l. 2 The Order Properties 11
1.2 The Order Properties
In working with real numbers we make frequent use of the concepts of less than
and greater than. But the field properties make no mention of "<" or ">."
To make these concepts available, we must make additional assumptions about
our field. We must assume that it is an "ordered" field, which we now define.Definition 1.2.1 A field F is said to be an ordered field with respect to a
particular subset P ~ F if the subset P satisfies the following axioms:
ORDER AXIOMS:
(01) '<Ix, y E P , x + y E P (Pis "closed" under+)
(02) 'r:/x,y E P , x · y E P (Pis "closed" under·)
(03) '<Ix E F , one and only one of the following holds:x E P , - x E P , or x = 0 (the "law of trichotomy")
While these three axioms do not appear very promising, they are entirely
sufficient to allow us to define the relations "<" and ">" and derive the prop-
erties usually associated with them. First, we observe that (01) - (03) allow
us to define "positive" and "negative."
Definition 1.2.2 If x E P we say that x is positive, and if - x E P we say
that x is negative. Thus, the law of trichotomy says that every element of
an ordered field is either positive, negative, or zero, but not more than one of
these.
Definition 1.2.3 Given x, yin an ordered field F, we say that x and y have
the same sign if x, y E P , or -x, - y E P. We say that x and y have opposite
signs if - x, y E P , or x, -y E P.
Definition 1.2.4 ( "Greater than,'' "Less than," etc.) We define the symbols
<, >, :::;, and ?: in an ordered field F as follows:
x < y if y - x E P ;
x > y if y < x;
x:::; y if x < y or x = y;
x ?: y if x > y or x = y.
The inequalities " <" and " >" are called "strict" inequalities, to distin-
guish them from "S::" and "?:."
Theorem 1.2.5 (Trivia) Let x and y be elements of an ordered field F. Then,
(a) x > 0 iff x E P ; x < 0 iff - x E P.
(b) One and only one of the following holds: x < y, x > y, or x = y.
(Alternate form of the law of trichotomy)