02count - EECS 501 COUNTABLE VS. UNCOUNTABLE SETS Fall 2001...

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Unformatted text preview: EECS 501 COUNTABLE VS. UNCOUNTABLE SETS Fall 2001 DEF: A set is finite if it has a finite number of elements. DEF: Two sets A, B are in one-to-one correspondence (1-1) if there exists a 1-1 mapping between elements of A and elements of B . NOTE: Two finite sets are 1-1 IFF they have same number of elements. EX: { a, b, c . . . z } and { 101 , 102 . . . 126 } are 1-1 (26 elements each). NOTE: An infinite set can be 1-1 with a proper subset of itself: A = { 1 , 2 , 3 , 4 . . . } and B = { 2 , 4 , 6 , 8 . . . } are 1-1: Mapping b = 2 a . Z = { . . .- 2 ,- 1 , , 1 , 2 . . . } and Y = Z + = { 1 , 2 , 3 . . . } are 1-1: 1-1 Mapping: z = y/ 2 if y is even; z = (1- y ) / 2 if y is odd. DEF: A set is countably infinite IFF it is 1-1 with { integers } . i.e.: You can count the elements of the set (this may take forever!). EX: { even integers } and { odd integers } are countably infinite. DEF: A set is countable IFF it is either finite or countably infinite. NOTE: A set is countable IFF it is 1-1 with another countable set. THM: The set of lattice points Z 2 = { ( i, j ) : i, j { integers }} is countable....
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This note was uploaded on 07/22/2011 for the course EECS 370 taught by Professor Lehman during the Spring '10 term at University of Florida.

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02count - EECS 501 COUNTABLE VS. UNCOUNTABLE SETS Fall 2001...

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