Finding complexity in case ratio of two compexity is constant.

Question

 Given two positive functions f(n) and g(n). If $\frac{f(n)}{g(n)}=c$, for some constant c ≥ 0 and c is non-negative but not infinite then which of the following is correct?

• f(n) = O(g(n))
• f(n) = θ (g(n))
• f(n) = Ω (g(n))
• None of these

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Here I strongly believe that Answer should be the f(n) = θ (g(n)). Other two are correct, but this is strongest statement !

As f(n) / g(n) = constant is only possible if they are of same order.

In that case they are both upper & lower bounds of each other !

Q 47

From Made Easy FLT 6-Practice Test 14

Comments

Then G(n) can be bigger than F(N). G(N) can never be 0.

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I think G(n) can be any positive function , But  Here fn can't be greator than gn but can be smaller

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f(n) can be as smaller as possible. It can not be ASYMPTOTICALLY bigger.

In sense,

it is allowed to be constant factor bigger !

f(n) = 2n, g(n) = n.

Answer 1

IT seems here I made mistake in analysis.

Here c = 0 is possible as pointed out by @Amar Sokt !

In that case f(n) can be 0, & in that case g(n) is always bigger than f(n).g(n) = 0 is not allowed, otherwise whole thing will go to infinity & it is said c can not be 0.

f(n) = θ (g(n)) will be false. As g(n) will not be lowerbound of f(n).

So A is correct answer.