document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1191577>Question 1191577</A>: <I><SPAN STYLE=\"word-break: break-all;\"> please help me solve this problem: Construct a proof of the following theorem: [(P → Q)&(R → ¬Q)] →<BR>\n" );
document.write( "¬(P&R) </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #823370 by </B><A HREF=/tutors/aboutme.mpl?userid=Edwin+McCravy><B>Edwin McCravy(20055)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Edwin+McCravy><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=Edwin+McCravy><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=823370\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <PRE STYLE=\"white-space: pre-wrap; white-space: -moz-pre-wrap; white-space: -pre-wrap; white-space: -o-pre-wrap; word-wrap: break-word;\">\r\n" );
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document.write( "[(P → Q)&(R → ¬Q)] → ¬(P&R)\r\n" );
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document.write( "Put TTTTFFFF under each P.\r\n" );
document.write( "Put TTFFTTFF under each Q.\r\n" );
document.write( "Put TFTFTFTF under each R.\r\n" );
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document.write( "[(P → Q)&(R → ¬Q)] → ¬(P&R)\r\n" );
document.write( "  T   T   T    T       T T      \r\n" );
document.write( "  T   T   F    T       T F\r\n" );
document.write( "  T   F   T    F       T T\r\n" );
document.write( "  T   F   F    F       T F\r\n" );
document.write( "  F   T   T    T       F T\r\n" );
document.write( "  F   T   F    T       F F\r\n" );
document.write( "  F   F   T    F       F T\r\n" );
document.write( "  F   F   F    F       F F\r\n" );
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document.write( "Under the ¬ of ¬Q write the opposite of what's under\r\n" );
document.write( "the Q.  Then erase what's under the Q\r\n" );
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document.write( "[(P → Q)&(R → ¬Q)] → ¬(P&R)\r\n" );
document.write( "  T   T   T   F        T T      \r\n" );
document.write( "  T   T   F   F        T F\r\n" );
document.write( "  T   F   T   T        T T\r\n" );
document.write( "  T   F   F   T        T F\r\n" );
document.write( "  F   T   T   F        F T\r\n" );
document.write( "  F   T   F   F        F F\r\n" );
document.write( "  F   F   T   T        F T\r\n" );
document.write( "  F   F   F   T        F F\r\n" );
document.write( "\r\n" );
document.write( "Under the → of P → Q write F for T → F\r\n" );
document.write( "and T for everything else. Then erase what's \r\n" );
document.write( "under P and Q:\r\n" );
document.write( "\r\n" );
document.write( "[(P → Q)&(R → ¬Q)] → ¬(P&R)\r\n" );
document.write( "    T     T   F        T T      \r\n" );
document.write( "    T     F   F        T F\r\n" );
document.write( "    F     T   T        T T\r\n" );
document.write( "    F     F   T        T F\r\n" );
document.write( "    T     T   F        F T\r\n" );
document.write( "    T     F   F        F F\r\n" );
document.write( "    T     T   T        F T\r\n" );
document.write( "    T     F   T        F F\r\n" );
document.write( "\r\n" );
document.write( "Under the → of R → ¬Q write F for T → F\r\n" );
document.write( "and T for everything else. Then erase what's \r\n" );
document.write( "under R and ¬:\r\n" );
document.write( "\r\n" );
document.write( "[(P → Q)&(R → ¬Q)] → ¬(P&R)\r\n" );
document.write( "    T       F          T T      \r\n" );
document.write( "    T       T          T F\r\n" );
document.write( "    F       T          T T\r\n" );
document.write( "    F       T          T F\r\n" );
document.write( "    T       F          F T\r\n" );
document.write( "    T       T          F F\r\n" );
document.write( "    T       T          F T\r\n" );
document.write( "    T       T          F F\r\n" );
document.write( "\r\n" );
document.write( "Under the & of P&R write T for T&T\r\n" );
document.write( "and F for everything else. Then erase what's \r\n" );
document.write( "under P and R:\r\n" );
document.write( "\r\n" );
document.write( "[(P → Q)&(R → ¬Q)] → ¬(P&R)\r\n" );
document.write( "    T       F           T      \r\n" );
document.write( "    T       T           F\r\n" );
document.write( "    F       T           F\r\n" );
document.write( "    F       T           F\r\n" );
document.write( "    T       F           F\r\n" );
document.write( "    T       T           F\r\n" );
document.write( "    T       T           F\r\n" );
document.write( "    T       T           F\r\n" );
document.write( "\r\n" );
document.write( "Under the & of (P → Q)&(R → ¬Q) write T for T&T\r\n" );
document.write( "and F for everything else. Then erase what's \r\n" );
document.write( "under (P → Q) and (R → ¬Q):\r\n" );
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document.write( "[(P → Q)&(R → ¬Q)] → ¬(P&R)\r\n" );
document.write( "        F               T      \r\n" );
document.write( "        T               F\r\n" );
document.write( "        F               F\r\n" );
document.write( "        F               F\r\n" );
document.write( "        F               F\r\n" );
document.write( "        T               F\r\n" );
document.write( "        T               F\r\n" );
document.write( "        T               F\r\n" );
document.write( "\r\n" );
document.write( "Under the ¬ of ¬(P&R) write the opposite of what's under\r\n" );
document.write( "the &.  Then erase what's under the &\r\n" );
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document.write( "[(P → Q)&(R → ¬Q)] → ¬(P&R)\r\n" );
document.write( "        F            F        \r\n" );
document.write( "        T            T   \r\n" );
document.write( "        F            T    \r\n" );
document.write( "        F            T   \r\n" );
document.write( "        F            T    \r\n" );
document.write( "        T            T    \r\n" );
document.write( "        T            T   \r\n" );
document.write( "        T            T   \r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Under the → of [(P → Q)&(R → ¬Q)] → ¬(P&R) write \r\n" );
document.write( "F for T → F and T for everything else. Then erase what's \r\n" );
document.write( "under [(P → Q)&(R → ¬Q)] and ¬(P&R):\r\n" );
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document.write( "[(P → Q)&(R → ¬Q)] → ¬(P&R)\r\n" );
document.write( "                   T          \r\n" );
document.write( "                   T     \r\n" );
document.write( "                   T      \r\n" );
document.write( "                   T     \r\n" );
document.write( "                   T      \r\n" );
document.write( "                   T      \r\n" );
document.write( "                   T     \r\n" );
document.write( "                   T    \r\n" );
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document.write( "Since we end up with all T's, the theorem is proved \r\n" );
document.write( "true in all 8 possible cases.\r\n" );
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document.write( "Edwin</PRE>\r<BR>\n" );
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document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );