25 - LECTURE 25: DENUMERABLE SETS ( § 10.1-10.2) We have...

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Unformatted text preview: LECTURE 25: DENUMERABLE SETS ( § 10.1-10.2) We have too many high sounding words, and too few actions that correspond with them. Abigail Adams 1. Review from last time Two sets have equal cardinality, or equal size, if there is a bijection between them. We showed this is an equivalence relation. If a set A has the same cardinality as the natural numbers we say A is denumerable. Another way of thinking about this is that we can list the elements of A one at a time: A = { a 1 ,a 2 ,a 3 ,... } . We make this list using a bijection f : N → A and setting f ( i ) = a i . The set Z is denumerable, since we can list its elements 0 , 1 ,- 1 , 2 ,- 2 , 3 ,- 3 ,... . We also proved a very technical result: any infinite subset of a denumerable set is denumerable (i.e. we can still list its elements one at a time). Example 1. Is Z- { 2 } a denumerable set? Yes, since it is an infinite subset of a denumerable set....
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This note was uploaded on 03/08/2011 for the course MATH 334 taught by Professor Smith during the Spring '11 term at Vanderbilt.

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25 - LECTURE 25: DENUMERABLE SETS ( § 10.1-10.2) We have...

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