1、New Words & Expressions: assert 断言，主张 predicate 谓词 conjunction 合取 quantifier 量词 connective 连词 quantification 量词化 disjunction 析取 statement 语句,2.11 数理逻辑入门 Elementary Mathematical Logic,Key points: introduction to predicates and quantifiers Difficult points: special terminology peculiar to probability
2、theory,Requirements:,1. 了解谓词和量词的基本表示方法。2 . 掌握概率论基本的表示方法。,Statements involving variables, such as“x3”, “x+y=3”, “x+y=z” are often found in mathematical assertion and in computer programs.,11-A Predicates,包含变量的语句，比如“x3”, “x+y=3”, “x+y=z” 常出现在数学论断和计算机程序中。,These statements are neither true nor false whe
3、n the values of the variables are not specified. In this section we will discuss the ways that propositions can be produced from such statements.,若未给语句中的所有变量赋值，则不能判定该语句是真是假，本节要讨论由这种语句生成命题的方法。,The statement “x is greater than 3” has two parts. The first part, the variables, is the subject of the stat
4、ement.,语句“x大于3”分成两部分，第一部分，变量，是语句的主语。,The second part-the predicate, “is greater than 3”-refers to a property that the subject of the statement can have.,第二部分，谓语，“大于3”，指的是语句主语具有的性质。,We can denote the statement “x is greater than 3” by P(x), where P denotes the predicate “is greater than 3” and x is t
5、he variable.,把语句“x大于3”记为P(x), 其中P表示谓词“大于3”，而x是变量。,The statement P(x) is also said to be the value of the propositional function P at x. Once a value has been assigned to the variable x, the statements P(x) becomes a proposition and has a truth value.,语句P(x)也称为命题函数P在x点处的值。一旦赋予变量x一个值，语句P(x)就成为一个命题,有了真
6、假值。,When all the variables in a propositional function are assigned values, the resulting statement has a truth value. However, there is another important way, called quantification, to create a proposition from a propositional function.,当命题函数所有变量都赋值时，结果语句就有了真假值。但是还有另外一种方式，称为量词化，可从命题函数中得到命题。,11-B Qu
7、antifiers,Two types of quantification will be discussed here, namely, universal quantification and existential quantification .,这里讨论两种量词化方法，也就是全称量词化和存在量词化。,Many mathematical statements assert that a property is true for all values of a variable in a particular domain, called the universe of discours
8、e.,许多数学语句认为，性质对论域这个特定领域内变量的所有值都成立。,Such a statement is expressed using a universal quantification.,这样的语句可用全称量词化表示。,The universal quantification of a propositional function is the proposition that assert that P(x) is true for all values of x in the universe of discourse. The universe of discourse spe
9、cifies the possible values of the variable x.,命题函数的全称量词化是一个命题，认为P(x)对论域中x的所有值P(x)都是真的。论域指定变量x的可能取值.,本小节重点掌握,本节要讨论由这种语句生成命题的方法。,The ways that propositions can be produced from such statements will be discussed in this section.,New Words & Expressions event 事件 sample 样本 population 总体 statistics 统计学 pr
10、obability 概率,2.12 概率论与数理统计 Probability Theory and Mathematical Statistics,In discussions involving probability, one often sees phrases from everyday language such as “two events are equally likely,” “an event is impossible,” or “an event is certain to occur.”,在讨论概率论时，会常常从日常用语中看到这样的语句：两个事件是同等可能的，一个事件
11、是不可能的，一个事件肯定发生。,Expressions of this sort have intuitive appeal and it is both pleasant and helpful to be able to employ such colorful language in mathematical discussions.,这种表达方式非常直观，在数学讨论中，乐于使用这样有色彩的语言，而且使用起来很有帮助。,Before we can do so, however, it is necessary to explain the meaning of this language
12、 in terms of the fundamental concepts of our theory. 但是，在我们这么做之前，有必要根据我们理论的基本概念来解释这种语句的含义。,Because of the way probability is used in practice, it is convenient to imagine that each probability space (S,B,P) is associated with a real or conceptual experiment.根据概率论实际应用的方式，把每一个概率空间(S,B,P)想象成对应于一个实际的或者概
13、念上的试验是很方便的。,The universal set S can then be thought of as the collection of all conceivable outcomes of the experiment, as in the example of coin tossing discussed in the foregoing section.全集S是试验中所有可能结果的集体，就像前面章节讨论的掷硬币的例子。,Each element of S is called an outcome or a sample and the subsets of S that
14、occur in the Boolean algebra B are called events. The reasons for this terminology will become more apparent when we treat some examples.,S的每一个元素称为结果或者样本，在布尔代数B中出现的S的子集称为事件，为什么使用这个术语在我们举例后就会很明显。,Assume we have a probability space (S,B,P) associated with an experiment. Let A be an event, and suppose
15、the experiment is performed and that its outcome is x. (In other words, let x be a point of S.),假设有一个对应于某一个试验的概率空间(S,B,P) 。A是一个事件，假设试验已经完成，结果是x(换句话说，x是S中的一个点)。,This outcome x may or may not belong to the set A. If it does, we say that the event A has occurred.,结果x可能属于集合A，也可能不属于A。如果属于，则称事件A发生。,否则，称事件
16、A不发生，那么余事件发生。,如果A等于空集，事件A称为不可能事件，因为在这种情况下试验的任何结果都不是A中的元素。,Otherwise, we say that the event A has not occurred, in which case , so the complementary event has occurred.,An event A is called impossible if , because in this case no outcome of the experiment can be an element of A.,The event A is said t
17、o be certain if A=S, because then every outcome is automatically an element of A.,如果A=S，则称事件A是必然事件，因为每一个结果必然是A中的元素。,Each event A has a probability P(A) assigned to it by the probability function P.,每一个事件A都通过概率函数P被赋予一个概率P(A)。,The number P(A) is also called the probability that an outcome of the exper
18、iment is one of the elements of A.数P(A)又称为试验的结果，是A的一个元素的概率。,We also say that P(A) is the probability that the event A occurs when the experiments is performed.也称P(A)是试验完成时事件A出现的概率。,The impossible event must be assigned probability zero because P is a finitely additive measure. However, there may be
19、events with probability zero that are not impossible.因为P是有限可加测度，所以不可能事件被赋予零概率。然而，也存在具有零概率的事件，但它并不是不可能事件。,In other words, some of the nonempty subsets of S may be assigned probability zero. The certain event S must be assigned probability 1 by the very definition of probability, but there may be othe
20、r subsets as well that are assigned probability 1.换句话说，S的某个非空子集也可能被赋予零概率。仅根据概率的定义就得把必然事件S的概率指定为1，但是可能有别的子集其概率也是1.,Two events A and B are said to be equally likely if P(A)=P(B). The event A is called more likely than B if P(A)P(B), and at least as likely as B if P(A) P(B).如果事件P(A)=P(B)，则A与B被称为同等可能。如果P(A)P(B), 则称事件A比事件B有更大的可能性，如果P(A) P(B), 则称事件A至少和事件B的可能性一样大。,本小节重点掌握,在试验中事件A出现的概率记做P(A).,The probability that the event A occurs in an experiment is denoted by P(A).,作业：,P95 2: (2)P87 2: (4),谢 谢！,
