Prb&Rand Proc EECS 501
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This 1 page Class Notes was uploaded by Ophelia Ritchie on Thursday October 29, 2015. The Class Notes belongs to EECS 501 at University of Michigan taught by Staff in Fall. Since its upload, it has received 13 views. For similar materials see /class/231536/eecs-501-university-of-michigan in Engineering Computer Science at University of Michigan.
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Date Created: 10/29/15
EECS 501 COUNTABLE VS UNCOUNTABLE SETS Fall 2001 DEF A set is nite if it has a nite number of elements DEF Two sets A7 B are in one to one correspondence 771 177 if there exists a 1 1 mapping between elements of A and elements of B NOTE Two nite sets are 1 1 IFF they have same number of elements EX a7 b7 0 and 1017 102 126 are 1 1 26 elements each NOTE An in nite set can be 1 1 with a proper subset of itself A 1727 374 and B 274767 8 are 1 1 Mapping b 2a Z 27 1707172 andY Z1L 17273 are 1 1 1 1 Mapping z y2 if y is even 2 1 y2 if y is odd DEF A set is countably in nite IFF it is 1 1 with integers ie You can 77count77 the elements of the set this may take foreverl EX even integers and odd integers are countably in nite DEF A set is countable IFF it is either nite or countably in nite NOTE A set is countable IFF it is 1 1 with another countable set THM The set of lattice points 22 i7j i7j E integers is countable Proof A 1 1 mapping between i7j i 07 17 2 7j 07 17 2 and nn17273isnij1ij22 j Can easily extend to negative values as shown above THM The set of Rationals Q i7j E integersj gt 0 is countable Proof 1 1 Mapping Q 3 q ij lt gt i7j E 22 is known to be countable In fact7 Q is 1 1 with a subset of ZQ Q is at most countable BUT Countable Z C Q7 so Q is at least countable gt Q is exactly countable THM A countable union of countable sets is countable Proof The 77countable union77 is 1 1 with 2 reindex it with i 17 2 The 77countable sets77 can similarly each be reindexed with j 17 2 UfiiAz U291ai17ai2 U521 U511 am lt M 6 32 Again7 possible duplications gtthis is at most countable This theorem is particularly useful for showing that a set is countable DEF An uncountable set is NOT 1 1 with any countable set THM Cantor 1890 07 1 Q for the wheel of fortune is an uncountable setl Proof Suppose 07 1 is countable Index all a E 07 1 as 3617 2 Let acn have binary expansion acn 0307113071236713 where mnj 0 or 1 Let ynj 1 mm Then 3 0y11y22y33 y acn for all n Since 9 07 1 for the wheel of fortune is uncountable7 the third axiom of probability does not hold in the 0 1 77proof77
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