week16 - Cardinality Part III Original Notes adopted from...

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Unformatted text preview: Cardinality Part III Original Notes adopted from January 22, 2002 (Week 16) c P. Rosenthal , MAT246Y1, University of Toronto, Department of Mathematics Eg . S = { a,b,c } Sets of Subsets of S: { , { a } , { b } , { c } , { a,b } , { a,c } , { b,c } , { a,b,c }} . Theorem. If S has n elements ( n natural number), then it has 2 n subsets. Proof: S = { a 1 ,a 2 ,a 3 a n } 2 choices: a i is either in a set or not. 2 * 2 * 2 * 2 ( n times) 2 n subsets Theorem. If S any set and P ( S ) is the set of all subsets of S (the power set of S ), then | P ( S ) | > | S | . Proof : Must show: 1) There exists a subset T of P ( S ) such that | T | = | S | (ie | P ( S ) | > | S | ) 2) | P ( S ) | 6 = | S | . For 1), Let T = { all subsets of S with exactly 1 element } Pair T with S by { s } s . Therefore | T | = | S | . For 2) Suppose g : S P ( S ). Suppose g 1:1. Ill show that g cant be onto all of P ( S )....
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week16 - Cardinality Part III Original Notes adopted from...

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