数学专业英语2.11A翻译

11-A predicatesStatements involving variables, such as “x>3”, ”x+y=3”, ”x+y=z” are often found in mathematical assertion and in computer programs. 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. 包含变量的语句，比如“x>3”, ”x+y=3”, ”x+y=z” 常出现在数学论断中和计算机程序中，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 statement. The second part-the predicate, “is greater than 3”-refers to a property that the subject of the statement can have. 语句“x大于3”分成两部分，第一部分，变量，是语句的主语。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 denote the predicate “is greater than 3” and x is the variab le. 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. 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。语句P(x)也称为命题函数P在x点处的值。一旦赋予x一个值，语句P(x)就成为一个命题,有了真值。11-B QuantifiersWhen 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. two types of quantification will be discussed here, namely, universal quantification and existential quantification . 当命题函数所有变量都赋值时，结果语句有真值，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。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 discourse. Such a statement is expressed using a universal quantification. 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. 许多数学语句认为，在特殊领域内对变量所有值都成立的性质称为论域。这样的语句可用全称量词化表示。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. 12-A Special terminology peculiar to probability theory 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.” expressions o f this sort have intuitive appeal and it is both pleasant and helpful to be able to employ such colorful language in mathematical discussions. 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，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. 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. 根据概率论实际应用的方式，很方便的把每一个概率空间（S,B,P）对应于一个实际的或者概念上的试验。全集S是试验中所有可信结果的集体，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 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。