UGC-NET | UGC NET CS 2015 Jun – III | Question 35

Let f(n) and g(n) be asymptotically non-negative functions. Which of the following is correct?

(A)

θ ( f (n)*g(n)) = min (f (n), g(n))

(B)

θ ( f (n)*g(n)) = max (f (n), g(n))

(C)

θ( f (n) + g(n)) = min (f (n), g(n))

(D)

θ ( f (n) + g(n)) = max (f (n), g(n))

Answer: (D)

Explanation:

• Case-1: When none of the f(n) and g(n) are constant functions – In this case max(f(n) , g(n)) <= f(n) * g(n) so max(f(n), g(n)) can not provide a upper bound for f(n) * g(n).
• Case-2: When both of the f(n) & g(n) are constant functions or when any one of the f(n) and g(n) is a non zero constant function, In this case f(n) * g(n) = theta(max(f(n), g(n))).
• Case-3: When at least any one of the f(n) and g(n) is 0, In this case f(n) * g(n) != theta(max(f(n), g(n))). Since max(f(n), g(n)) COULD BE unable to give a lower bound.

Option (D) is correct.

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