Prove or disprove each of the following. Either give a proof using the definitions of order notation, or give a counter example.

$1$

$($a$)$ If f$1($n$)$ in $\backslash$Theta $($g$1($n$))$ and f$2($n$)$ in $\backslash$Theta $($g$2($n$)),$ then f$1($n$)+$ f$2($n$)$ in $\backslash$Theta $($g$1($n$)+$ g$2($n$)).$