Let the propositional atoms $e_A,e_B$ represent that $Mr.A($resp$.$ Mr$.$ B$)$ will be elected and let $t_A,t_B$ represent that $Mr.A($resp$.$ Mr$.B)$ speaks the truth. Let us denote $P=\{e_A,e_B,t_A,t_B\}.$

$($a$)$ Write propositions ${}_1,{}_2$ in the form of equivalence and a proposition ${}_3$ in CNF expressing $($in this order$)(i),(ii),(iii),$ all over the language $P.(20$p$)$

$($b$)$ Let $T=\{{}_1,{}_2,{}_3\}.$ Prove by tableau method that $T=e_B*(40p).$

$($c$)$ Give an example of a proposition over $P$ that is independent in theory $T,$ or show that such proposition does not exist. $(20$p$).$

$($d$)$ Find a theory $S$ over $\{e_A,e_B\}$ such that $T$ is a conservative extension of $S.(20$p$)$

In the presidental elections we have two candidates, Mr$.$ A and Mr$.$ B$.$

$($i$)$ Mr$.$ A says: "I will be elected or Mr$.$ B lies."

$($ii$)$ Mr$.$ B says: $"$Mr$.$ A will not be elected or I lie."

$($iii$)$ Exactly one candidate will be elected.