week13 - Cardinality Part I Original Notes adopted from...

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Cardinality Part I Original Notes adopted from December 3, 2001 (Week 13) c ± P. Rosenthal , MAT246Y1, University of Toronto, Department of Mathematics S and T have the same cardinality if there exists f : S T one-to-one onto (i.e. a “pairing” ) or one-to-one correspondence. We showed that | N | = | E | = | Q + | | S | = | N | iff S is an infinite set whose elements can be listed. We call such sets “countably infinite”, or say they have cardinality 0 . | S | = 0 means | S | = | N | . | [0 , 1] | 6 = 0 Proof We’ll show no list can contain all numbers in [0,1]. a ij ∈ { 0 , 1 , 2 , 3 , 4 ,.... 9 } Suppose we have a list c 1 ,c 2 ,c 3 , ··· , write them as c 1 = .a 11 a 12 a 13 a 14 a 15 ··· c 2 = .a 21 a 22 a 23 a 24 a 25 ··· c 3 = .a 31 a 32 a 33 a 34 a 35 ··· ······ In ambiguous cases, pick representation with all 9’s. e.g. . 34999 ··· = . 3500000. Let x = .b 1 b 2 b 3 b 4 ··· where b j any digit other than 0, 9 or a jj Then x isn’t among numbers listed for it differs from the k th number listed in its k th place. Therefore | [0 , 1] | 6 = 0 We say [0,1] has the cardinality of the continuum, or | [0 , 1] | = c Definition. | S | 6 | T | (“The cardinality of
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week13 - Cardinality Part I Original Notes adopted from...

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