Thread: Geometry - Rays, logical statement

Show that, given a ray AB, there is a coordinate system f on the line AB such that: ray AB = {P|f(P) >= 0}.


Here is what I have:

By definition ray AB is the union of the segment AB and the set of all points C such that A-B-C. Now, let L be a line and let P and Q be points on L. Let f be any coordinate system for L. Then let a=f(P) and for each point T of L, let g(t)=f(t)-a. Then this means that f is a coordinate system for L, and f(p)>=0. Therefore ray AB = {P|f(p)>=0}.

How is this? Is there any way that I can improve it?


Thanks for the help.

Originally Posted by mathprincess88

Show that, given a ray AB, there is a coordinate system f on the line AB such that: ray AB = {P|f(P) >= 0}.
How is this? Is there any way that I can improve it?

Any answer depends on how strong your Ruler Postulate is and upon the sequence of betweenness theorems.
And we do not know the answer to that.

In general here is a outline of a proof.
Suppose f is a coordinate system on line AB.
So f(A) is the coordinate of point A.
If f(B)-f(A)>0 then define g(x)=f(x)-f(A) otherwise define g(x)=f(A)-f(x).
By use of the betweenness theorems the result should follow.