Let f (n) an= g(n) be asymptotically positive functions. Prove or disprove each of the following conjectures.

a. f (n) = O(g(n)) implies g(n) = O(f (n)).

b. f (n) + g(n) = Θ(min(f (n), g(n))).

c. f (n) = O(g(n)) implies lg(f (n)) = O(lg(g(n))), where lg(g(n)) ≥ 1 and f (n) ≥ 1 for all sufficiently large n.

d. f (n) = O(g(n)) implies 2^f(n) = O (2^g(n)).

e. f (n) = O ((f (n))²).

f. f (n) = O(g(n)) implies g(n) = Ω(f (n)).

g. f (n) = Θ(f (n/2)).

h. f (n) + o(f (n)) = Θ(f (n)).