41 - terms: A = { a 1 , a 2 , a 3 , . . . } . E XAMPLE 4.28...

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are ‘bigger’ than the set of natural numbers. The set of natural numbers is one of the simplest inFnite sets. We can build it up by stages: 0 0 , 1 0 , 1 , 2 . . . in such a way that any number n will appear at some stage. InFnite sets which can be built up in Fnite portions by stages are particularly nice for computing. We therefore distinguish those set which are either Fnite or which have a bijection to N . D EFINITION 4.27 (C OUNTABLE ) ±or any set A , A is countable if and only if A is Fnite or A N . The elements of a countable set A can be listed as a Fnite or inFnite sequence of distinct
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Unformatted text preview: terms: A = { a 1 , a 2 , a 3 , . . . } . E XAMPLE 4.28 The integers Z are countable, since they can be listed as: , 1 ,-1 , 2 ,-2 , 3 ,-3 , . . . This counting function can be deFned formally by the bijection g : Z N deFned by g ( x ) = 2 x, x =-1-2 x, x < E XAMPLE 4.29 The set of integers Z is like two copies of the natural numbers N . We can even count N 2 , which is like N copies of N , as illustrated by the following diagram: 1 2 3 4 1 2 3 4 41...
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This note was uploaded on 03/02/2010 for the course MATH Math2009 taught by Professor Koskesh during the Spring '09 term at SUNY Empire State.

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