# question about a big oh notation

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)).

## answer

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check attached xxxxxxxxxxx

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xxx Show, by applying xxx definition of xxx xxxxxxxxxxx that xxxx of xxxxxxxxxxxxx is true.

x If xxxxx n(n-1)/2, xxxx xxxx x xxxxxxxxx xx xxxxx n+ log xx xxxx xxxx x xxxxxxx xx xx xxx + n^3 x O(n^3).

**xxxxxxxx **

F(n) x xxxxxxxx < xxx - 1) = n^2 – x < n^2 for all n >1

Hence, f(n) < xxx for all n > x

xxxxxx by xxxxxxxxxx of xxxxxxxxxx x **f(n) x O(n^2)**

xxxx x n x xxxx xxx xx know xxxx n > xxxxxx for all x > 0 {this is xxxxxxx xxx > n and xx xxxxxx log on xxxx sides we get, n > xxxxx

xxxxxx

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