Question

6) This question is part of Exercise 3-4 on page 62 of CLRS, but the letters aren't all the same. Answer TRUE or FALSE for each of the following statements. TRUE means that the statement is TRUE for any asymptotically positive functions f(n) and g(n). Otherwise, answer FALSE. You don't have to Prove or Disprove these statements... but you should learn how to do that. f(n)- O( g(n)) implies g(n) O( f(n)) b) This is part a in Exercise 3-4. This is part b in Exercise 3-4. This is part g in Exercise 3-4. This is part h in Exercise 3-4. a) fin) + g(n)-Θ( min(f(n), g(n)) ) f(n) = Θ( f(n/2) ) c) d) fin) + 0( f(n) )= Θ( f(n) ) Part d of Question 6 warrants some explanation. What it asks is whether f(n) plus any function that is o(Nn)) must always be Θ( f(n) ), for any asymptotically positive function f(n).

Answer 1

As per Chegg answering guidelines, I'm answering only the first 4 sub-parts:-

6)

a) FALSE

Take f(n) = 1 and g(n) = n .

Here f(n) = O(g(n)) but g(n) is not O(f(n)).

b) FALSE

As, f(n) + g(n) = θ(max(f(n) g(n)))

c) FALSE

Let f(n) = 2²ⁿ then f(n/2) = 2ⁿ and 2²ⁿ is not O(2ⁿ).

d) TRUE