Transactions of the American Mathematical Society Volume 1-5. American Mathematical Society
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Author: American Mathematical Society
Number of Pages: 152 pages
Published Date: 06 Aug 2012
Publisher: Rarebooksclub.com
Publication Country: Miami Fl, United States
Language: English
ISBN: 9781130303254
Download Link: Transactions of the American Mathematical Society Volume 1-5
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This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1904 edition. Excerpt: ...of 3 when (c) is defined as follows: Definition. We shall write a (c) b (or b (c) a) when and only when a (c) b = b. The proof of 6, for example, is as follows: By F, B, C and A, f-(o (c) b) (c) a = a (c) (a (c) 6) = (a (c) a) (c) 6 = a (c) 6, and similarly (a(c)6)(c)6 = a(c)6; whence, a (c) b (c) a and a (c) b (c) 6. Further, if y (c) a and y(r)b, then a(r)y = y and b(r)y = y, whence (a (c) y) (c) (6 (c) y) = y (c) y, or(a(c)6)(c)y = y, or y (c) (a (c) 6). Postulate 7 follows as in 256, and we may define a (c) b thus: Definition, a Q b = a (r)b. The equivalence between the algebra of 3 and the algebras of 1 and 2 is thus readily established. Independence of the postulates of the third set. The independence of the nine postulates of the third set is shown by the following systems (iT, (c)), each of which satisfies all the other postulates, but not the one for which it is lettered. (A) The class of positive integers, and 0 and oo; with a (c) b = a + b. Here postulates B-F clearly hold. In G take a = oo when a oo, and a = any other element when a = oo. Postulate His satisfied vacuously. (i?) A class of two or more elements, with a (c) b = a. Here B clearly fails. In Z), F, and F, any elements will answer as A and V-Postulate G is satisfied vacuously, since A and V are not uniquely determined. In H, x = any element. (C) A class of six elements, 0,1,2, -,5, with (c) defined as in the accompanying table. Here C fails, since (2(c)4)(c)3=4(c)3 = 3, while 2(c)(4(c)3) = 2(c)3 = 1. In D and E, take A = 0 and 7 = 1. Postulate
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