Express each of the following statements symbolically, using quantifiers, variables, and the predicates P$($s$),$ J$($s$),$ and C$($s$).$

$($a$)$ All peanut butter sandwiches have jelly on them.

$$s in S$,$ P$($s$)->$ J$($s$)$

$($b$)$ Some peanut butter sandwiches have jelly on them.

$$s in S$,$ P$($s$)$ J$($s$)$

$($c$)$ No cheese sandwiches have peanut butter on them.

$$s in S$,$ C$($s$)->-$P$($s$)$

$($d$)$ A sandwich with neither peanut butter nor jelly must have cheese on it$.$

$$s in S$,(-$P$($s$)-$ J$($s$))->$ C$($s$)$

$($e$)$ Some cheese sandwiches have jelly on them and some do not.