document.write( "Question 188873: I need to solve using Transportation, Material Implication, Material Equivalence, Exportation, Tautology, Double Negation, Commutation, Association, Distribution, Demorgan's Theorem, Modus Ponems, Modus Tollens, Hypothetical Syllogism, Conjunstion, Simplification, Addition, Constructive Dilemma, Absorption and Disjuntive Syllogism. \r
\n" ); document.write( "\n" ); document.write( "1. (p <--> q) -> s
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\n" ); document.write( "3. ~q v ~s Therefore: (t v p) -> (~t & ~q) \r
\n" ); document.write( "\n" ); document.write( "AND this is the other problem\r
\n" ); document.write( "\n" ); document.write( "1. p v (q & r)
\n" ); document.write( "2. ~r
\n" ); document.write( "3. p -> (s -> ~t) Therefore: ~(s & t) \n" ); document.write( "

Algebra.Com's Answer #141970 by jim_thompson5910(35256) $\"\"$ $\"About$

You can put this solution on YOUR website!
This is probably too late, but here are the solutions anyway...\r
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\n" ); document.write( "\n" ); document.write( "# 1\r
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\n" ); document.write( "\n" ); document.write( "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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\n" ); document.write( "\n" ); document.write( "# 2\r
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\n" ); document.write( "\n" ); document.write( "This derivation mainly involves isolating atomic components through modus ponens and a disjunction syllogism.\r
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document.write( "1. p v (q & r)\r\n" );
document.write( "2. ~r\r\n" );
document.write( "3. p -> (s -> ~t) Therefore: ~(s & t)\r\n" );
document.write( "-----------------------------------------\r\n" );
document.write( "4.  (p v q) & (p v r)                    1     Distribution\r\n" );
document.write( "5.  (p v r) & (p v q)                    4     Commutation\r\n" );
document.write( "6.  p v r                                5     Simplification\r\n" );
document.write( "7.  p                                  6,2     Disjunctive Syllogism     \r\n" );
document.write( "8.  s -> ~t                            3,7     Modus Ponens\r\n" );
document.write( "9.  ~s v ~t                              8     Material Implication   \r\n" );
document.write( "10. ~(s & t)                             9     De Morgan's Law\r\n" );
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