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bernstein

# bernstein - The Cantor-Bernstein-Schrder Theorem o The...

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The Cantor-Bernstein-Schr¨ oder Theorem The following theorem is a precise formulation of the (obvious-sounding) fact that if “ A has fewer elements than B ” and “ B has fewer elements than A ”, then A and B are equinumerous. Theorem (Bernstein 1897). Let A and B be two non-empty sets. Assume that there exist two one-to-one maps f : A B and g : B A . Then, there exists a bijection h : A B . In fact, there are partitions A = A 1 ] A 2 and B = B 1 ] B 2 such that f restricted to A 1 is a bijection from A 1 to B 1 , and g restricted to B 2 is a bijection from B 2 to A 2 . Hence the function h ( x ) = f ( x ) if x A 1 g - 1 ( x ) if x A 2 is a bijection from A to B . http://en.wikipedia.org/wiki/Cantor-Bernstein-Schroeder theorem http://en.wikipedia.org/wiki/Cardinal number http://www.math.rutgers.edu/ useminar/bernstein.pdf Here’s a surprising application: Bernstein’s theorem implies that we can dissect a square into two disjoint parts S 1 and S 2 and a circular disk into C 1 and C 2 in such a way that S 1 is

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bernstein - The Cantor-Bernstein-Schrder Theorem o The...

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