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

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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 , …
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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
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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
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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
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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...

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