Note: (1) n assumes to be natural numbers. (2) All log functions outside asymptotic notations assume to have a positive constant base that is larger than 1.

Prove :

1 - Using the definition of O, prove: 3n^2+nn = O(n^2)

2 - Using the definition of o, prove: 2(n+100 n)(log n)^2 = o(nn / log n)

3 - Using the definition of , prove: 10n^3 + 7n log n = (n^3 )

4 - Using the definition of , prove: 2n^2 + 5nn = (n log n)

5 - Let f(n) and g(n) be increasing positive functions. Using the definition of prove the claim f(n) + g(n) = (max { f(n), g(n) }) is always true.

6 - Let f(n) be a positive increasing function. Is the claim f(n) = (f(n)) always true? if so prove it, otherwise show example function for f on which the claim fails.