$(20$ pts$.)$ Big$-$O notation. Assume you have functions $f$ and $g$ such that $f(n)$ is $O(g(n)).$ For each of

the following statements, decide whether it is true or false. Give a proof for the true statements and a

counterexample for the false ones. $($Hint: Think carefully before committing to an answer. Sometimes the

answer that seems "obvious" might be wrong.$)$

$($a$)2^{f(n)}$ is $O(2^{g(n)})$

$($b$)f(n{)}^2$ is $O(g(n{)}^2)$

$($c$)2^{f(n)}$ is $O(2^{O(g(n))})$

$($d$)lo{g}_{2}f(n)$ is $O(lo{g}_{2}g(n))$ where $g(n)>1$