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Question

I apologize in advance for my poor math skills...

I'm trying to understand how the math behind Big O Notation works. I understand from this that 2n^2 = O(n^3) and have proved that n = O(n^2) , but I also seem to have proved that n^3 = O(n^2) which doesn't make sense and I'm pretty sure is wrong. Here's how I'm "proving" this:

n^3 = O(n^2)
n^3 <= c*n^2
n <= c     #n^2 cancels out
1 <= c/n
c = 1; n0 = 1

What am I doing wrong?

Answer 1

1 <= c/n does not hold for all n > n0 , for example, for n=2 (with your n0=1,c=1), you get:

1 <= 1/2

and this is a false statement.

The key in big O notation is you need to prove that for ALL n > n0 , the equation f(n) <= C*g(n) holds (for some C,n0 ) , in order to show f(n) is in O(g(n))

Answer 2

Normally Big O notation is defined as a set of functions with common asymptotic behaviour. That is, as n grow large how will the function grow.

In this case, we clearly see that n^3 outgrows n^2, so they are in fact in different O classes.