1. The integers and the rational numbers. 1.1. The integers. Let Summary: 1. The integers and the rational numbers. 1.1. The integers. Let Z = {((m, n), (p, q)) (N × N) × (N × N) : m + q = p + n}. It is a simple matter to verify that Z is an equivalence relation on N × N. Let Z = (N × N)/Z and call its members integers. Let m - n = (m, n)/Z for (m, n) N × N. One easily verifies that N n (n - 0)/Z is univalent. In what follows we will not distinguish between a member of N and its image under this mapping. One verifies that there is a unique unary operation - on Z such that -(m - n) = n - m and that there are unique binary operation + and on Z such that (m-n)+(p-q) = (m+p)-(n+q) and (m-n)(p-q) = (mp+nq)-(mq+np) for (m, n), (p, q) N × N; on the right hand side of these equations + and are the operations on N. One easily verifies that (Z, +, 0, -) is an Abelian group and that (Z, +, 0, ) is an integral domain in which 1 is the neutral element with respect to . 1.2. The rational numbers. Let Collections: Mathematics