thinking about bijections

..of the unit interval.
the unit interval is the segment of the number line between 0 and 1, although any segment comes out to pretty much the same thing, and a bijection is a transformation of the interval so that every point is goes to a unique point and each point has exactly one mapped to it. basically the point (npi!) of all this is that if it's a bijection, it has an inverse; given this transformation you can make another which sends every point back to where it came from. You can't do that to a transformation if two points go to the same point - because you don't know which one to send it back to, or if some point doesn't have any points mapped to it - because you've got nowhere to send it back to.

Ways of making a bijection:

The pictures show graphs of each type of mapping. The unit interval goes left to right along the bottom border and a copy of it goes up the left edge from bottom to top. The colours show where specific points on the interval are mapped to on the copy.

Other ways...

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