数学专业英语2-11,2-12

包含变量的语句，比如“x>3”, “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 when the values of the variables are not specified. In this section we will discuss the ways that propositions can be produced from such statements.
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 particularse.
Key points:
introduction to predicates and quantifiers
Difficult points:
special terminology peculiar to probability
theory
Requirements:
1. 了解谓词和量词的基本表示方法。
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。
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 specifies 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.
2.11 数理逻辑入门 Elementary Mathematical Logic
New Words & Expressions:
assert 断言，主张 predicate 谓词
conjunction 合取
connective 连词 disjunction 析取
quantifier 量词
quantification 量词化 statement 语句
The event A is said to 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)。
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念上的试验 是很方便的。
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 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。
11-B Quantifiers
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。
Otherwise, we say that the event A has not occurred, in which case x A' , so the complementary event A' has occurred.
否则，称事件A不发生，那么余事件发生。 An event A is called impossible if A , because in this case no outcome of the experiment can be an element of A. 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。
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 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
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。
2 . 掌握概率论基本的表示方法。
11-A Predicates
Statements involving variables, such as “x>3”, “x+y=3”, “x+y=z” are often found in mathematical assertion and in computer programs.
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 the 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,有了真假值。
New Words & Expressions event 事件 sample 样本
population 总体
probability 概率
statistics 统计学
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，一个事件是不可能的， 一个事件肯定发生。