Download A treatise on trigonometry by George Biddell Airy PDF

By George Biddell Airy

ISBN-10: 1418179310

ISBN-13: 9781418179311

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Show that a set of four elements can be partially ordered in 219 ways; show that isomorphism classifies these 219 equivalent sets into 16 abstract par­ tially ordered sets. 36. A dyadic relation Q which is reflexive and transitive is called a relation of quasi-order. Let 5 be a set quasi-ordered by such a relation β ; define a new relation R between the elements of S by the following: aRb if and only if aQb and bQa; show that Ä is an equivalence relation. } is the set of equi­ valence classes into which R partitions S, and if we lay down that X^ Y if and only if xQy for ;c in A'and 7 in Y, show that with this relation Γ is a partially ordered set.

This set consists of sixteen elements: (4,1,1), (3,1,1), (2,1,1), (1,1,1), (4,1,2), (3,1,2), (2,1,2), (1,1,2), (4,2,1), (3,2,1), (2,2,1), (1,2,1), (4,2,2), (3,2,2), (2,2,2), (1,2,2); a partially ordered set with least or zero element (1, 1, 1) and greatest or unity element (4, 2, 2). For a diagram of this set see Chapter 2, §9, Fig. 12c. Exercises 37. Prove the statements made about Example 31. 38. Show that finite chains are self-dual but infinite chains not necessarily so. (Cf. Fig. ) 39. Prove that the chains of Examples 32 and 33 are dually isomorphic.

M. f. Example 10. ) Let (0, b), (c, d) denote ordered pairs of integers with 6, d not zero; we construct an algebra A from all such pairs by defining two binary operations: (i)(a, 6) + (c,d) = (úíd + 6c, bd); (H)(a,b)x{c,d) = {ac, bd). 28 L A T T I C E THEORY Thus (7, 2) + (5, 8) = (66, 16); (7, 2) χ (5, 8) = (35, 16). We now define a relation C over A by the rule: (iii) (a, b) C (c, d) if and only if ad = be. Thus (7, 2) = (21, 6). C is clearly reflexive and symmetric; transitivity is readily proved, for if {a, b) C(c, d) and (c, d) C(e,f),ad = be and cf = de, adf = ¿ c / = bde, hence α/ = and (a, b) C {e, f); therefore C is an equivalence relation.

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