Countable listed a 0 ,a 1 ,a 2 ,. N same as bij A or A finite - PowerPoint PPT Presentation

Mathematics for Computer Science Countable Sets MIT 6.042J/18.062J A is countable iff can be Countable listed a 0 ,a 1 ,a 2 ,…. N same as bij A or A finite Sets so Z , Z countable + Albert R Meyer, March 4, 2015 Albert R Meyer, March 4, 2015 countable.1 countable.2 N N × Binary words are countable is countable {0,1} ::= finite binary words start with (0,0) * list the (empty) string of length 0 then (0,1), (1,0) list the 2 length-1 bit strings then (0,2), (2,0), (1,1) then list the 2 2 length-2 bit strings then (0,3), (3,0), (1,2), (2,1) (in binary notation order) then the 2 3 length-3 bit strings then all pairs with sum n ⋮ Albert R Meyer, March 4, 2015 Albert R Meyer, March 4, 2015 countable.3 countable.4 1

Proving Countability Rationals are countable m Lemma: A is countable iff map (m,n) to can list A allowing repeats: n N surj A N × N surj Q ≥ 0 Corollary: A is countable iff ! ! C surj A so countable countable for some countable C Albert R Meyer, March 4, 2015 Albert R Meyer, March 4, 2015 countable.5 countable.6 Reals are uncountable But {0,1} ω and the R real numbers are not countable: next lecture. Albert R Meyer, March 4, 2015 countable.7 2

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