Q1) Show, by applying the definition of the O-notation, that each of the
following is true.
– If f(n)= n(n-1)/2, then f(n) = O(n^2).
– If f(n)= n+ log n, then f(n) = O(n).
– 1+ n+ n^2 + n^3 = O(n^3).
Q2) State without proof whether each of the following is True or False.
– 7 = O(1).
– n + n^4 = O(n^3).
– For any polynomial T(n), T(2n) = O(T(n)).
– For any function T(n), T(2n) = O(T(n)).
Q3) Show, by the definition of the O-notation, that n^3 != O(n^2).
(Note != means not-equal.)
Q4) Let T1(n)= O(f(n)) and T2(n)= O((g(n)). Prove by the definition of
the O-notation, this implies T1(n) + T2(n)= O(f(n) + g(n)).
Q5) Let T1(n)= O(f(n)) and T2(n)= O((g(n)). Prove by the definition of
the O-notation, this implies T1(n) * T2(n)= O(f(n) * g(n)).
