Given functions f$1,$ f$2,$ g$1,$ g$2$ such that f$1($n$)=$ O$($g$1($n$))$ and f$2($n$)=$ O$($g$2($n$)).$ For each of the

following statements, decide whether you think it is true or false and give a proof or

counterexample. $(12$ points$)$

$($a$)$ f$1($n$)$ f$2($n$)=$ O $($g$1($n$)$ g$2($n$))$

$($b$)$ f$1($n$)+$ f$2($n$)=$ O $($max $($g$1($n$),$ g$2($n$)))$

$($c$)$ f$1($n$)2=$ O g$1($n$)2$

$($d$)$ log$2$ f$1($n$)=$ O $($log$2$ g$1($n$))$