Consider the following function complexities

f$1($n$)=5$n$2-$n$+3$ f$2($n$)=5$lgn $+3$ f$3($n$)=4$nlgn

f$4($n$)=3$n $+4$ f$5($n$)=$ lg$($n$2)$ f$6($n$)=$ en

$($a$)$ List all the elements of the following Big $$O$$ sets, given the following $5$ functions. Remember that Big $$O$$ contains proportional and slower$-$growing functions. All you need to do is add all appropriate fi into the sets, denoted by the $\{\}.$ It is possible that a set will not have elements in it$.$

O$($n$3)=\{\}$

O$($n$)=\{\}$

O$($lgn$)=\{\}$

O$(3$n$)=\{\}$

$($b$)$ Do the same for the Big $\backslash$theta and Little $$o$$ sets. It is possible that a set will not have elements in it$.$

$\backslash$theta $($n$2)=\{\}$

$\backslash$theta $($lgn$)=\{\}$

o$($nlgn$)=\{\}$

o$(3$n$)=\{\}$

$($c$)$ If Suppose f$($n$)$ is a function which belongs in sets O$($n$2)$ as well as in $\backslash$Omega $($n$).$ List any $4$ functions that would belong in both sets.

$\_\_\_\_\_\_\_\_,\_\_\_\_\_\_\_\_,\_\_\_\_\_\_\_\_,\_\_\_\_\_\_\_\_$

$($d$)$ Suppose a function f$($n$)$ belongs in both O$($g$($n$))$ and $\backslash$Omega $($g$($n$)).$ Clearly explain why this function also must belong in $\backslash$theta $($g$($n$)).$