Question

Need help with 1,2,3 thank you.

1. Order of growth (20 points) Order the following functions according to their order of growth from the lowest to the highest. If you think that two functions are of the same order (Le f(n) E Θ(g(n))), put then in the same group. log(n!), n., log log n, logn, n log(n), n2 V, (1)!, 2", n!, 3", 21 2. Asymptotic Notation (20 points) For each pair of functions in the table below, deternme whether f(n) E O(g(n), f(n) € Ω(g(n)), or f(n) Θ(g(n)). Complete the table following the examples in the first three lines. o gin /n n+2n On 2n a. d.log n 10n 2n g.log(n n log n log(n +1) log n" log(n!) j.logi0(3n +1) 3. Prove asymptotic notation by definition (36 points) Using the basic definition of O, Ω, and Θ, show that (a) 10722 + 2n + 1 E 0(n2) (b) n2-9n + 5 e Ω(n) (d) Let f(n) and g(n) be asymptotically positive functions. Prove or disprove each of the following conjectures. (To prove, use definitions of asymptotic notations. To disprove, find a counter example.) i. f(n) E O((n) implies that g(n) E O(f(n)+g(n)) ii, f(n) E 0(g(n) implies that g(n) E Ω(f(n) + g(n)). iii. f(n) E Ω(g(n) implies that 2f(n) E Ω(2g(n)). Continued on the back→

Answer 1

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