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Consider functions f (n) and g (n) as given below. Use the most precise asymptotic notation to show how function f is related to function g in each case (i. e., f ∈? (g)). For example, if you were given the pair of functions f (n) = n and g (n) = n2 then the correct answer would be: f ∈ o (g). To avoid any ambiguity between O (g) and o (g) notations due to writing, use Big-O (g) instead of O (g).

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  1. 24 December, 10:40
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    1.

    The relation is: f = Θ (g)

    For c1 = 1 and c2 = 200, and n > = 0, we have

    0 < = g (n) < = f (n) < = 200*g (n)

    Hence, f = Θ (g)

    2.

    The relation is: f = Ω (g)

    For very large values of n,

    0 < = g (n) < = f (n)

    Hence, f = Ω (g)

    3.

    The relation is: f = Ω (g)

    Since, for n > = 0,

    log2n > log3n

    Hence, f = Ω (g)

    4.

    The relation is: f = Big-O (g)

    For all n > = 0,

    2n < = 3n

    Hence, f = Big-O (g)

    5.

    The relation is: f = Big-O (g)

    For n > = 0,

    0.5n < 1

    Hence, f = Big-O (g)
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