Question $5(2$pt$)$

Remember the following grammar for propositional logic:

$$ VT $=\{$p$1,$ p$2,...,$ pn$,,,,,(,)\}$

$$ Vn $=\{$S$\}$

$$ S

$$ R $=$

$($i$)$ S $->$ pn

$($ii$)$ S $->$S

$($iii$)$ S $->($S $$ S$)$

$($iv$)$ S $->($S $$ S$)$

$($v$)$ S $->($S $$ S$)$

Using the grammar, provide a derivation of the following sentences of propositional logic.

$(($p $$ q$)($p $$ q$))$

$($p $$p$)$

$(($p $$ q$)$q$)$

$($p $$ q$)$

$($p $$q$)$