Question

I have received the assignment to prove 1/O(n) = Ω(n)

However, this would mean that n element of O(n) => 1/n element of Ω(n) which is clearly wrong.

So my question is: Is the statement 1/O(n) = Ω(n) correct?

EDIT: I send an Email to the assistant who wrote the questions. And used the example of f(n) = 1.
He then said, that the statement is indeed incorrect.

Answer 1

The notation 1/O(n) = Ω(n) is a bit vague. There is really no O(n) on it’s own, there is only f(n) ~ O(n), which is a statement about values of function f (there is a constant C so that f(n) < Cn for each n).

And the statement to prove, if I understand it correctly, is “if function f(n) is O(n) than 1/f(n) is Ω(n)“, formally:

f(n) ~ O(n) => 1/f(n) ~ Ω(n)

Edit: Except I don’t think I understand it correctly, because f(n) = 1 ~ O(n), but 1/f(n) = f(n) = 1 is clearly isn’t Ω(n). Weren’t the assignment f(n) ~ O(n) => 1/f(n) ~ Ω(1/n) instead?

Edit: Different people tend to use different operators. Most common is f(n) = O(n), but that is confusing because the right hand side is not a function, so it can’t be normal equality. We usually used f(n) ~ O(n) in school, which is less confusing, but still inconsistent with common use of that operator for general equivalence relations. Most consistent operator would be f(n) ∈ O(n), because the right hand side can reasonably be treated as set of functions.