12 CHAPTER 1. PRELIMINARIES

With this definition, the familiar properties of order can be proved. The followingproposition lists many of these familiar properties. The relation ‘a > b’ has the samemeaning as ‘b < a’.

Proposition 1.4.2 The following are obtained. Recall −x is the symbol for the additiveinverse of x.

1. If x < y and y < z, then x < z.

2. If x > 0 and y > 0, then x+ y > 0.

3. If x > 0, then −x < 0.

4. If x ̸= 0, either x or −x is > 0.

5. If x < y, then −x > −y.

6. If x ̸= 0, then x2 > 0.

7. If 0 < x < y then x−1 > y−1.

Proof: First consider 1, called the transitive law. Suppose that x < y and y < z. Thenfrom the axioms, x+ y < y + z and so, adding −y to both sides, it follows

x < z

Next consider 2. Suppose x > 0 and y > 0. Then from 2,

0 = 0 + 0 < x+ y.

Next consider 3. It is assumed x > 0 so

0 = −x+ x > 0 + (−x) = −x

Now consider 4. If x < 0, then

0 = x+ (−x) < 0 + (−x) = −x.

Consider the 5. Since x < y, it follows from 2

0 = x+ (−x) < y + (−x)

and so by 4 and Proposition 1.3.10,

(−1) (y + (−x)) < 0

Also from Proposition 1.3.10 (−1) (−x) = − (−x) = x and so

−y + x < 0.

Hence−y < −x.

Consider 6. If x > 0, there is nothing to show. It follows from the definition. If x < 0,then by 4, −x > 0 and so by Proposition 1.3.10 and the definition of the order,

(−x)2 = (−1) (−1)x2 > 0