document.write( "Question 189101: I need to construct a proof using all of the rules: (ex. commutation, association, material implication, exportation, hypothetical syllogism, DeMorgans, Double Negation, Distribution, etc.)\r
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Algebra.Com's Answer #141917 by jim_thompson5910(35256) $\"\"$ $\"About$

You can put this solution on YOUR website!
This is one tricky derivation, so you need to be a little creative with this one. \r
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\n" ); document.write( "\n" ); document.write( "Note: each premise is used and when a new premise is derived from, I separated it to keep things looking clean.\r
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document.write( "1. (p <--> q) -> s\r\n" );
document.write( "2. ~(~r -> t)\r\n" );
document.write( "3. ~q v ~s Therefore: (t v p) -> (~t & ~ q)\r\n" );
document.write( "--------------------------------------------\r\n" );
document.write( "4.  ~(~~r v t)                               2     Material Implication\r\n" );
document.write( "5.  ~(r v t)                                 4     Double Negation\r\n" );
document.write( "6.   ~r & ~t                                 5     De Morgan's Law\r\n" );
document.write( "7.   ~t & ~r                                 6     Commutation\r\n" );
document.write( "8.   ~t                                      7     Simplification\r\n" );
document.write( "---\r\n" );
document.write( "9.  ~s v ~q                                  3     Commutation\r\n" );
document.write( "10.  s -> ~q                                 9     Material Implication\r\n" );
document.write( "---\r\n" );
document.write( "11. (p <--> q) -> ~q                         1,10  Hypothetical Syllogism \r\n" );
document.write( "12. [(p & q) v (~p & ~q) ] -> ~q             11    Material Equivalence\r\n" );
document.write( "13. ~[(p & q) v (~p & ~q) ] v ~q             12    Material Implication\r\n" );
document.write( "14. [~(p & q) & ~(~p & ~q) ] v ~q            13    De Morgan's Law\r\n" );
document.write( "15. ~q v [~(p & q) & ~(~p & ~q) ]            14    Commutation\r\n" );
document.write( "16. [~q v (~p v ~q)] & [~q v (p v q) ]       15    Distribution\r\n" );
document.write( "17. ~q v (~q v ~p)                           16    Simplification\r\n" );
document.write( "18. (~q v ~q) v ~p                           17    Association\r\n" );
document.write( "19. ~q v ~p                                  18    Tautology\r\n" );
document.write( "20. ~p v ~q                                  19    Commutation\r\n" );
document.write( "---\r\n" );
document.write( "21. ~t & (~p v ~q)                           8,20  Conjunction\r\n" );
document.write( "22. (~t & ~p) v (~t & ~q)                    21    Distribution\r\n" );
document.write( "23. ~(t v p) v (~t & ~q)                     22    De Morgan's Law\r\n" );
document.write( "24. (t v p) -> (~t & ~q)                     23    Material Implication\r\n" );
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