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# lecture14 - Lecture 14 Countability(recap Algorithms Recap...

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Lecture 14 Countability (recap) Algorithms

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Recap Sequences Arithmetic progression a, a+d , a+2d, …, a+n *d, …. Geometric progression a, a*r, a*r 2 , …, a* r n , …
Recap: Summations 1 i n i 1 , 0 , 1 1 1 0 r r r r n n i i

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Recap: Cardinality Sets A and B have the same cardinality iff there is a 1-1 correspondence between A and B A set that is either finite or has the same cardinality as the set of positive integers is countable
Example Show that the set of positive even numbers is countable Use the mapping f(n) = 2n Need to show that f is 1-1 and onto 1-1: Suppose f(n) = f(m). Then 2n = 2m and n =m Onto: Take any integer even x. x = 2k f(k) = x

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Set of positive rational numbers is countable Image from: http://www.homeschoolmath.net/teaching/rational-numbers-countable.php
True or False The (cartesian) product of two countable sets is countable

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An uncountable set Set of real numbers between 0 and 1 is uncountable Hint: every real number has a unique decimal expansion Cantor’s diagonalization argument
Proof by contradiction Suppose real numbers between 0 and 1 were countable. Let’s write down the list! 0.x 11 x 12 x 13 0.x 11 x 12 x 13 0.x 11 x 12 x 13 ..

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lecture14 - Lecture 14 Countability(recap Algorithms Recap...

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