For the following proof (of equivalence of 2 formulae) provide the justification

Question

For the following proof (of equivalence of 2 formulae) provide the justifications at each step, using the following equivalences. Use the following key: a Idempotent b Law Double Negation c De Morgan's Law d Commutative Properties e Associative Properties f Distributive Properties e Equivalence of Contrapositive h Definition of Implication i Definition of Equivalence j Identity Laws (p F = p T = p) k Tautology (p p = T) l Contradiction (p p = F) p rightarrow (p q) = p (p q) by = (p p) (p q) by = (p p) (p q) by = T (q q) by = (p q) T by = p q by

Explanation / Answer

p -> (p ^ q)

= ~p v (p ^ q) by definition of implication (h)

= (~p v p) ^ (~p v q) by distributive properties (f)

= (p v ~p) ^ (~p v q) by commutative properties (d)

= T ^ (~p v q) by tautology (k)

= (~p v q) ^ T by commutative properties (d)

= ~p v q by identity laws (j)

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