Infinite Numbers, again

Factual Questions
1

OK. I know all mostly about infinite sets. Mama always said, “Infinite is as infinite does.” I have always loved Cantor’s sieve and the correspondence that it refers to. But I have a question for those numerical theory types.
As one looks at pairs of infinite sets and sets them up into one-to-one relationships with each other it is possible to show that each is bigger than the other. I say that of course with tongue firmly in cheek. It is to me one of the best proofs of transfinite analysis (if that is the term) that if you can show that each set is bigger than the other, than they are clearly the same size. I am thinking here of the fun you can have with the natural numbers and integers, to be specific.

I only recently found out about the method to correspond points on a real line to all points on a two-dimensional plane, or any number of real valued axes. I was really tickled to finally hear about it. It made my day.

I am curious if the same sort of fun exists for C. Can you show with naïve math that two sets of size Care each bigger than the other?

2

A set is infinite if and only if you can set up a 1 to 1 correspondence with a proper subset. I wouldn’t use the word bigger the way you do. If you show a 1 to 1 correspondence from A to a proper subset of B, then B is greater than or equal to A, not that B is bigger than A.

The answer to your question is yes. If you take a semi-circle sitting on the number line at the origin. You can create a 1 to 1 map from the points from the semi-circle to the whole line by drawing a line from the center of the circle to the number line. Every point on the line maps to one point on the semi-circle, You have a line of finite length mapping onto an infinite line.

3

Here’s a simple exapmle of a bijection from C to a subset thereof:

F(r×e[sup]i theta[/sup]) = tanh®×e[sup]i theta[/sup]. This maps C to the unit open disk in the complex plane.