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I have some examples for both Asymptotic Bounds: Upper and Lower and I can't understand why we are considering the dominant terms or the n terms in each of them. Can someone please explain them to me?

Asymptotic Upper Bound:

12n^3 + 8n + 20 = O(n^3)

12n^3 + 8n + 20 = O(n^5) [ I think it should be 12n^5 ]

Asymptotic Lower Bound:

12n^3 + 8n + 20 = Omega(n^3)

12n^3 + 8n + 20 = Omega(n)

I read upon the definitions but cannot understand why it changes in the lower bound and why it is like in the upper bound.

tribrick
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  • probably better for http://math.stackexchange.com/ – Carlos Bribiescas Apr 01 '15 at 18:49
  • "12n^3 + 8n + 20 = O(n^3) [ I think it should be O(n^5) ]" Why would you think that? As `n` grows, the dominant value in the sum is the `n^3` term, which is only proportional to, well, `n^3`... not `n^5`... You only consider the highest term, because, for example, when `n = 1e6`, the contribution to the sum of `8n` is only `8e6`, while the first term contributes `12e18`, which is large enough to say that the other terms don't really matter much at all... Also, the `Omega(n)` line is highly suspect... – twalberg Apr 01 '15 at 19:51
  • @twalberg my mistake, i have edited it. the answer is O(n^5) and not O(n^3). – tribrick Apr 01 '15 at 20:07
  • Again, though, why should `12n^3 + ... = O(n^5)`? That doesn't make sense. – twalberg Apr 01 '15 at 20:12
  • @twalberg, yes that is what i also thought as I read upon it online and it must have been a mistake. Can you please explain the lower bound cases also? – tribrick Apr 01 '15 at 20:34
  • I'm not very good explaining mistakes I didn't make, no... Although, looking at it again, were these some sort of "choose the right answer" kind of question? Because in both cases, there's one equation that's clearly right, and one that's just as clearly wrong... – twalberg Apr 01 '15 at 20:58

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