(Solved Homework): Asymptotic relations like O, Ohm, and Theta represent relationships between functions, and these relationships are transitive….

Asymptotic relations like O, Ohm, and Theta represent relationships between functions, and these relationships are transitive. That is, if some f(n) = Ohm(g(n)), and g(n) = Ohm(h(n)), then it is also true that f(n) = Ohm(h(n)). This means that we can sort functions by their asymptotic growth.^1 Sort the following functions by order of asymptotic growth such that the final arrangement of functions g_1, g_2, , g_12 satisfies the ordering constraint g_1 = Ohm (g_2), g_2 = Ohm (g_3), , g_11 = Ohm (g_12). Give the final sorted list and identify which pair(s) functions f(n), g(n), if any, are in the same equivalence class, i.e., f(n) = Theta (g(n)).

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1.the functions that are given are and we have to sort them in decreasing order

n²

( $sqrt$ 2)^logn

2(^{logn)^*}

(logn)!

(3/2)ⁿ

n^1/logn

nlogn

log(n!)

eⁿ

taking log of all the functions

logn, 2logn, lognlog( $sqrt$ 2), logn^*log2, nlogn, logn(loglogn), nlog(3/2), 1/logn*logn, logn+log(logn), logn+log(logn), nloge , 0

n!>eⁿ>(3/2)ⁿ>n²>2(^{logn)^*}> ( $sqrt$ 2)^logn >nlogn>log(n!)>(logn!)>n>n^1/logn>1

eⁿ,(3/2)ⁿ belong to same class

nlogn,log(n!)belong to same class

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