document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1041047>Question 1041047</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Note that ‘->’ is used for conditionals, ‘~’ is used for negations, ‘v’ is used for disjunctions, ‘&’ is used for conjunctions, ‘<->’ is used for biconditionals, and ‘/’ is used as the conclusion indicator.\r<BR>\n" );
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document.write( "Prove the following three arguments to be valid using the method of Natural Deduction. Do not use Conditional Proof in these proofs.\r<BR>\n" );
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document.write( "1. ~(P v Q)<BR>\n" );
document.write( "2.  (R v S) -> P       / ~S \r<BR>\n" );
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document.write( "1. P v (Q v R)<BR>\n" );
document.write( "2. (~T & ~R)           / (Q v P)\r<BR>\n" );
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document.write( "1. (P v Q) -> R<BR>\n" );
document.write( "2. (S v R) -> T<BR>\n" );
document.write( "3. ~(Q -> E)           / T </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #656058 by </B><A HREF=/tutors/aboutme.mpl?userid=robertb><B>robertb(5830)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=robertb><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/cgi-bin/show-face.mpl?user=robertb><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=656058\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> 1. ~(P v Q)<BR>\n" );
document.write( "2.  (R v S) -> P       / ~S <BR>\n" );
document.write( "3. ~P&~Q ............De Morgan's<BR>\n" );
document.write( "4. ~P   .............Simplification<BR>\n" );
document.write( "5. ~(R v S) v P .....Material implication on 2<BR>\n" );
document.write( "6. ~(R v S) v ~~P ....Double negation<BR>\n" );
document.write( "7. ~(R v S)  .........Disjunctive syllogism on 4 and 6<BR>\n" );
document.write( "8. ~R&~S   ...........De Morgan's on 7<BR>\n" );
document.write( "9. ~S  ...............Sinplification\r<BR>\n" );
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document.write( "1. P v (Q v R)<BR>\n" );
document.write( "2. (~T & ~R)           / (Q v P)<BR>\n" );
document.write( "3. ~R  .............Simplification on 2.<BR>\n" );
document.write( "4. ( P v Q) v R .... Associativity on 1<BR>\n" );
document.write( "5. P v Q  ..........Disjunctive syllogism on 3 and 4<BR>\n" );
document.write( "6. Q v P ............Commutativity\r<BR>\n" );
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document.write( "1. (P v Q) -> R<BR>\n" );
document.write( "2. (S v R) -> T<BR>\n" );
document.write( "3. ~(Q -> E)           / T<BR>\n" );
document.write( "4. ~(P v Q) v R .......MI on 1<BR>\n" );
document.write( "5. (~P & ~Q) v R ......Dl on 4<BR>\n" );
document.write( "6. (~P v R) & (~Q v R) ...Distributivity on 5<BR>\n" );
document.write( "7. ~(S v R) v T .......MI on 2<BR>\n" );
document.write( "8. (~S & ~R) v T ......Dl on 7<BR>\n" );
document.write( "9. (~S v T) & (~R v T) ...Distributivity on 5<BR>\n" );
document.write( "10. ~Q v R .............simplification on 6<BR>\n" );
document.write( "11. Q -> R ............MI on 10<BR>\n" );
document.write( "12. ~R v T .............simplification on 9<BR>\n" );
document.write( "13. R -> T ............MI on 12<BR>\n" );
document.write( "14. Q -> T ............hypothetical syllogism on 11 and 13<BR>\n" );
document.write( "15. ~Q v T ............MI on 14.<BR>\n" );
document.write( "16. ~(~Q v E) .........MI on 3.<BR>\n" );
document.write( "17.~~Q & ~E ............distribution<BR>\n" );
document.write( "18. Q & ~E ............double negation<BR>\n" );
document.write( "19. Q .................simplification on 18<BR>\n" );
document.write( "20. ~~Q  .............DN<BR>\n" );
document.write( "21.  T  ...............disjunctive syllogism on 15 and 20.\r<BR>\n" );
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