- #1
Jamin2112
- 986
- 12
Homework Statement
I'm asked to prove that
If F is an ordered field, then the following properties hold for any elements a, b, and c of F:
(a) a<b if and only if 0<b-a
(b) ...
...
Right now I'm working on (a)
Homework Equations
We're supposed to draw from the basic properties (closure, associativity, commutativity, etc.) and also from the following definition.
Definition 1.4. A field F is ordered if it has an ordering < so that:
For all a, b in F, exactly of of these holds: a<b, a=b, a>b.
For all a, b in F, if a<b, then a+c<b+c.
For all a, b in F, if a>0 and b>0, then a+b>0 and ab>0.
The Attempt at a Solution
So far I wrote:
(a) Assume a < b.
Using Definition 1.6, we can add c to both sides of the inequality. Let c = -a, the additive inverse of a, so that the left side of the inequality equals 0.
0 < b + (-a)
But I'm wondering, is it a "given" that + (-a) = -a? I'm wondering whether I can just change this to b - a to complete the proof (actually, half the proof because we're dealing with an "if and only if" proof) or if I need to draw from some property.