This document is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. The copyright holder grants you permission to redistribute this document freely as a verbatim copy. Furthermore, the copyright holder permits you to develop any derived work from this document provided that the following conditions are met. a) The derived work acknowledges the fact that it is derived from this document, and maintains a prominent reference in the work to the original source. b) The fact that the derived work is not the original OpenMath document is stated prominently in the derived work. Moreover if both this document and the derived work are Content Dictionaries then the derived work must include a different CDName element, chosen so that it cannot be confused with any works adopted by the OpenMath Society. In particular, if there is a Content Dictionary Group whose name is, for example, `math' containing Content Dictionaries named `math1', `math2' etc., then you should not name a derived Content Dictionary `mathN' where N is an integer. However you are free to name it `private_mathN' or some such. This is because the names `mathN' may be used by the OpenMath Society for future extensions. c) The derived work is distributed under terms that allow the compilation of derived works, but keep paragraphs a) and b) intact. The simplest way to do this is to distribute the derived work under the OpenMath license, but this is not a requirement. If you have questions about this license please contact the OpenMath society at http://www.openmath.org. setname2 http://www.openmath.org/cd http://www.openmath.org/cd/setname2.ocd 2017-12-31 2004-03-30 3 1 Author: OpenMath Consortium SourceURL: https://github.com/OpenMath/CDs experimental This CD defines some common sets of mathematics. Written by J.H. Davenport on 1999-04-18. Revised to add Zm, GFp, GFpn on 1999-11-09. Revised to add QuotientField and A on 1999-11-19. Boolean constant This symbol represents the set of Booleans. That is the truth values, true and false. for all b in the booleans | (there exists an nb in the booleans | nb not= b implies nb = not b) A constant This symbol represents the set of algebraic numbers. The algebraic numbers are a proper subset of the reals The rationals are a proper subset of the algebraic numbers Zm application This symbol represents the set of integers modulo m, where m is not necessarily a prime. It takes one argument, the integer m. for all x in the integers modulo m | there exists an n such that n is an integer and n <= m and x^n = x The integers mod 12: 12 The integers mod m: 4*5=8 in Z mod 12 12 4 12 5 12 8 GFp application This symbol represents the finite field of integers modulo p, where p is a prime. x^p = x mod p GFpn application This symbol represents the finite field with p^n elements, where p is a prime. QuotientField application This symbol represents the quotient field of any integral domain. The rationals equals QuotientField(Integers) R is a field iff QuotientField(R)=R H constant This symbol represents the set of quaternions. 1 is a quaternion and there exists i,j,k s.t. i,j,k are quaternions and i^2 = j^2 = k^2 = ijk = -1 with abs(i) not = abs(j) not = abs(k) not = 1 implies for all q, q a quaternion implies there exists r_0, r_1, r_2, r_3 reals s.t. q = r_0 + r_1*i + r_2*j + r_3*k 2 2 2 There exists a,b in the quaternions s.t. a*b neq b*a