document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=162438>Question 162438</A>This question is from textbook <A HREF=http://www.amazon.com/exec/obidos/ASIN/0915144506/algebrahomeworkh></A><BR>\n" );
document.write( ": <I><SPAN STYLE=\"word-break: break-all;\"> problem 1f:\r<BR>\n" );
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document.write( "-(PvQ) -||- -P&-Q </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #119708 by </B><A HREF=/tutors/aboutme.mpl?userid=vleith><B>vleith(2983)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=vleith><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=119708\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> One way to do all these logic type problems is to make a table. Make two rows for each variable (in this case 4 rows, one for P and one for Q). Then make a column for each logical step you take. The spacing will be messed up, but you can line them up by taking out all the fillers (+).<BR>\n" );
document.write( "Start by adding both true (1) and false (0) for each variable. Do this until you have each combination for all variables. Since we have two variables, that makes 4 total combinations (2 * 2). See the table below\r<BR>\n" );
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document.write( "+P++Q+++(-P)+++(-Q)++(PvQ)+++(-P&-Q)+++-(PvQ)<BR>\n" );
document.write( "+0++0<BR>\n" );
document.write( "+1++0<BR>\n" );
document.write( "+0++1<BR>\n" );
document.write( "+1++1\r<BR>\n" );
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document.write( "Now, using the info about P and Q, you can easily find (-P) and (-Q). Fill them in as follows.\r<BR>\n" );
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document.write( "+P++Q+++(-P)+++(-Q)++(PvQ)+++(-P&-Q)+++-(PvQ)<BR>\n" );
document.write( "+0++0+++++1++++++1+<BR>\n" );
document.write( "+1++0+++++0++++++1+<BR>\n" );
document.write( "+0++1+++++1++++++0+<BR>\n" );
document.write( "+1++1+++++0++++++0+\r<BR>\n" );
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document.write( "Next, using logical properties for AND, OR, etc, continue completing the table Fill them in as follows.\r<BR>\n" );
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document.write( "+P++Q+++(-P)+++(-Q)++(PvQ)+++(-P&-Q)+++-(PvQ)<BR>\n" );
document.write( "+0++0+++++1++++++1+++++0++++++++1<BR>\n" );
document.write( "+1++0+++++0++++++1+++++1++++++++0<BR>\n" );
document.write( "+0++1+++++1++++++0+++++1++++++++0<BR>\n" );
document.write( "+1++1+++++0++++++0+++++1++++++++0\r<BR>\n" );
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document.write( "Finally, using the logical negative, complete the table. Then note that the columns for the two entities in the original problem are identical \r<BR>\n" );
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document.write( "+P++Q+++(-P)+++(-Q)++(PvQ)+++(-P&-Q)+++-(PvQ)<BR>\n" );
document.write( "+0++0+++++1++++++1+++++0++++++++1+++++++++1<BR>\n" );
document.write( "+1++0+++++0++++++1+++++1++++++++0+++++++++0<BR>\n" );
document.write( "+0++1+++++1++++++0+++++1++++++++0+++++++++0<BR>\n" );
document.write( "+1++1+++++0++++++0+++++1++++++++0+++++++++0\r<BR>\n" );
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document.write( "So what you need to know are:<BR>\n" );
document.write( "The rules for ANA, OR, negation, implication, etc. Your book will have all that stuff. Then you need to be methodical about starting with a truth table and gradually working your way through each logical step. Once you get two columns that have the same entries, you have proven those columns are logical equivalents.\r<BR>\n" );
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document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );