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CHAPTER 8 o H.. 1) E H. E D S E T S A. Order relations 1、he concept of order in elemcntary mathematics and in daily lifc‘ is so familiar to everybody that a motivation seems hardly to be necessary here. In fact we have discussed at some Icngth the usual order relation of natural numbers. In this chapter wc shall develop the general theory of order relations within the framework of 泊t theory. The familiar results of the usual order relation of natural numbers mav now serve as examples to illustrate the more abstract concepts of this chapter. DEFINITION 8.1. Let A be a set. Then a relation R d拼ned ùz A is said to be an order rclation in A lf and only if the following cOllditions are satisfied: (i) aRb and bRa ~f and only if a = b; and (ii) if aRb alld bRc, then aRc. The condition (i) means that R is refiexive and anti-synznzetric; (ii) means that R is transitive. It is c1ear that if R is an order relation in A , then the inverse relation R-l is also an order relation in A. DEFINITION 8.2. An ordered set is an ordered pair (A ,R) where A is a set and R is an order relatioll in A. When dealing with one ordered set (A ,R) at a time, wefind thefollowing abbreviations and notations convcnient: (a) We replace the symbols R and R-l respectively by the familiar inequality signs 三 and 這; (b) we write A for the ordercd set (A ， 三) if no confusion is possible, and in this case an clement or a subset of A is understood to be respectively an element or a subsct of the set A; (c) we say that two elements x and y of A are 三-comparable if x~三 y or y ~三 x; (d) for x 豆 y， we say, as the case may be, that x is less than or equal to y , x is smaller than or equal to y , x precedes y , or that x is inferior to y; Sec. A] ORDER RELATIONS 103 (e) for y 呈几 we say, as the case may be, that y is greater than or equal to x, y is larger than or equal to x, y follows x, or that y is superior to x; and (f) for x 三 y and x 手 y， we write x x. When several order re1ations are under consideration, signs such as 豆， 逗， 三三， 三三， etc. can be used. EXAMPLE 8.3. The usual order relation of natural numbers is clearly an order rdation in the set N of all natural numbers. EXAMPLE 8.4. Let A be a set. For any two subsets X and Y of A ,we put X 豆 Y if and only if X c Y. It can immediate1y be seen that 三 is an order relation in the power set ~(A) of A , and we say that 帶(A) is ordered by inc/usioll. EXAMPLE 8.5. Let .4 and B be scts and consider the set E of all mappings f:A' • B, where A I is any subset of A. We can define an ordcr relation in E as follows: for any two mappings f :A ' • B andg :.4"• B of E, f 三 g if and 01砂 if A' c A" alld g : A' = J. We also say that E is oTdered by extension. EXAMPLE 8.6. Let 阱，三) be an ordered sct and B a subset of the set A. We define an order re1ation S in B as follows: for any two dcments x and y of B xSy if and only if x 歪 y. This order re1ation S in B is said to be induced by the order relation 豆 m A, and the ordered set B is ca11ed a subset of the ordered set A. If no confusion is possible, we sha11 again denote S by 三:: • From the above examples, we see that in thc ordered set of 8.3 any two e1ements are 三 -comparable， whcreas this is not the case for thc ordered sets of 8.4 and 8.5 when A consists of more than one e1ement. DEFINITION 8.7. An ordered set (A ， 三) is said to be tota11y ordered, or a chain, if and only if any two elements of A aTe 豆 -comparable. It fo11ows immediate1y from the definition that the familiar law of trichotomy holds in any totally ordered set A: for any two e1ements x and y of A , it is...

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