MIT6_042JF10_rec13_sol

MIT6_042JF10_rec13_sol - 6.042/18.062J Mathematics for...

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6.042/18.062J Mathematics for Computer Science October 22, 2010 Tom Leighton and Marten van Dijk Notes for Recitation 13 1 Asymptotic Notation Which of these symbols Θ O Ω o ω can go in these boxes? (List all that apply.) 2 n + log n = ( n ) Θ , O, Ω log n = ( n ) O, o n = (log 300 n ) Ω , ω n 2 n = ( n ) Ω , ω n 7 = (1 . 01 n ) O, o
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Recitation 13 2 2 Asymptotic Equivalence Suppose f, g : Z + Z + and f g . 1. Prove that 2 f 2 g . Solution. 2 f 2 g = f g , so they have the same limit as n infty . 2. Prove that f 2 g 2 . Solution. f ( n ) 2 f ( n ) f ( n ) f ( n ) f ( n ) lim = lim = lim lim = 1 1 = 1 . n →∞ g ( n ) 2 n →∞ g ( n ) · g ( n ) n →∞ g ( n ) · n →∞ g ( n ) · 3. Give examples of f and g such that 2 f �∼ 2 g . Solution. f ( n ) = n + 1 g ( n ) = n. Then f g since lim( n + 1) /n = 1, but 2 f = 2 n +1 = 2 2 n = 2 2 g so · · 2 f lim = 2 = 1 . 2 g 4. Show that is an equivalence relation Solution. (a) Reflexive: f f since f ( x ) /f ( x ) = 1 for all x (assuming f ( x ) = 0), so lim x →∞ f ( x ) /f ( x ) = 1 (b) Symmetric: f g implies g f since if lim x →∞ f ( x ) /g ( x ) = 1, then by the laws
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