Bridge to Abstract Mathematics: Mathematical Proof and Structures

306 PROPERTIES OF NUMBER SYSTEMS Chapter 9

Definition 1, (z - y) + (y - x) E 9. But (z - y) + (y - x) =

z+(-y+y)-x=z-x,soz-x~9,asdesired.

For (c) and (d), we note, by Definition 2 (using specialization), 0 < x if

and only if x - 0 E 9, that is, x E 9. Similarly, x < 0 if and only if

0 - x E @ that is, -x E 9. Hence (d) follows directly from (c) of Def-

inition 1. Cl

Part (c) of Theorem 2 confirms that the positive part of the real number
field corresponds to our usual notion of positive real number, that is, a
number greater than zero, or graphically, a number to the right of zero on
the real number line.
The relation "less than," in an ordered field, leads in a natural way, to
an associated partial ordering relation on F.

DEFINITION 3

Let F be an ordered field. We define a relation "less than or equal to," denoted

I, by the rule x I y (x is less than or equal to y) if and only if either x < y or

x = y.

THEOREM 3

In an ordered field F:

(a) I is a partial ordering on F, so that (F, I) is a poset.

(6) I is a total ordering on F.

Proof (a) Clearly x I x for any x E F, since x = x, so that the relation

I is reflexive. For antisymmetry, assume x, y E F with x 5 y and y I x.

If x # y, then x < y and y < x. But this contradicts (e) of Theorem 2.

Hence I is antisymmetric. Finally, if x, y, z E F with x I y and y I z,

an analysis of all the logical possibilities for cases [e.g., x = y and y < z;

you should determine how many cases there are and verify each one, see

Exercise 3(a)] leads to the desired conclusion x I z, so that 5 is

transitive.

(b) Let x, y E F be given. We must prove that either x I y or y 5 x.

Suppose x I y is false. Then it is not the case that either x < y or x = y;

that is, x is not less than y and x is not equal to y. By (e) of Theorem 2,

we have y < x, which implies y I x. El

The proof of the transitivity of I in (a) of Theorem 3 suggests a fact
that is worth noting. By the definition of < in an ordered field, many prop-
erties of that relation are best proved by arguments involving division into
cases [e.g., Exercise 3(b)(iii)]. (Recall Article 5.3.)
A number of properties that are familiar from the algebra of inequalities
involving real numbers carry over to an arbitrary ordered field.