# Thread: piecewise and non-piecewise isomorphic?

1. ## piecewise and non-piecewise isomorphic?

Ok, let me give the problem first:

Consider the following interpretation of these terms: "points" are ordered pairs (x,y) in the xy-plane, "lines" are the functions

$\displaystyle y = \begin{cases} mx+b & x < 0 \\ 2mx+b & x \ge 0 \end{cases}$

And say a "point" "lies on" a "line" if the point (x,y) satisfies the equation. Show that this interpretation is a model for incidence geometry and determine which, if any, parallel property it has.
When visualizing the problem, I usually imagine the y-axis is the surface of a body of water, and the lines are light rays being refracted.

I'm already nearing a solution for the problem. However, there is one related thing I'd like to do for my own edification: prove that $\displaystyle y = \begin{cases} mx+b & x < 0 \\ 2mx+b & x \ge 0 \end{cases}$ is isomorphic with $\displaystyle y = mx+b$ Intuitively it seems true, but I don't have a clue as to how to go about proving it. Can anyone provide me with such a clue?

Notes:

1. If anyone wants to see my solution to the actual problem (once I finish it) let me know and I'll make an edit to this post.
2. The actual problem says "undefined terms" instead of "these terms" in the first sentence.

2. This is not incidence geometry. Consider $\displaystyle (0,1)$ and $\displaystyle (0,2)$. What is the "line" joining them?

3. Originally Posted by ThePerfectHacker
This is not incidence geometry. Consider $\displaystyle (0,1)$ and $\displaystyle (0,2)$. What is the "line" joining them?
Doh! Thanks! I was starting to bog down in the middle of my proof.