Asymptotic Notations with condition

Question

Identify the FALSE statement:
$A)$ $f(n)=\theta(f(\frac{n}{2})$  $implies$  $f(n^{2})=\theta(f(\frac{n^{2}}{2}))$

$B)$ $f(n)=O(g(n))$  $implies$  $log(f(n))=O(log(g(n)))$ where $n\geq2$

$C)$ $f(n)=O(g(n))$  $implies$  $2^{f(n)}=O(2^{g(n)})$

$D)$ $f(n)+g(n)=\theta(Max(f(n),g(n)))$

Comments

ok c is the answer

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$C$ is the answer.

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suppose F(n)=2logn and G(n)=logn​​​​​​  here f(n)=O(g(n))

but 

2^f(n)=2^2logn

2^g(n)=2^logn

now assume 2^logn=x

then 2^f(n)= x²

and 
2^g(n)=x

so 2^f(n)=O(2^g(n)) is not hold

this was expalnation ??

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f(n)=2logn and g(n)=logn​​​​​​  here f(n)=O(g(n))??

i think f(n)=logn and g(n)=2logn​​​​​​  here f(n)=O(g(n))
how you write this?

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take f(n)=2n, g(n)=n  then, f(n)<=c.g(n) for c>=2; i.e., f(n)=O(g(n))

but,here 2^f(n) != O( 2^g(n) )

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I think

take f(n)=2n, g(n)=n  then, f(n)<=c.g(n) for c>=2; i.e., f(n)=O(g(n))

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yes..corrected

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yes

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For Option 1 below link might be helpful

https://math.stackexchange.com/questions/819186/fn-thetafn-2-prove-or-disprove

Answer 1

option c is wrong.

suppose F(n)=logn and G(n)=2logn​​​​​​ then

2^logn=2^2logn​​​​​​ now assume 2^logn=x

then it becomes x=x^2(which are not equal)