Asymptotic notation properties
Let f(n) and 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 2f(n) = O(2g(n)).
e. f(n)=O((f(n)2).
f. f(n) = O(g(n)) implies g(n) = Ω(f(n)).
g. f(n) = O(f(n/2)).
h. f(n)=Θ(f(n))=Θ(f(n).
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