asymptotic notation

Question

if

f(n) ≠  O(g(n))  then, g(n)  ≠ O(f(n)) 

state true or false with explanation.

Answer 1

This is false .

f(n) ≠  O(g(n)) since this is given so f(n) and g(n) cannot equal asymptoticaly.

so f(n) = n² , g(n) = n so f(n) ≠  O(g(n)) satisfy

now  g(n)  =  O(f(n)) always i.e. n =O(n²)

Comments

correct...!!

Answer 2

False, take G(n) = 10 and F(n) = 100

Then, 100 != O(10) (TRUE)

but, 10 = O(100) (Also true, which it is saying in the question as false.)

Comments

@Kapil. For the case which you have taken, F = $O(g)$ which isnt the case as given. The asymptotic notation is different than the actual magnitude.

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if we trignometry fuction then in this case it is true.f(n) ≠  O(g(n))  then, g(n)  ≠ O(f(n))  both can we true.