document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=242905>Question 242905</A>: <I><SPAN STYLE=\"word-break: break-all;\"> How do I solve these proofs?\r<BR>\n" );
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document.write( "Number one<BR>\n" );
document.write( "1.~B v[(C>D)&(E>D)]<BR>\n" );
document.write( "2. B&(C v E)   /therefore, D\r<BR>\n" );
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document.write( "Number two<BR>\n" );
document.write( "1. W&(A&M)<BR>\n" );
document.write( "2. (A&W)>([(N v(R v H)]<BR>\n" );
document.write( "3. ~N & (~P&~H)   /therefore, R\r<BR>\n" );
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document.write( "Number three<BR>\n" );
document.write( "1. (O&T)>(S&M)<BR>\n" );
document.write( "2. R>~M<BR>\n" );
document.write( "3. T&R<BR>\n" );
document.write( "4. O&S  /therefore, V\r<BR>\n" );
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document.write( "Number 4. last one<BR>\n" );
document.write( "1. F>W /therefore, (F&S)>W<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #177901 by </B><A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><B>jim_thompson5910(35256)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><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/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=177901\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> # 1\r<BR>\n" );
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document.write( "1.  ~B v [(C -> D) & (E -> D)]\r\n" );
document.write( "2.  B & (C v E) /therefore   D\r\n" );
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document.write( "3.  (C v E) & B                2     Commutation\r\n" );
document.write( "4.  B                          2     Simplification\r\n" );
document.write( "5.  C v E                      3     Simplification\r\n" );
document.write( "6.  ~~B                        4     Double Negation\r\n" );
document.write( "7.  (C -> D) & (E -> D)        1,6   Disjunctive Syllogism\r\n" );
document.write( "8.  D v D                      7,5   Constructive Dilemma\r\n" );
document.write( "9.  D                          8     Tautology\r\n" );
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document.write( "# 2\r<BR>\n" );
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document.write( "1.  W & (A & M)\r\n" );
document.write( "2.  (A & W) -> [N v (R v H)]\r\n" );
document.write( "3.  ~N & ( ~P & ~H )    /therefore, R\r\n" );
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document.write( "4.  (W & A) & M                     1      Association\r\n" );
document.write( "5.  W & A                           4      Simplification\r\n" );
document.write( "6.  A & W                           5      Commutation\r\n" );
document.write( "7.  N v (R v H)                     2,6    Modus Ponens\r\n" );
document.write( "8.  ~N                              3      Simplification\r\n" );
document.write( "9.  ( ~P & ~H ) & ~N                3      Commutation\r\n" );
document.write( "10. ~P & ~H                         9      Simplification\r\n" );
document.write( "11.  ~H & ~P                        10     Commutation\r\n" );
document.write( "12.  ~H                             11     Simplification\r\n" );
document.write( "13.  R v H                          7,8    Disjunctive Syllogism\r\n" );
document.write( "14.  H v R                          13     Commutation\r\n" );
document.write( "15.  R                              14,12  Disjunctive Syllogism\r\n" );
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document.write( "# 3\r<BR>\n" );
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document.write( "1.  (O & T)>(S & M)\r\n" );
document.write( "2.  R -> ~M\r\n" );
document.write( "3.  T & R\r\n" );
document.write( "4.  O & S          /therefore, V \r\n" );
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document.write( "5.  R & T                         3        Commutation\r\n" );
document.write( "6.  R                             5        Simplification\r\n" );
document.write( "7.  O                             4        Simplification\r\n" );
document.write( "8.  T                             3        Simplification\r\n" );
document.write( "9.  O & T                         7,8      Conjunction\r\n" );
document.write( "10. S & M                         1,9      Modus Ponens\r\n" );
document.write( "11. M & S                         10       Commutation\r\n" );
document.write( "12. M                             11       Simplification\r\n" );
document.write( "13. ~~M                           12       Double Negation\r\n" );
document.write( "14. ~R                            2,13     Modus Tollens\r\n" );
document.write( "15. R & ~R                        6,14     Conjunction\r\n" );
document.write( "16. F                             15       Contradiction\r\n" );
document.write( "17. F v V                         16       Addition\r\n" );
document.write( "18. V                             17       Tautology\r\n" );
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document.write( "note: the truth value of R & ~R is ALWAYS false (since R can't be both true and false at the same time). I'm denoting 'false' as the letter 'F'. Also, the truth value of F v V is dependent on the truth value of V (since F is a constant). So F v V is equivalent to V\r<BR>\n" );
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document.write( "# 4\r<BR>\n" );
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document.write( "1.  F -> W          /therefore, (F & S) -> W\r\n" );
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document.write( "2.  ~F v W                             1      Material Implication\r\n" );
document.write( "3.  (~F v W) v ~S                      2      Addition\r\n" );
document.write( "4.  ~F v (W v ~S)                      3      Association\r\n" );
document.write( "5.  ~F v (~S v W)                      4      Commutation\r\n" );
document.write( "6.  (~F v ~S) v W                      5      Association\r\n" );
document.write( "7.  ~(F & S) v W                       6      De Morgan's Law\r\n" );
document.write( "8.  (F & S) -> W                       7      Material Implication\r\n" );
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