Q1) Show by applying the definition of the O-notation that each of the follow

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)).

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