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(18)    [(p Aq) Ar] £Ąp A(q Ar)] (the associative property

of conjunction)

(19)    [(p Vq) Vr] &{p V(q Vr)] (the associative property

of disjunction)

(20)    p =>{p Vq)

(21)    (p Aq) =>p

(22)    ~(~p) &p (law of double negation)

(23)    {p =>q) =A(p Aq) <=>p]

(24)    {~p) =*[~(p Aq)]

(25)    {[{-p) =K~q)] A[(~p) =>q]} =>p

(26)    [p A(p =>q)] =>q (the law of "modus ponens")

(27)    [{p =>q) A(~q)] =K~p) (the law of "modus tollens”)

(28)    [(pA~q)^q] <=*p=>q)

(29)    (p =>q) =Ą(p Aq) <=>p]

(30)    'i £*), -o <=n,_P \Ą~p) <=n,_P A(~p) m

(properties of negation)

To prove whether the given statement is a tautology, contradiction or neither, we can use the truth table. For example, let's verify the tautology (P ==>q) ^1(P Aq) <=*p] (*) pqp==>qp Aq(p Aq) <==%(*)

1 1 1 1 1 i 100001 011011