Let f be a one-to-one correspondence of the points in a plane. Prove or disprove the following statement:

"If f maps circles to circles, then it maps straight lines to straight lines."

(In reply to re(2): Not a proof, but ... by Steve Herman)

Actually, I disagree.  There is only one circle that definitely does not map to a circle.  However, there are many circles which may or may not map to circles, depending on how I define the function (since I did not fully define it previously).

I have some reasoning below, but after typing it all out, I feel like I've wasted a lot of time on a mapping that certainly does not work (on this point, I agree with you).  Read, if you want.

First of all, I wish to modify the function so it is more in line with what I was actually thinking.  Allow me to restate, so that I may clear up any confusion. 
Consider all circles that intersect the points (0,1) and (0,-1).  Each circle intersects (0,1) at a different angle from horizontal.  However, since no circle intersects at 90 degrees, we will use a y-axis in place of a circle.  Map each circle/line to the one that intersects (0,1) at a perpendicular angle.
With this modification, it defines all points.  However, as I said (or at least was thinking) previously, the unit-circle maps to the y-axis and vice versa, so this is, already, not a valid mapping.

Second of all, let me make clear that I did not fully define a function.  One circle can be mapped to another in infinitely many ways.  I only said which circle mapped to which, and I didn't even include all circles.  It is not clear, since I did not fully define a function, whether the other circles map to circles or not.  For you to assert that "most circles [map] into something that is not a circle," you must provide some reasoning since it is not at all obvious, though you may be entirely correct.

Posted by Tristan on 2006-11-24 02:07:22