Bridge to Abstract Mathematics: Mathematical Proof and Structures

9.2 ORDERED FIELDS 309

THEOREM 6

Let F be an ordered field with x, y, ZE F. Then:

We conclude this article with the reminder that the concept of ordered

field has now given us the ability to distinguish between R and C. But R

and Q are both ordered fields; what we need next is an abstract property of

ordered fields that is satisfied by one, but not the other. Completeness in

an ordered field, the subject of the next article, is just such a property.

Exercises

1. Suppose (F, 9) is an ordered field. Prove:

*(a) If x E F, x # 0, and n E N, then x2" E 9
(b) If a, b E 9, then alb E 9
(c) If x E 9 and n E N, then nx E 9. (Note: If F is any field, if x E F, and n E N, we
define the quantity nx recursively by the rules 1. x = x and nx = (n - l)x + x if
n > 1.)

2. Suppose (F, 9) is an ordered field. Prove:
   (a) x < x is false for any x E F (< is irreflexive)
   (b) xO
   (c) Given any x, y E F, precisely one of the three possibilities x < y, x = y, or x > y
   is true.


3. Suppose (F, 9) is an ordered field.
   (a) Prove that if x, y, z E F with x 2 y and y I z, then x I z [recall Theorem 3(a)].
   (b) Prove that if a, x, y E F, then:
   (i) x < y implies a + x < a + y (ii) x < y and a > 0 imply ax < ay
   (iii) x I y implies a + x I a + y (iv) x I y and a 2 0 imply ax I ay
   *(v) xly and a I 0 imply ax^2 ay
   (c) Prove that if a, b, x, y E F, then:
   (i) a I x arid b < y imply a + b < x + y
   (ii) a^5 x and b^5 y imply a + b I x + y
   (iii) 0 5 a < x and 0 I b < y imply ab < xy
   (iv) 0 5 a I x and 0 I b < y imply ab < xy
   (v) 0 I a I x and 0 I b I y imply ab I xy


4. Let (F, 9) be an ordered field.
   (a) (i) Prove that the product of two negative elements is positive.
   (ii) Prove that the product of a positive and a negative element is negative.
   (iii) Verify the remaining cases of the proof of Theorem 5.