数学专业英语(吴炯圻-第2版)2-4总结归纳.pptVIP

New Words Expressions: conversely 反之 geometric interpretation 几何意义 correspond 对应 induction 归纳法 deducible 可推导的 proof by induction 归纳证明 difference 差 inductive set 归纳集 distinguished 著名的 inequality 不等式 entirely complete 完整的 integer 整数 Euclid 欧几里得 interchangeably 可互相交换的 Euclidean 欧式的 intuitive直观的 the field axiom 域公理 irrational 无理的 ;;There exist certain subsets of R which are distinguished because they have special properties not shared by all real numbers. In this section we shall discuss such subsets, the integers and the rational numbers.;To introduce the positive integers we begin with the number 1, whose existence is guaranteed by Axiom 4. The number 1+1 is denoted by 2, the number 2+1 by 3, and so on. The numbers 1,2,3,…, obtained in this way by repeated addition of 1 are all positive, and they are called the positive integers.;Strictly speaking, this description of the positive integers is not entirely complete because we have not explained in detail what we mean by the expressions “and so on”, or “repeated addition of 1”. ;Although the intuitive meaning of expressions may seem clear, in careful treatment of the real-number system it is necessary to give a more precise definition of the positive integers. There are many ways to do this. One convenient method is to introduce first the notion of an inductive set.;DEFINITION OF AN INDUCTIVE SET. A set of real numbers is called an inductive set if it has the following two properties: The number 1 is in the set. For every x in the set, the number x+1 is also in the set. For example, R is an inductive set. So is the set . Now we shall define the positive integers to be those real numbers which belong to every inductive set.;Let P denote the set of all positive integers. Then P is itself an inductive set because (a) it contains 1, and (b) it contains x+1 whenever it contains x. Since the members of P bel