document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1199080>Question 1199080</A>: <I><SPAN STYLE=\"word-break: break-all;\"> <BR>\n" );
document.write( "Use ordinary truth tables to answer the following problem. Construct the truth tables as per the instructions in the textbook.\r<BR>\n" );
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document.write( "Given the statement: [K  •  (P  v ~ R))] • [K  > (R  •  ~ P)] \r<BR>\n" );
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document.write( "The statement is:<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 #833029 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=833029\">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" );
document.write( "Learn the four rules ~, v, •, >\r\n" );
document.write( "1. ~ means the opposite of what follows the ~\r\n" );
document.write( "2. FvF is the only false case of v, all others T.\r\n" );
document.write( "3. T•T is the only true case of •, all others F.\r\n" );
document.write( "4. T>F is the only false case of >, all others T.\r\n" );
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document.write( "Write the expression across the paper.\r\n" );
document.write( "Under the K's put TTTTFFFF\r\n" );
document.write( "Under the P's put TTFFTTFF\r\n" );
document.write( "Under the R's put TFTFTFTF\r\n" );
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document.write( " [K • (P v ~ R)] • [K > (R • ~ P)] \r\n" );
document.write( "  T    T     T      T    T     T\r\n" );
document.write( "  T    T     F      T    F     T\r\n" );
document.write( "  T    F     T      T    T     F\r\n" );
document.write( "  T    F     F      T    F     F\r\n" );
document.write( "  F    T     T      F    T     T\r\n" );
document.write( "  F    T     F      F    F     T\r\n" );
document.write( "  F    F     T      F    T     F\r\n" );
document.write( "  F    F     F      F    F     F\r\n" );
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document.write( "Under the ~'s, put F if ~ is before a T\r\n" );
document.write( "and F if ~ is before a T\r\n" );
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document.write( " [K • (P v ~ R)] • [K > (R • ~ P)] \r\n" );
document.write( "  T    T   F T      T    T   F T\r\n" );
document.write( "  T    T   T F      T    F   F T\r\n" );
document.write( "  T    F   F T      T    T   T F\r\n" );
document.write( "  T    F   T F      T    F   T F\r\n" );
document.write( "  F    T   F T      F    T   F T\r\n" );
document.write( "  F    T   T F      F    F   F T\r\n" );
document.write( "  F    F   F T      F    T   T F\r\n" );
document.write( "  F    F   T F      F    F   T F\r\n" );
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document.write( "Erase the T's and F's in the columns you just use\r\n" );
document.write( "to get the last columns you put in.\r\n" );
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document.write( " [K • (P v ~ R)] • [K > (R • ~ P)] \r\n" );
document.write( "  T    T   F        T    T   F \r\n" );
document.write( "  T    T   T        T    F   F \r\n" );
document.write( "  T    F   F        T    T   T \r\n" );
document.write( "  T    F   T        T    F   T \r\n" );
document.write( "  F    T   F        F    T   F \r\n" );
document.write( "  F    T   T        F    F   F \r\n" );
document.write( "  F    F   F        F    T   T \r\n" );
document.write( "  F    F   T        F    F   T \r\n" );
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document.write( "Staying within the first innermost parentheses (),\r\n" );
document.write( "under the v, put T everywhere except where v is\r\n" );
document.write( "between two F's. This is the only time we put F. \r\n" );
document.write( "[You will notice that this his rule is the exact \r\n" );
document.write( "opposite of what we will put under •].\r\n" );
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document.write( " [K • (P v ~ R)] • [K > (R • ~ P)] \r\n" );
document.write( "  T    T T F        T    T   F \r\n" );
document.write( "  T    T T T        T    F   F \r\n" );
document.write( "  T    F F F        T    T   T \r\n" );
document.write( "  T    F T T        T    F   T \r\n" );
document.write( "  F    T T F        F    T   F \r\n" );
document.write( "  F    T T T        F    F   F \r\n" );
document.write( "  F    F F F        F    T   T \r\n" );
document.write( "  F    F T T        F    F   T  \r\n" );
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document.write( "Erase the two columns of T's and F's that we used\r\n" );
document.write( "to get the last column we made:\r\n" );
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document.write( " [K • (P v ~ R)] • [K > (R • ~ P)] \r\n" );
document.write( "  T      T          T    T   F \r\n" );
document.write( "  T      T          T    F   F \r\n" );
document.write( "  T      F          T    T   T \r\n" );
document.write( "  T      T          T    F   T \r\n" );
document.write( "  F      T          F    T   F \r\n" );
document.write( "  F      T          F    F   F \r\n" );
document.write( "  F      F          F    T   T \r\n" );
document.write( "  F      T          F    F   T \r\n" );
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document.write( "Staying within the other innermost parentheses, ()\r\n" );
document.write( "under the •, put F everywhere except where • is\r\n" );
document.write( "between two T's. This is the only time we put T. \r\n" );
document.write( "[This rule is the exact opposite of what we put \r\n" );
document.write( "under v].\r\n" );
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document.write( " [K • (P v ~ R)] • [K > (R • ~ P)] \r\n" );
document.write( "  T      T          T    T F F \r\n" );
document.write( "  T      T          T    F F F \r\n" );
document.write( "  T      F          T    T T T \r\n" );
document.write( "  T      T          T    F F T \r\n" );
document.write( "  F      T          F    T F F \r\n" );
document.write( "  F      T          F    F F F \r\n" );
document.write( "  F      F          F    T T T \r\n" );
document.write( "  F      T          F    F F T \r\n" );
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document.write( "Erase the two columns of T's and F's that we used\r\n" );
document.write( "to get the last column we made:\r\n" );
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document.write( " [K • (P v ~ R)] • [K > (R • ~ P)] \r\n" );
document.write( "  T      T          T      F  \r\n" );
document.write( "  T      T          T      F  \r\n" );
document.write( "  T      F          T      T  \r\n" );
document.write( "  T      T          T      F  \r\n" );
document.write( "  F      T          F      F  \r\n" );
document.write( "  F      T          F      F  \r\n" );
document.write( "  F      F          F      T  \r\n" );
document.write( "  F      T          F      F  \r\n" );
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document.write( "Staying within the first innermost brackets, [],\r\n" );
document.write( "under the •, as before, put F everywhere except \r\n" );
document.write( "where • is between two T's. This is the only \r\n" );
document.write( "time we put T. \r\n" );
document.write( "\r\n" );
document.write( " [K • (P v ~ R)] • [K > (R • ~ P)] \r\n" );
document.write( "  T T    T          T      F  \r\n" );
document.write( "  T T    T          T      F  \r\n" );
document.write( "  T F    F          T      T  \r\n" );
document.write( "  T T    T          T      F  \r\n" );
document.write( "  F F    T          F      F  \r\n" );
document.write( "  F F    T          F      F  \r\n" );
document.write( "  F F    F          F      T  \r\n" );
document.write( "  F F    T          F      F \r\n" );
document.write( "\r\n" );
document.write( "Erase the two columns of T's and F's that we used\r\n" );
document.write( "to get the last column we made:\r\n" );
document.write( "\r\n" );
document.write( " [K • (P v ~ R)] • [K > (R • ~ P)] \r\n" );
document.write( "    T               T      F  \r\n" );
document.write( "    T               T      F  \r\n" );
document.write( "    F               T      T  \r\n" );
document.write( "    T               T      F  \r\n" );
document.write( "    F               F      F  \r\n" );
document.write( "    F               F      F  \r\n" );
document.write( "    F               F      T  \r\n" );
document.write( "    F               F      F\r\n" );
document.write( "\r\n" );
document.write( "Staying within the other innermost brackets, [],\r\n" );
document.write( "under the >, put T everywhere except where > has \r\n" );
document.write( "T on the left and F on the right. This is the \r\n" );
document.write( "only time we put T. \r\n" );
document.write( "\r\n" );
document.write( " [K • (P v ~ R)] • [K > (R • ~ P)] \r\n" );
document.write( "    T               T F    F  \r\n" );
document.write( "    T               T F    F  \r\n" );
document.write( "    F               T T    T  \r\n" );
document.write( "    T               T F    F  \r\n" );
document.write( "    F               F T    F  \r\n" );
document.write( "    F               F T    F  \r\n" );
document.write( "    F               F T    T  \r\n" );
document.write( "    F               F T    F \r\n" );
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document.write( "Erase the two columns of T's and F's that we used\r\n" );
document.write( "to get the last column we made:\r\n" );
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document.write( " [K • (P v ~ R)] • [K > (R • ~ P)] \r\n" );
document.write( "    T                 F      \r\n" );
document.write( "    T                 F      \r\n" );
document.write( "    F                 T      \r\n" );
document.write( "    T                 F      \r\n" );
document.write( "    F                 T      \r\n" );
document.write( "    F                 T      \r\n" );
document.write( "    F                 T      \r\n" );
document.write( "    F                 T     \r\n" );
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document.write( "Now that we have finished all the parentheses\r\n" );
document.write( "and brackets, we are dow to just 2 columns of T's \r\n" );
document.write( "and F's. Under the final • outside all parentheses\r\n" );
document.write( "and brackets we put F's for everything but T•T.\r\n" );
document.write( "There are none, so we put F's for everything.\r\n" );
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document.write( " [K • (P v ~ R)] • [K > (R • ~ P)] \r\n" );
document.write( "    T            F    F      \r\n" );
document.write( "    T            F    F      \r\n" );
document.write( "    F            F    T      \r\n" );
document.write( "    T            F    F      \r\n" );
document.write( "    F            F    T      \r\n" );
document.write( "    F            F    T      \r\n" );
document.write( "    F            F    T      \r\n" );
document.write( "    F            F    T\r\n" );
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document.write( "We erase the two columns of T's and F's that we used\r\n" );
document.write( "to get the last column we made:\r\n" );
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document.write( " [K • (P v ~ R)] • [K > (R • ~ P)] \r\n" );
document.write( "                 F          \r\n" );
document.write( "                 F          \r\n" );
document.write( "                 F          \r\n" );
document.write( "                 F          \r\n" );
document.write( "                 F          \r\n" );
document.write( "                 F          \r\n" );
document.write( "                 F          \r\n" );
document.write( "                 F    \r\n" );
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document.write( "Since all the values are F, the statement is a \r\n" );
document.write( "logical contradiction, which means it is always\r\n" );
document.write( "false.\r\n" );
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document.write( "[In other problems, when all the values are T, the \r\n" );
document.write( "statement is a logical tautology, or always true.]\r\n" );
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document.write( "[In other problems when some are T and some are F,\r\n" );
document.write( "the statement is a contingency, or sometimes true\r\n" );
document.write( "and sometimes false.]\r\n" );
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document.write( "Edwin</PRE><BR>\n" );
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