$($i$)$ Define recursively the set Prop$()$ of propositional symbols that occur

in a formula $$ from $L_M(P).$ Thus, for example, your definition should

establish that Prop$(p$diamond$(qvvr))=\{p,q,r\}.$

$($ii$)$ Suppose that $M=(W,R,)$ and $M^{\text{'}}=(W,R,{}^{\text{'}})$ are models. Note

that the relational structures of $M$ and $M^{\text{'}}$ are the same. Show that

for every formula $in{L}_M(P),$ if $(w)$Prop$()={}^{\text{'}}(w)$Prop$()$ for

all winW, then

$M,w,>,M^{\text{'}},w$

for each winW.