document.write( "Question 160091: Prove the argument:
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Algebra.Com's Answer #118258 by Edwin McCravy(20054)\"\" \"About 
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document.write( "Prove the argument:\r\n" );
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document.write( "1.  p -> q\r\n" );
document.write( "2.  r \/ s \r\n" );
document.write( "3.  ~s -> ~t\r\n" );
document.write( "4.  ~q \/ s\r\n" );
document.write( "5.  ~s\r\n" );
document.write( "6.  (~p \/ r) -> u\r\n" );
document.write( "7.  w \/ t\r\n" );
document.write( "therefore u /\ w\r\n" );
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document.write( "Change the disjunctions in 2, 4, and 7, to their equivalent conditionals \r\n" );
document.write( "by this rule  a \/ b <=> ~a -> b\r\n" );
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document.write( "1.  p -> q\r\n" );
document.write( "2.  ~r -> s \r\n" );
document.write( "3.  ~s -> ~t\r\n" );
document.write( "4.  q -> s\r\n" );
document.write( "5.  ~s\r\n" );
document.write( "6.  (~p \/ r) -> u\r\n" );
document.write( "7.  ~w -> t\r\n" );
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document.write( "Now write all the equivalent contrapositives of the conditionals:\r\n" );
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document.write( " 8.  ~q -> p\r\n" );
document.write( " 9.  ~s -> r \r\n" );
document.write( "10.    t -> s\r\n" );
document.write( "11.  ~s -> ~q\r\n" );
document.write( "12.  ~s\r\n" );
document.write( "13.  ~u -> ~(~p \/ r) \r\n" );
document.write( "14.  ~t -> w\r\n" );
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document.write( "Start with 5\r\n" );
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document.write( "      ~s \r\n" );
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document.write( "By 9, we have\r\n" );
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document.write( "      ~s -> r\r\n" );
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document.write( "By the conditional r -> (r \/ ~p), we have\r\n" );
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document.write( "      ~s -> r -> (r \/ ~p)\r\n" );
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document.write( "By the commutative law, (r \/ ~p) <=> (~p \/ r), we have:\r\n" );
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document.write( "      ~s -> r -> (r \/ ~p) <=> (~p \/ r)\r\n" );
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document.write( "By 6, we have \r\n" );
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document.write( "      ~s -> r -> (r \/ ~p) <=> (~p \/ r) -> u\r\n" );
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document.write( "That is one part of the conjunction conclusion\r\n" );
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document.write( "Start again with 5\r\n" );
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document.write( "      ~s\r\n" );
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document.write( "By 3, we have\r\n" );
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document.write( "      ~s -> ~t\r\n" );
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document.write( "By 14, we have\r\n" );
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document.write( "      ~s -> ~t -> w\r\n" );
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document.write( "So by syllogism we have both parts\r\n" );
document.write( "of the conjunction conclusion u and w.\r\n" );
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document.write( "Therefore u /\ w\r\n" );
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document.write( "q was not involved so we did not need 1, 4,\r\n" );
document.write( "or their contrapositives 8, 11.\r\n" );
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document.write( "Edwin
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