lec5 - Math 117 – April 12, 2011 5 Cardinality Definition...

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Unformatted text preview: Math 117 – April 12, 2011 5 Cardinality Definition 5.1. Two sets S and T are called equinumerous, denoted by S ∼ T , if there exists a bijection from S onto T . Equinumerosity defines an equivalence relation on families of sets. The equivalence classes separate sets into classes of equal size, and we associate cardinal numbers to each class for the size of the sets in that particular class. Definition 5.2. A set S is said to be finite if there exists n ∈ N and a bijection f : I n = { 1 , 2 ,...,n } → S . If a set is not finite, it is called infinite A set S is called denumerable, if there exists a bijection f : N → S . If a set is finite or denumerable, it is called countable. Countable sets are those whose elements can be indexed by the natural numbers, i.e. S = { s 1 ,s 2 ,... } . The indexing is given by the bijection f : I n → S or f : N → S , in which case s 1 = f (1), s 2 = f (2) ,...,s n = f ( n ) ,... . One can show that any subset of a countable set is also countable.....
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This note was uploaded on 05/31/2011 for the course MATH 117 taught by Professor Akhmedov,a during the Spring '08 term at UCSB.

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