Today I Learned

Some of the things I've learned every day since Oct 10, 2016

1: The Cantor-Schröder-Bernstein Theorem

The Cantor-Schröder-Bernstein Theorem states that if there are two sets A, B and injective functions f: A \rightarrow B, g: B \rightarrow A, then there exists a bijective function h: A \rightarrow B, and so A and B are isomorphic.

This is intuitive when you think of it in terms of cardinality. If there is a bijection f: A \rightarrow B, then that means |A| \leq |B|, and the reverse means |B| \leq |A|, so the two combined imply |A| = |B|.

This can be helpful when trying to show two sets are isomorphic to each other, particularly when those sets are uncountable, because proving mutual bijection is often much easier than constructing a bijection.

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