setname2 http://www.openmath.org/cd/setname2.ocd 2003-04-01 2000-02-02 2 0 experimental alg1 arith1 logic1 quant1 relation1 setname1 set1 sts 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 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 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 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 This symbol represents the finite field of integers modulo p, where p is a prime. x^p = x mod p GFpn This symbol represents the finite field with p^n elements, where p is a prime. QuotientField This symbol represents the quotient field of any integral domain. The rationals equals QuotientField(Integers) R is a field iff QuotientField(R)=R H 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