352 CONSTRUCTION OF NUMBER SYSTEMS Chapter 10
- (a) Prove that the relation < from Definition 2 is well defined. That is, prove
that if (a, b) - (a', b') and (c, d) - (c', d'), where all the variables involved repre-
sent positive integers, then a + d < b + c if and only if a' + d' < b' + c'.
(b) In Z, prove that:
(i) 1 > 0
*(ii) If x E Z, x = [(a, b)], then x >^0 if and only if a > b, x < 0 if and only if
a<bandx=Oifandonlyifa=b
(iii) If XEZ, then x > 0 if and only if -x < 0
(iv) Ifx,y~Z, thenx<yifand onlyif -y< -x
(v) If x, y E Z, then x < y if and only if y - x > 0; x = y if and only if
y-x=o - (a) Generalize part (c) of Theorem 8, Article 10.1, to Z; that is, prove that the
relation <, defined on Z in Definition 2, is transitive.
(b) Generalize Theorem 10, Article 10.1, to Z; that is, prove the trichotomy prop-
erty for the relation < on Z.
(c) Prove parts (b)(c)(d) of Theorem 6.
(d) Generalize parts (a)(b)(c) of Exercise 6, Article 10.1, to Z [adding, however,
whatever additional hypotheses might be necessary in (b)]. - (a) Prove that there is no integer x such that 0 < x < 1. [Hint: Apply the well-
ordering principle to the set {x E ZIO < x < 1) under the assumption that this
set is nonempty. Keep in mind the result in Theorem 6(a).]
(b) Prove that there is no integer between x and x + 1, for any integer x. [Hint:
Use an argument by cases, approaching the case x > 0 by induction.) - Prove that the operations of addition and multiplication on Q, from Definition 3,
are well defined. - Prove that the mapping x -* [[x]] of Z into Q, defined in the discussion follow-
ing Definition 3 (see the fourth paragraph following Definition 3) is one to one, not
onto, and preserves the operations of addition and multiplication. Prove that the
image of this mapping, a proper subset of Q, is closed under the operations of ad-
dition and multiplication in Q. - Prove that the subset Q+ of Q, defined by Q+ = {[[(a, b)]] la, b E Z, ab > 0)
satisfies the conditions of Definition 1, Article 9.2, for the positive part of an ordered
field. - Using Articles 9.1 and 9.2, make a list of algebraic and order properties that are
valid in the structure (Q, +, -), by virtue of the fact that this structure is an ordered
field.
10.3 Outline of the Construction of the Reals
We now deal with the problem of constructing R from Q. The difficulties
in presenting this material are pedagogical as well as mathematical. As
has happened at several previous stages of the text (e.g., Articles 6.4 and 8.3),
we have a situation in which the details of the mathematics we would like