶数的集合可表示为2, 4, 6, ，其中三个点代表“等等”。 注意：也可用for convenience来代替when convenient.,7、 The method of listing the members of a set within braces is sometimes referred to as the roster notation. 译文：在大括号中列出集合中的元素的方法有时被称为花名册标记法。,8、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. 译文：称两个集合A和B是相等的(或等同的)， 若它们严格地由相同的元素组成，此时我们记A=B. 注：be said to be常在下定义时用，表示“称.是”。,9、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. 译文：根据这个定义可知，这两个集合2, 4, 6, 8 和2, 8, 6, 4 是相等的，因为它们都由4个整数2, 4, 6,和8组成。,课文3B Subsets(子集),1、From a given set S we may form new sets, called subsets of S. 译文：由给定的集合S可形成新的集合，其称为S的子集。 2、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. 译文：当集合A的每个元素也属于集合B，则称A是B的子集，并记为A B 。,3、We also say that A is contained in B or that B contains A. 译文：我们也称A被B包含或者B包含A。 注：We say，表示“称”，注意这是另一种下定义的方式。 4、The relation is referred to as set inclusion 译文：关系 被称为集合包含。,5、The statement A B does not rule out the probability B A. 译文：陈述A B并不排除B A的可能性。,6、In fact, we have both A B and B A, but this happens only if A and B have the same elements. 译文：事实上，可以得到A B和B A, 但只有A和B具有同样的元素时才能发生这种情况。 7、In other words, A=B if and only if A B and B A. 译文：换句话说，A=B 当且仅当 A B，B A 。,8、If A B but BA, then we say that A is a proper subset of B, we indicate this by writing A B 译文：若A B 但 BA, 则称A是B的一个真子集, 并表示为A B。 9、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. 译文：在所有的集合理论应用中，我们预先给定一个固定的集合，而我们仅仅关心的是这个给定集合的子集。 注：we have given in advance，表示“预先给定”，主要用于写论文时，事先给定整篇论文的通用条件。,10、The underlying set S may vary from one application to another; it will be referred to as the universal set of each particular discourse. 译文：底集S可能随着不同的应用而变；称它为每个特定应用的全集 。 11、The notation x|xS and x satisfies P will designate the set of all elements x in S which satisfy the property P. 译文：记号x|xS and x satisfies P 表示S中所有满足性质P的x的集合。,12、Sets designated in this way are said to be described by a defining property. 译文：用这种方式表示的集合被称为通过定义性质描述集合。 13、For example, the set of all positive real numbers could be designated as x|x0. 译文：例如，所有正实数的集合可表示为x|x0.,14、Of course, the letter x is a dummy and may be replaced by any other convenient symbol. 译文：当然，字母x是一个哑标并可用任何方便的符号代替它。,15、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. 译文：一个集合不包含任何元素是可能的。这个集合称为空集，并用符号来表示。,16、To avoid logical difficulties, we must distinguish between the element x and the set x whose only element is x. 译文：为了避免逻辑上的一些麻烦，我们必须将元素x和只有一个元素x的集合x区别开来。,17、In particular, is not the same as. 注：“ In particular”表示“特别地”，用于强调，引起注意。“is not the same as”表示“与.不一样”,18、Subsets of S may then be thought of as collections of points within S. 译文：也可将S的子集看成S里面一些点的构成的集合。 注意：be thought of as可替换为be considered as,2. 汉译英,(1) 由小于10且能被3整除的正整数组成的集是自然数集的一个子集。,译文: The set consisting of those positive integers less than 10 which are divisible by 3 is a subset of the set of all natural numbers.,(2) 如果方便，我们通过在括号中列举元素的方法来表示集。,译文：When convenient, we shall designate sets by displaying the elements in braces.,(3) 我们用符号“”表示集的包含关系。,译文: We use the symbol “” to denote the relation of set inclusion.,(5) 基础集可根据使用场合不同而改变。,译文: The underlying set can vary from one application to another.,(4) 命题AB并不排除BA的可能。,译文: The proposition AB does not rule out the possibility that BA.,(6) 为了避免逻辑上的困难，我们必须把元素x与仅含有元素x的集x区分开来。,译文: To avoid logical difficulties, we must distinguish between the element x and the set x with only one element x.,译文: Diagrams help us visualize relations between sets.,(7) 图解法有助于将集合之间的关系形象化。,(8) 定理的证明仅仅依赖于概念和已知的结论，而不依赖于图形。,译文: The proofs of the theorems only rely on the concepts and the conclusions known and not on the diagrams .,数词的一些用法,数词的词性与用法 数词分基数词和序数词两类,它们具有两种词性:形容词和名词性. There are two variables in this function. 在这个函数中有两个变量. Two of its inputs are irrelevant. 它的两个输入是互不相关的.,若表示一个大约数, 可用tens of(数十), dozens of(几十), scores of (好几十), hundreds of(数百), thousands of (数千, 成千上万), millions of(数百万)等等.,2. 数词可用作数量状语 The sum becomes two less than the total of the operands. 和数比运算数的总数少了“2”.,3. 序数词不强调次序时的用法 序数词从“第二”开始, 若不再强调次序而表示“另一个”时,科技文献中常用a second, a third等等. A second method for describing the behavior of this system is the use of a state diagram. 描述这个系统性能的第二种 (另一种)方法是采用状态图.,4. First可以与复数名词连用 First与复数名词连用时, 应翻译成“开头的, 首批的, 初始的, 最初的, 最早的”等词义. 这是first实际上已经属于形容词了. The first methods of proving the security of the digital signatures are not formal. 最初的数字签名安全证明方法并不正规. The firs