We denote propositional symbols with p,q,r and

equivalence as "=",

conjunction as '/\',

disjunction with '\/',

negation with '~',

implication with '=>'.

1. Using truth-tables, prove the following tautologies (axioms or theorems of propositional logic): p V (Ar) = (p V ) (p Vr) pi (qV r) = (p /\ a) V (p /\ r) ~(p/\ q) = ~p V -q -(p V q) -p / -a p/\ (p => q) q (p => q) = ((p V 9)=q) ((p => q) = (p V q)) = q (p => q) = ((p /\ q) = p) ((p => q) = ( pq)) = p (p => q) / ( => r) (p => r) (p q) / (q => r) (p => r) (p => q) (q=r) (p => r) (p => 9) (p /\ r => q /\ r)