Bridge to Abstract Mathematics: Mathematical Proof and Structures

10.2 DEVELOPMENT OF THE INTEGERS AND RATIONAL NUMBERS 351

main; this F is called the field of quotients of D. Viewed in this more general

context, the field of rationals is the field of quotients of the integral domain

of integers.

Finally, let us deal with the issue of order in Q. We do this by noting

that the subset Q+ of Q defined by Q+ = ([[(a, b)]] la, b E Z, ab > 0) sat-

isfies the properties of 9, the positive part of an ordered field, in Defini-

tion 1, Article 9.2. Once again, verification of this fact is left to you in Ex-

ercise 9.

With this, all the properties of ordered fields listed in Article 9.2 can

now be seen as valid in Q. We remind you of the deficiency in Q, discussed

in Article 9.3 (recall Example 3 of that article), that causes us to carry our

construction beyond Q. The ordered field (Q, +, .) is incomplete, in the

sense of Definition 4, Article 9.3. In Article 10.3 we will outline a con-

struction of a complete ordered field, the real numbers R, that can be built

from Q using equivalence classes of Cauchy sequences. As in the two

constructions of this article, well-definedness of operations will be an im-

portant issue; analogous to those two constructions we will see that R

contains a substructure that is essentially identical to Q.

Exercises

*I. Prove Theorem 2(b); that is, prove that the operation of multiplication on Z, as
defined in Definition l(b), is well defined.

2. This exercise relates to the "embedding" of N in Z, outlined in the text following
   the proof of Theorem 2.
   (a) Prove that the mapping n -, [n] of N into Z is one to one.
   (b) Prove that the mapping of N into Z described in (a) is not onto. Prove,
   in addition, that an equivalence class [(p, q)] is in the image of this mapping if
   and only if p > q.
   (c) Prove that the mapping described in (a) "preserves the operations" of addition
   and multiplication in N; that is [nl] + [n,] = [n, + n,] and [nr][n2] = [nlnz]
   for any positive integers n, and n,.
   *(d) Prove that the subset Z+ = {[(p, q)] 1 p E N, q E N, p > q} is closed under
   addition and multiplication.


3. Verify the following algebraic properties for Z:
   (a) Closure, associativity, and commutativity for addition
   (b) Additive cancellation

• (c) Distributivity of multiplication over addition
  (d) Closure, associativity, commutativity and identity for multiplication
  (e) (i) Foralla~Z,(-l)a= -a
  (ii) For alla,b~Z,a(-b)= -(ab)=(-a)b
  (iii) For all aeZ, -(-a)=a
  (iv) Foralla,b~Z,(-a)(-b)=ab
  (f) The results of Theorem 9, Article 10.1, generalized to Z (dropping, however,
  any assumptions about order relationships between integers a, b, and c).