asymptotic_Notations Self_Doubt

Question

Please give an example case for which all the three conditions

$f(n)\neq O(g(n))$,

$f(n)\neq \Theta (g(n))$ and

$f(n)\neq \Omega (g(n))$

holds true.

Comments

I think what he is trying to say is that time taken to calculate them will be O(1) for both f(n) and g(n), and i cant deny with him as well.

But strictly going by definition and (even from the graphs) of asymptotic notation we can see that at certain point sin x upperbounds cosx and vice versa.

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@Hirak

It would be much better if write your answer and discuss there in comments.

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How does sin and cos functions satisfy those 3 conditions? If I take a constant then one function will be completely above the other function so it doesn't satisfy the conditions.

Answer 1

n^(1+sin n) and n^(1+cos n) these two functions are incomparable so satisfy all three properties given . Correct me if wrong