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Math Help - piecewise and non-piecewise isomorphic?

  1. #1
    wil
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    Question 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

    <br />
y = \begin{cases}<br />
  mx+b & x < 0 \\<br />
  2mx+b & x \ge 0<br />
\end{cases}<br />

    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 <br />
y = \begin{cases}<br />
  mx+b & x < 0 \\<br />
  2mx+b & x \ge 0<br />
\end{cases}<br />
is isomorphic with <br />
y = mx+b<br />
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.
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  2. #2
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    This is not incidence geometry. Consider (0,1) and (0,2). What is the "line" joining them?
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  3. #3
    wil
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    Quote Originally Posted by ThePerfectHacker View Post
    This is not incidence geometry. Consider (0,1) and (0,2). What is the "line" joining them?
    Doh! Thanks! I was starting to bog down in the middle of my proof.
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