b$)$ Suppose T$1($N$)$ in O $($f$($N$))$ and T$2($N$)$ in O $($f$($N$)).$ Which of the following are true?

$($i$)$ T$1($N$)+$ T$2($N$)$ in O $($f$($N$))($ii$)$ T$1($N$)$ T$2($N$)$ in O $($f$($N$))($iii$)$ T$1($N $)$ T$2($N $)$ in o $($f $($N $))$

$($iv$)$ T$1($N$)$ in O$(1)$ T$2($N$)$

$($v$)$ T$1($N$)$ in O $($T$2($N$))$

$($c$)$ Find a counterexample to the following claim:

f$($n$)=$ O$($s$($n$))$ and g$($n$)=$ O$($r$($n$))$ imply that f$($n$)$ g$($n$)=$ O$($s$($n$)$ r$($n$)).$