Tuesday, February 5, 2008

1.2 12D In Progress

1 [(p v q) ^ (p => r) ^ (q => r)] => r
2 [(p v q) ^ (~p v r) ^ (~q v r)] => r by OULE
3 ~[(p v q) ^ (~p v r) ^ (~q v r)] v r by OULE

[~(p v q) ^ ~(~p v r) v ~(~q v r)] v r by De Morgan's Law (I think this line should replace step 4.)
[(~p ^ ~q) ^ (~p ^ ~r) v (q ^ ~r)] v r by De Morgan's Law and Double Negation
[(~p ^ ~p) ^ (~q ^ ~r) v (q ^ ~r)] v r by Commutative and Association Laws
[~p ^ (~q ^ ~r) v (q ^ ~r)] v r by Idempotent Law
[~p ^ ((~q ^ ~r) v q) ^ ((~q ^ ~r) v ~r))] v r by Distributive Law
[~p ^ ((~q v q) ^ (~r v q)) ^ ((~q v ~r) ^ (~r v ~r))] v r by Distributive Law
[~p ^ ((T) ^ (~r v q)) ^ ((~q v ~r) ^ ~r)] v r by Commutative and Negation Law
[~p ^ (~r v q) ^ ((~q v ~r) ^ ~r)] v r by Identity Law
[((~p ^ ~r) v (~p ^ q)) ^ ((~q v ~r) ^ ~r)] v r by Distributive Law
(r v ((~
p ^ ~r) v (~p ^ q))) ^ (r v ((~q v ~r) ^ ~r)) by Commutative and Distributive Laws (Review this line.)
(r v (~p ^ ~r))

4 [(~p v ~q) v (p v ~r) v (q v ~r)] v r by DeMorgans
5 [(~p v p) v (~q v ~r) v (q v ~r)] v r by Associative
6 [(T) v (~q v ~r) v (q v ~r)] v r by Domination
7 [T v (q v ~r)] v r by Domination
8 T v r = T by Domination

No comments: