$1.$ Propositional Logic $[12$ Marks$].$

$($a$)(6$ marks$)$ Consider a propositional language where A $=$Anna participates in the tournament$,$ B $=$Bianca participates in the tournament$,$ C $=$Carol participates in the tournament$,$ D $=$Dahlia participates in the tournament$.$ Formalise the following sentences in this language.

i$.$ Carol participates in the tournament provided that Dahlia doesn$$t$,$ but, Dahlia participates unless Bianca doesn$$t$.$

ii$.$ If Dahlia participates in the tournament, then, if Carol doesn$$t participate then Anna or Bianca do$.$

iii. Anna, Bianca and Carol participate in the tournament if and only if Dahlia doesn$$t$,$ but, if neither

Anna nor Bianca participate, then Dahlia participates only if Carol does.

$($b$)(6$ marks$)$ Indiana Jones finds a cave with three trunks A$,$ B$,$ C $-$ one contains the Holy Grail, the other two are filled with snakes. On Trunks A and B is inscribed the message "The Holy Grail is not here". On Trunk C is inscribed "The Holy Grail is in Trunk B$".$ Indiana Jones knows the legend that only one inscribed message is true, while the other two are false. Help Indiana Jones find the Holy Grail by formalizing the puzzle in Propositional Language.