數(shù)學(xué)專業(yè)英語2-11,2-12

1、New Words & Expressions:assert 斷言，主張斷言，主張 predicate 謂詞謂詞 conjunction 合取合取 quantifier 量詞量詞connective 連詞連詞 quantification 量詞化量詞化 disjunction 析取析取 statement 語句語句2.11 數(shù)理邏輯入門數(shù)理邏輯入門Elementary Mathematical LogicKey points: 1. 了解謂詞和量詞的基本表示方法。了解謂詞和量詞的基本表示方法。 2 . 掌握概率論基本的表示方法。掌握概率論基本的表示方法。Statements invol

2、ving variables, such as “x3”, “x+y=3”, “x+y=z” are often found in mathematical assertion and in computer programs.11-A Predicates包含變量的語句，比如包含變量的語句，比如“x3”, “x+y=3”, “x+y=z” 常出現(xiàn)在數(shù)常出現(xiàn)在數(shù)學(xué)論斷和計算機(jī)程序中。學(xué)論斷和計算機(jī)程序中。These statements are neither true nor false when the values of the variables are not specified.

3、In this section we will discuss the ways that propositions can be produced from such statements.若未給語句中的所有變量賦值，則不能判定該語句是真是假，若未給語句中的所有變量賦值，則不能判定該語句是真是假，本節(jié)要討論由這種語句生成命題的方法。本節(jié)要討論由這種語句生成命題的方法。 The statement “x is greater than 3” has two parts. The first part, the variables, is the subject of the statement

4、.語句語句“x大于大于3”分成兩部分，第一部分，變量，是語分成兩部分，第一部分，變量，是語句的主語。句的主語。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，謂語，第二部分，謂語，“大于大于3”，指的是語句主語具有的，指的是語句主語具有的性質(zhì)。性質(zhì)。We can denote the statement “x is greater than 3” by P(x), where P denotes the

5、predicate “is greater than 3” and x is the variable.把語句把語句“x大于大于3”記為記為P(x), 其中其中P表示謂詞表示謂詞“大于大于3”，而而x是變量。是變量。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 trut

6、h value.語句語句P(x)也稱為命題函數(shù)也稱為命題函數(shù)P在在x點處的值。一旦賦予變點處的值。一旦賦予變量量x一個值，語句一個值，語句P(x)就成為一個命題就成為一個命題,有了真假值。有了真假值。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 propositi

7、on from a propositional function.當(dāng)命題函數(shù)所有變量都賦值時，結(jié)果語句就有了真假當(dāng)命題函數(shù)所有變量都賦值時，結(jié)果語句就有了真假值。但是還有另外一種方式，稱為量詞化，可從命題值。但是還有另外一種方式，稱為量詞化，可從命題函數(shù)中得到命題。函數(shù)中得到命題。11-B QuantifiersTwo types of quantification will be discussed here, namely, universal quantification and existential quantification .這里討論兩種量詞化方法，這里討論兩種量詞化方法，也就

8、是也就是全稱量詞化和存在全稱量詞化和存在量詞化。量詞化。Many mathematical statements assert that a property is true for all values of a variable in a particular domain, called the universe of discourse. 許多數(shù)學(xué)語句認(rèn)為，性質(zhì)對論域這個特定領(lǐng)域內(nèi)變量許多數(shù)學(xué)語句認(rèn)為，性質(zhì)對論域這個特定領(lǐng)域內(nèi)變量的所有值都成立。的所有值都成立。Such a statement is expressed using a universal quantification.

9、這樣的語句可用全稱量詞化表示。這樣的語句可用全稱量詞化表示。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 specifies the possible values of the variable x.命題函數(shù)的全稱量詞化是一個命題，認(rèn)為命題函數(shù)的全稱量詞化是一個命題，認(rèn)為P(x)

10、對論域?qū)φ撚蛑兄衳的所有值的所有值P(x)都是真的。論域指定變量都是真的。論域指定變量x的可能取的可能取值值. 本小節(jié)重點掌握本小節(jié)重點掌握本節(jié)要討論由這種語句生成命題的方法。本節(jié)要討論由這種語句生成命題的方法。The ways that propositions can be produced from such statements will be discussed in this section.New Words & Expressionsevent 事件事件 sample 樣本樣本population 總體總體 statistics 統(tǒng)計學(xué)統(tǒng)計學(xué) probability 概

11、率概率2.12 概率論與數(shù)理統(tǒng)計概率論與數(shù)理統(tǒng)計Probability Theory and Mathematical StatisticsIn 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.”在討論概率論時，會常常從日常用語中看到這樣的語在討論概率論時，會常常從日常用語中看到這樣的

12、語句：兩個事件是同等可能的，一個事件是不可能的，句：兩個事件是同等可能的，一個事件是不可能的，一個事件肯定發(fā)生。一個事件肯定發(fā)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.這種表達(dá)方式非常直觀，在數(shù)學(xué)討論中，樂于使用這這種表達(dá)方式非常直觀，在數(shù)學(xué)討論中，樂于使用這樣有色彩的語言，而且使用起來很有幫助。樣有色彩的語言，而且使用起來很

13、有幫助。Before we can do so, however, it is necessary to explain the meaning of this language in terms of the fundamental concepts of our theory.但是，在我們這么做之前，有必要根據(jù)我們理論的基但是，在我們這么做之前，有必要根據(jù)我們理論的基本概念來解釋這種語句的含義。本概念來解釋這種語句的含義。 Because of the way probability is used in practice, it is convenient to imagine that

14、 each probability space (S,B,P) is associated with a real or conceptual experiment.根據(jù)概率論實際應(yīng)用的方式，把每一個概率空間根據(jù)概率論實際應(yīng)用的方式，把每一個概率空間(S,B,P)想象成對應(yīng)于一個實際的或者概念上的試驗想象成對應(yīng)于一個實際的或者概念上的試驗是很方便的。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

15、example of coin tossing discussed in the foregoing section.全集全集S是試驗中所有可能結(jié)果的集體，就像前面章節(jié)是試驗中所有可能結(jié)果的集體，就像前面章節(jié)討論的擲硬幣的例子。討論的擲硬幣的例子。Each element of S is called an outcome or a sample and the subsets of S that occur in the Boolean algebra B are called events. The reasons for this terminology will become more

16、 apparent when we treat some examples.S的每一個元素稱為結(jié)果或者樣本，在布爾代數(shù)的每一個元素稱為結(jié)果或者樣本，在布爾代數(shù)B中中出現(xiàn)的出現(xiàn)的S的子集稱為事件，為什么使用這個術(shù)語在的子集稱為事件，為什么使用這個術(shù)語在我們舉例后就會很明顯。我們舉例后就會很明顯。Assume we have a probability space (S,B,P) associated with an experiment. Let A be an event, and suppose the experiment is performed and that its outcome

17、 is x. (In other words, let x be a point of S.)假設(shè)有一個對應(yīng)于某一個試驗的概率空間假設(shè)有一個對應(yīng)于某一個試驗的概率空間(S,B,P) 。A是一個事件，假設(shè)試驗已經(jīng)完成，結(jié)果是是一個事件，假設(shè)試驗已經(jīng)完成，結(jié)果是x(換句話說，換句話說，x是是S中的一個點中的一個點)。This outcome x may or may not belong to the set A. If it does, we say that the event A has occurred.結(jié)果結(jié)果x可能屬于集合可能屬于集合A，也可能不屬于，也可能不屬于A。如果屬于，。如果屬

18、于，則稱事件則稱事件A發(fā)生。發(fā)生。否則，稱事件否則，稱事件A不發(fā)生，那么余事件發(fā)生。不發(fā)生，那么余事件發(fā)生。如果如果A等于空集，事件等于空集，事件A稱為不可能事件，因為在這稱為不可能事件，因為在這種情況下試驗的任何結(jié)果都不是種情況下試驗的任何結(jié)果都不是A中的元素。中的元素。Otherwise, we say that the event A has not occurred, in which case , so the complementary event has occurred.xAAAn event A is called impossible if , because in thi

19、s case no outcome of the experiment can be an element of A.A The event A is said to be certain if A=S, because then every outcome is automatically an element of A.如果如果A=S，則稱事件，則稱事件A是必然事件，因為每一個結(jié)果是必然事件，因為每一個結(jié)果必然是必然是A中的元素。中的元素。Each event A has a probability P(A) assigned to it by the probability functi

20、on P. 每一個事件每一個事件A都通過概率函數(shù)都通過概率函數(shù)P被賦予一個概率被賦予一個概率P(A)。The number P(A) is also called the probability that an outcome of the experiment is one of the elements of A.數(shù)數(shù)P(A)又稱為試驗的結(jié)果，是又稱為試驗的結(jié)果，是A的一個元素的概率。的一個元素的概率。We also say that P(A) is the probability that the event A occurs when the experiments is perfor

21、med.也稱也稱P(A)是試驗完成時事件是試驗完成時事件A出現(xiàn)的概率。出現(xiàn)的概率。The impossible event must be assigned probability zero because P is a finitely additive measure. However, there may be events with probability zero that are not impossible.因為因為P是有限可加測度，所以不可能事件被賦予零是有限可加測度，所以不可能事件被賦予零概率。然而，也存在具有零概率的事件，但它并不概率。然而，也存在具有零概率的事件，但它并不是不可能事件。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