## asymptotic notation informal definitions

I was wondering if the O notation definition could be exchanged with the Ω notation and o could be exchanged with the ω notation.

I ask this because of this:
2nē O(nē) means that 2nē <= c*nē which it is true for c=3 and n>=1 for example

Instead, it would be like this:

c*2nē <= nē which would be true for c=1/3 and n>=1 for example

and in the Ω case, 2nē Ω nē means that c*2nē <= nē ,which is true to c=1/3 and n>=1 for example. Instead could it be defined so it would means that 2nē < c*nē, which would be true for for c=3 and n>=1, for example. Basically changing the meaning of these notations

The same would be applicable to o and ω notations.

And in the case of nē-2n = θ(nē), instead of c1*nē<=nē-2n<=c2*nē could it be c1*nē-2n<=nē<=c2*nē-2n adjusting the coefficients as needed?

It is all of this valid, or there is some restriction that forces these definitions to be that way?
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 I don't think it's possible to know what you are talking about. Could you please try to explain your question over?
 I will try starting with only the O notation then for a function f(x) to be O g(x) is necessary that f(x) <= c * g(x) for a certain n >= n0 right? for f(x) = 2nē to be <= than g(x) = nē, you could call c of 3 and n >= 1 for instance. However, if the definition of O notation were c* f(x) <= g(x), for a certain n>= n0, c could be 1/3 and n>=1 that the inequality would keep being true. Instead to make g(x) bigger, the idea is to make f(x) smaller. But if you pay attention to the Ω notation would have noted that it is c* f(x) <= g(x) for n>= n0, exactly the "new" notation for O. So Ω would have to change for not be equal to O. And so would became f(x) <= c * g(x), that it is the "old" definition of O So what I am doing here is basically swapping their definitions. The original Ω is equal to the new O notation

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