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41 - terms A = a 1 a 2 a 3 E XAMPLE 4.28 The integers Z are...

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are ‘bigger’ than the set of natural numbers. The set of natural numbers is one of the simplest infinite 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. Infinite sets which can be built up in finite portions by stages are particularly nice for computing. We therefore distinguish those set which are either finite or which have a bijection to N . D EFINITION 4.27 (C OUNTABLE ) For any set A , A is countable if and only if A is finite or A N . The elements of a countable set A can be listed as a finite or infinite 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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