We denote propositional symbols with p, q, r and equivalence as "=". conjunction as 'l'. disjunction with 'V', negation with 'x', implication with '=>'. 1. Using truth-tables, prove the following tautologies (axioms or theorems of propositional logic): p V ( ar) = (p V 9) (p V r) p(q V ) = (p /\ a) V ( pr) ~(p / 9 = ~p / wa ~(p V 9) = ~p Ana p (p => a) 9 (p => a) = ((p V 9)=9) ((p => q) = (p V 9)) = 9 (p => q) = ((p /\ a) = p) ((p => q) = (p q)) = p. (p => a) A (g => r) => (p => r) (p = a) A (a => ) => (p => r) (p => 9) A (q=r) => (p => r) (p => 9) => (p A r => q ) 2. Using truth-tables, show that the following are not tautologies: (p => q) => r = p => (a => r) (p / 9) = ~p /\ ~ ~(p V 9) = ~p V ~ / wp