Question

Using the axioms of propositional logic equivalence outlined in lecture and available in your book, simplify the following compound statements as much as possible. For every derivation you make, write down the axiom you apply on the right of the derivation (e.g law of associativity, De Morgan's, etc). (i) (k Lambda l) ((k Lambda m) Lambda k Lambda (l ~ m)) (ii) z Lambda (q Lambda (~w Lambda q)) Lambda (~z ~q)

Answer 1

(i)

(k ^l ) v ((k ^ m) ^ k ^ (l v ~m))

(k ^ l) v ((k ^ k) ^ m ^ (l v ~m)) Associative law ((A^B) ^ C = (A^C) ^ B)

(k ^l ) v (k ^ m ^ (l v ~m)) Idempotent law (k ^ k =k)

(k ^ l) v (k ^ (m ^ l) v (m ^ ~m)) Distibutive law   ((A^B) v C = (A^C) v ( B ^C))

(k ^ l) v (k ^ (m ^ l) v 0) Compliment law ( A ^ ~A = 0)

(k ^ l) v (k ^ (m ^ l)) Identity law (A v 0 = A)

(k ^ l) v ((k ^ l) ^ m) Associative law ((A^B) ^ C = (A^C) ^ B)

(k ^ l) Absorption law (A v (A ^ B) = A)

(ii)

z ^ (q v (~w ^ q )) ^ (~z v ~q)

z ^ q ^ (~z v ~q) Absorption law (A v (A ^ B) = A)

(z ^ q) ^ ~(z ^ q) Demogan Law (~(A ^ B) = (~A v ~B ) )

0 Compliment law ( A ^ ~A = 0)