Asymptotic Analysis For each of the following statements dec

Asymptotic Analysis For each of the following statements, decide whether it is always true, never true, or sometimes true for asymptotically nonnegative functions f and g. If it is always true or never true, give a proof. If it is sometimes true, give one example for which itis true, and one for which it is false.

(a) f(n) + g(n) = (max(f(n), g(n)))

(b) f(n) = (g(n)) and f(n) = O(g(n))

(c) f(n) = O(g(n)) implies 2^f(n) = O(2^g(n))

Solution

(a) f(n) + g(n) = (max(f(n), g(n)))

Ans: True

(b) f(n) = (g(n)) and f(n) = O(g(n))

Ans : False

(c) f(n) = O(g(n)) implies 2f(n)  = O(2g(n))

Ans : False