I have to solve the following $n^2+n \in O(n^2−n)$.

I did it this way: $n^2+n \in O(n^2−n) = n^2 \in O(n^2−n) = n^2 / n^2-n = 2n / 2n-n$ which is infinite.

I don't know if this is correct because the $-n$ in the $O$ notation is confusing me and I haven't found anything about it online. I know if this would be at example -1000 that $n^2+n \in O(n^2−n)$ would be correct.

Could someone explain it to me ?

  • $\begingroup$ Duplicate of our reference question. $\endgroup$ – Raphael May 15 '17 at 5:21
  • $\begingroup$ "I haven't found anything about it online" -- You need to work on your searching skills. Given the definition of $O$, it's clear that the $-n$ is irrelevant, so you'll not find anything if you search for "minus in big-oh" or something like that. $\endgroup$ – Raphael May 15 '17 at 5:22

$f \in O(g)$ means there's $n_0$ and $c>0$ such that $n>n_0$ implies $f(n) \leq cg(n)$.

Note that for $n>2$, we have $n^2 \geq 3n$.

Then, for $n>2$, we have: $$ \begin{align} 2(n^2 - n) & = n^2 - 2n + n^2 \\ & \geq n^2 - 2n + 3n\\ & = n^2 + n\\ \end{align} $$

Summarizing: for $n>2$, we have $n^2+n \leq 2(n^2-n)$. Thus $n^2+n \in O(n^2-n)$.

  • $\begingroup$ Where do you get the 3n from ? $\endgroup$ – Mattjo May 2 '17 at 16:45
  • $\begingroup$ If you follow the line "Then...", then you see that n^2 >= 3n is exactly the term needed to make the inequalities work. Working backwards from there, one gets the lower limit for n: n>=3 or equivalently n>2. $\endgroup$ – Paul Hankin May 2 '17 at 17:01
  • $\begingroup$ I understand. Thank you very much for the quick answers ! $\endgroup$ – Mattjo May 2 '17 at 17:07

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