Cardinal Arithmetic

Cardinal Arithmetic - Math 100 Cardinal Arithmetic Martin...

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Unformatted text preview: Math 100 Cardinal Arithmetic Martin H. Weissman 1. A few countable sets Suppose that A and B are sets. The following are basic definitions of cardinality: # A # B if and only if there exists an injective function from A to B . # A = # B if and only if there exists a bijective function from A to B . # A # B if and only if there exists a surjective function from A to B . The following are basic properties of cardinality: # A # B if and only if # B # A . The relations , , = are reflexive and transitive. (The Cantor-Schroeder-Bernstein Theorem): If # A # B and # B # A , then # A = # B . If A and B are sets, then either # A # B or # B # A (or both). In addition to the usual natural numbers, there are various infinite cardinal numbers: = # N , and c = # R . Some intuition about finite sets breaks down with infinite sets. Here is a basic example: Theorem 1. The natural numbers and the integers are sets with the same cardinality. In other words, # Z = # N = . Proof. The inclusion function from N to Z is an injective function. Hence # N # Z . Thus # Z ....
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This note was uploaded on 04/17/2008 for the course MATH 100 taught by Professor Weissman during the Fall '08 term at UCSC.

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Cardinal Arithmetic - Math 100 Cardinal Arithmetic Martin...

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