document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1175639>Question 1175639</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Construct a truth table for ( p ---> ~ r ) v ( ~ q ^ r )  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #801326 by </B><A HREF=/tutors/aboutme.mpl?userid=Edwin+McCravy><B>Edwin McCravy(20056)</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=801326\">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;\"></font>\r\n" );
document.write( " ( p ---> ~ r ) v ( ~ q ^ r ) \r\n" );
document.write( "\r\n" );
document.write( "Start with this.  \r\n" );
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document.write( "| p | q | r | -q | -r | p ---> ~r | ~q ^ r || ( p ---> ~r ) v ( ~q ^ r ) |\r\n" );
document.write( "|---|---|---|----|----|-----------|------------------------------------------\r\n" );
document.write( "| T | T | T |    |    |           |        ||                            | \r\n" );
document.write( "| T | T | F |    |    |           |        ||                            | \r\n" );
document.write( "| T | F | T |    |    |           |        ||                            | \r\n" );
document.write( "| T | F | F |    |    |           |        ||                            | \r\n" );
document.write( "| F | T | T |    |    |           |        ||                            | \r\n" );
document.write( "| F | T | F |    |    |           |        ||                            | \r\n" );
document.write( "| F | F | T |    |    |           |        ||                            | \r\n" );
document.write( "| F | F | F |    |    |           |        ||                            | \r\n" );
document.write( "\r\n" );
document.write( "Fill in the ~q and the ~r column with the opposite truth values of the q and r\r\n" );
document.write( "columns:\r\n" );
document.write( "\r\n" );
document.write( "| p | q | r | -q | -r | p ---> ~r | ~q ^ r || ( p ---> ~r ) v ( ~q ^ r ) |\r\n" );
document.write( "|---|---|---|----|----|-----------|------------------------------------------\r\n" );
document.write( "| T | T | T |  F |  F |           |        ||                            | \r\n" );
document.write( "| T | T | F |  F |  T |           |        ||                            | \r\n" );
document.write( "| T | F | T |  T |  F |           |        ||                            | \r\n" );
document.write( "| T | F | F |  T |  T |           |        ||                            | \r\n" );
document.write( "| F | T | T |  F |  F |           |        ||                            | \r\n" );
document.write( "| F | T | F |  F |  T |           |        ||                            | \r\n" );
document.write( "| F | F | T |  T |  F |           |        ||                            | \r\n" );
document.write( "| F | F | F |  T |  T |           |        ||                            | \r\n" );
document.write( "\r\n" );
document.write( "To fill in the next column, p ---> ~r, using the rule for --->.\r\n" );
document.write( "Put T except in case of \"T ---> F\", [T on the left and F on the right\r\n" );
document.write( "of --->) and put F in that case. (Notice that only the 1st and 3rd rows have T\r\n" );
document.write( "under p and F under ~r.  So put F's in those two rows and T's everywhere else:\r\n" );
document.write( "\r\n" );
document.write( "| p | q | r | -q | -r | p ---> ~r | ~q ^ r || ( p ---> ~r ) v ( ~q ^ r ) |\r\n" );
document.write( "|---|---|---|----|----|-----------|------------------------------------------\r\n" );
document.write( "| T | T | T |  F |  F |    F      |        ||                            | \r\n" );
document.write( "| T | T | F |  F |  T |    T      |        ||                            | \r\n" );
document.write( "| T | F | T |  T |  F |    F      |        ||                            | \r\n" );
document.write( "| T | F | F |  T |  T |    T      |        ||                            | \r\n" );
document.write( "| F | T | T |  F |  F |    T      |        ||                            | \r\n" );
document.write( "| F | T | F |  F |  T |    T      |        ||                            | \r\n" );
document.write( "| F | F | T |  T |  F |    T      |        ||                            | \r\n" );
document.write( "| F | F | F |  T |  T |    T      |        ||                            | \r\n" );
document.write( "\r\n" );
document.write( "To fill in the next column, ~q^r, using the rule for ^.\r\n" );
document.write( "Put F except in case of \"T ---> T\", [T on the both sides of ^] and put T in\r\n" );
document.write( "that case. (Notice that only the 3rd and 7th rows have T under both ~q and r.\r\n" );
document.write( "So put T's in those two rows and F's everywhere else:\r\n" );
document.write( "\r\n" );
document.write( "| p | q | r | ~q | ~r | p ---> ~r | ~q ^ r || ( p ---> ~r ) v ( ~q ^ r ) |\r\n" );
document.write( "|---|---|---|----|----|-----------|------------------------------------------\r\n" );
document.write( "| T | T | T |  F |  F |    F      |    F   ||                            | \r\n" );
document.write( "| T | T | F |  F |  T |    T      |    F   ||                            | \r\n" );
document.write( "| T | F | T |  T |  F |    F      |    T   ||                            | \r\n" );
document.write( "| T | F | F |  T |  T |    T      |    F   ||                            | \r\n" );
document.write( "| F | T | T |  F |  F |    T      |    F   ||                            | \r\n" );
document.write( "| F | T | F |  F |  T |    T      |    F   ||                            | \r\n" );
document.write( "| F | F | T |  T |  F |    T      |    T   ||                            | \r\n" );
document.write( "| F | F | F |  T |  T |    T      |    F   ||                            | \r\n" );
document.write( "\r\n" );
document.write( "To fill in the last column, (p--->~r)v(~q^r), using the rule for v.\r\n" );
document.write( "Put T except in case of \"F ---> F\", [F on the both sides of v] and put F in\r\n" );
document.write( "that case. (Notice that only the 1st row has F under both p--->~r and ~qvr.  So\r\n" );
document.write( "put F's on the first row and T's everywhere else:\r\n" );
document.write( "\r\n" );
document.write( "| p | q | r | ~q | ~r | p ---> ~r | ~q ^ r || ( p ---> ~r ) v ( ~q ^ r ) |\r\n" );
document.write( "|---|---|---|----|----|-----------|------------------------------------------\r\n" );
document.write( "| T | T | T |  F |  F |    F      |    F   ||               F            | \r\n" );
document.write( "| T | T | F |  F |  T |    T      |    F   ||               T            | \r\n" );
document.write( "| T | F | T |  T |  F |    F      |    T   ||               T            | \r\n" );
document.write( "| T | F | F |  T |  T |    T      |    F   ||               T            | \r\n" );
document.write( "| F | T | T |  F |  F |    T      |    F   ||               T            | \r\n" );
document.write( "| F | T | F |  F |  T |    T      |    F   ||               T            | \r\n" );
document.write( "| F | F | T |  T |  F |    T      |    T   ||               T            | \r\n" );
document.write( "| F | F | F |  T |  T |    T      |    F   ||               T            | \r\n" );
document.write( "\r\n" );
document.write( "Edwin</PRE> </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );