Using the notation of Example $1\mathrm{.}13,$ find grammars for the languages below. Assume $=\{$a$,$ b$,$ c$\}.$

$($a$)$ L $=\{$w : na $($w$)=$ nb $($w$)+1\}.$

$($b$)$ L $=\{$w : na $($w$)>$ nb $($w$)\}.$

$*($c$)$ L $=\{$w : na $($w$)=2$nb $($w$)\}.$

$($d$)$ L $=\{$w in $\{$a$,$ b$\}*$ : $|$na $($w$)$ nb $($w$)|=1\}.$