28 - LECTURE 28: INJECTIONS AND BIJECTIONS (10.5) Things...

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LECTURE 28: INJECTIONS AND BIJECTIONS ( § 10.5) Things are only impossible until they’re not. Jean-Luc Picard 1. Genealogy and comparing cardinalities Recall that | A | ≤ | B | means there is an injection from A to B . From the notation, we might expect that if | A | ≤ | B | and | B | ≤ | A | then | A | = | B | . In other words, if we have injections both ways between A and B , then there is a bijection. Near the 1900’s this was proved, and is known as the Schr¨ oder-Bernstein theorem. Theorem 1. If A and B are sets with | A | ≤ | B | and | B | ≤ | A | then | A | = | B | . Proof sketch. Assume | A | ≤ | B | and | B | ≤ | A | . This means there is an injective function f : A B and an injective function g : B A . We need to find a bijection between A and B . Notice that if a A there is at most one element in b B that could map to a . We can think of b as being the parent of a (but notice that some a ’s have no parents). Similarly, there is as most one element in A that could map to any b B . While not every element has a parent, every element does
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28 - LECTURE 28: INJECTIONS AND BIJECTIONS (10.5) Things...

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