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

This preview shows pages 1–2. Sign up to view the full content.

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

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### Page1 / 3

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

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online