Understanding big O notation based on the examples given

Question

I have some questions about the examples I came by in a book that helps me to explore the world of programming.

I understand what Big-O notation is and interested in the following:

1. To prove if $g(n) \in O(f)$, everything we need is to find only one set of ($c$ and $n_0$) that $g(n) \leq c\cdot f(n)$ , $n>n_0$. So, for example, if I found 1 set that holds this inequality and 10 that don't, $g(n) \in (O(f))$ anyway. Other words, it's sufficient to find only ONE set of($c$,$n_0$) that holds the inequality to prove it or not?

2. To disprove it, we need to find $N$ that after $c$ and $n_0$ have been chosen doesn't hold the inequality. If I can't do it with the chosen set of $c$ and $n_0$ does it mean that I can't disprove it?

Here is the example I am perplexed about:

We know that $n$ is in $O(n^2)$ but $(n^2)$ is not in $O(n)$.

To prove the first statement we have to prove that $n \leq c\cdot n^2$ and it's clear enough that doesn't need any explanation.

But if we reverse it and try to prove that $n^2$ is in $O(n)$, which is obviously not true, there is some mystery that I can't find an answer to. We pick $c=n$ and the inequality becomes $n^2 \leq n\cdot n$, so they become equal and from that is seems that $n^2$ is IN $O(n)$, because we can't find any values of n that the inequation doesn't hold.

Thanks in advance and sorry if the question is very stupid.

Answer 1

The question is not stupid and arises from an understandable point of view. The key point is that $c$ must be a constant. On the other hand, $n$ is the parameter that is tending towards $\infty$. Thus you cannot pick $c = n$ because $n$ keeps increasing.

Everything else you said is right. To prove it is in $O(f)$ it suffices to find one set of ($c$, $n_0$). To disprove, you must show that regardless of ($c$, $n_0$), the function grows too fast.

Answer 2

Your argument for $n = O(n^2)$ is correct.

For $n^2 \ne O(n)$, you can't pick $c=n$; because $c$ must be a constant.

Suppose you pick the constant $c=17$. Then you would need to also pick $n_0$ so that $n^2 < 17n$ for all $n>n_0$. But you can't do this because there is no such $n_0$: whenever $n$ is at least $17$, then $n^2 < 17n$ is false. Okay, $c=17$ didn't work, perhaps $c=1019$ will work instead. Then you would need to pick $n_0$ so that $n^2 < 1019n$ for all $n>n_0$. This is also impossible.