document.write( "Question 297357: I'm stumped on this problem, could someone please help me construct a proof?\r
\n" ); document.write( "\n" ); document.write( "1. (G * P) -> K
\n" ); document.write( "2. E -> Z
\n" ); document.write( "3. ~P -> ~Z
\n" ); document.write( "4. G -> (E v L) conclusion is (G * ~L) -> K \n" ); document.write( "

Algebra.Com's Answer #214152 by jim_thompson5910(35256) $\"\"$ $\"About$

You can put this solution on YOUR website!
It really helps to fully know the 19 rules of inference.\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "

\r\n" );
document.write( "1.  (G * P) -> K\r\n" );
document.write( "2.  E -> Z\r\n" );
document.write( "3.  ~P -> ~Z\r\n" );
document.write( "4.  G ->(E v L)  :. (G * ~L) -> K\r\n" );
document.write( "--------------------------------\r\n" );
document.write( "5.  ~~Z -> ~~P                             3         Transposition\r\n" );
document.write( "6.  Z -> P                                 5         Double Negation\r\n" );
document.write( "7.  E -> P                                 2,6       Hypothetical Syllogism\r\n" );
document.write( "8.  (P * G) -> K                           1         Commutation\r\n" );
document.write( "9.  P -> (G -> K)                          8         Exportation\r\n" );
document.write( "10. E -> (G -> K)                          7,9       Hypothetical Syllogism\r\n" );
document.write( "11. ~G v (E v L)                           4         Material Implication\r\n" );
document.write( "12. (~G v E) v L                           11        Association\r\n" );
document.write( "13. (E v ~G) v L                           12        Commutation\r\n" );
document.write( "14. (~~E v ~G) v L                         13        Double Negation\r\n" );
document.write( "15. ~(~E * G) v L                          14        De Morgan's Law\r\n" );
document.write( "16. (~E * G) -> L                          15        Material Implication\r\n" );
document.write( "17. ~E -> (G -> L)                         16        Exportation\r\n" );
document.write( "18. ~(G -> L) -> ~~E                       17        Transposition\r\n" );
document.write( "19. ~(G -> L) -> E                         18        Double Negation\r\n" );
document.write( "20. ~(G -> L) -> (G -> K)                  19,10     Hypothetical Syllogism\r\n" );
document.write( "21. (G -> L) v (G -> K)                    20        Material Implication\r\n" );
document.write( "22. (~G v L) v (~G v K)                    21        Material Implication\r\n" );
document.write( "23. ~G v (L v ~G) v K                      22        Association\r\n" );
document.write( "24. ~G v (~G v L) v K                      23        Commutation\r\n" );
document.write( "25. (~G v ~G) v (L v K)                    24        Association\r\n" );
document.write( "26. ~G v (L v K)                           25        Tautology\r\n" );
document.write( "27. (~G v L) v K                           26        Association\r\n" );
document.write( "28. (~G v ~~L) v K                         27        Double Negation\r\n" );
document.write( "29. ~(G * ~L) v K                          28        De Morgan's Law\r\n" );
document.write( "30. (G * ~L) -> K                          29        Material Implication\r\n" );
document.write( "\r\n" );
document.write( "

\n" ); document.write( "

\n" );