Solution to (a) Suppose f and g are functions whose domains are subsets of z+, the set … - Sikademy
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Archangel Macsika

(a) Suppose f and g are functions whose domains are subsets of z+, the set of positive integers. Give the definition of 'f is \OmicronO(g)' (b) Use the definition of 'f is \OmicronO(g)' to show that: (I) 2n+27 is \OmicronO(3n) (ii) 5n is not \OmicronO(4n)

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Let f and g are functions whose domains are subsets of \Z_+, the set of positive integers. We say that f is O(g) if there exist a positive real number c and a positive integer n_0 such that f(n)\le cg(n) for all n\ge n_0.


(b) Let us use the definition of 'f is \Omicron(g)' to show that the following statements.


(I) Since 2^n+27<3^n+27\cdot 3^n=28\cdot 3^n for each positive integer n, we put c=28 and conclude that 2^n+27 is O(3^n).


(ii) Let us show that 5^n is not O(4^n). Let us prove using the method by contradiction. Suppose that there exist a positive real number c and a positive integer n_0 such that 5^n\le c\cdot 4^n for all n\ge n_0.

It follows that (\frac{5}{4})^n\le c for all n\ge n_0. Since \lim\limits_{n\to+\infty}(\frac{5}{4})^n=+\infty, we conclude that there exist m\in\Z_+ such that (\frac{5}{4})^n>c for all n\ge m. This contradiction proves the statement.


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