document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1083625>Question 1083625</A>: <I><SPAN STYLE=\"word-break: break-all;\"> P &#8729; (Q &#8835; R) ; (P &#8729; Q) &#8835; (P &#8729; R)<BR>\n" );
document.write( "are logically equivalent to each other, or whether they are contradictory to each other by making a truth table for them.  If they are neither of those, determine whether they are consistent with each other, or whether they are inconsistent with each other </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #697656 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=697656\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=\"black\" face=\"times\" size=\"4\">T = True<BR>\n" );
document.write( "F = False\r<BR>\n" );
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document.write( "I'm using the ampersand in place of the dot (to make it more visible).<BR>\n" );
document.write( "Also I'm using the symbol \">\" without quotes in place of the horsehoe. \r<BR>\n" );
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document.write( "Truth Table for <font color=blue>P & (Q > R)</font>\r<BR>\n" );
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document.write( "<tr><th>P</th><th>Q</th><th>R</th><th>Q &gt; R</th><th>P &amp; (Q &gt; R)</th></tr><tr><TD >T</TD><TD >T</TD><TD >T</TD><TD >T</TD><TD >T</TD></tr><tr><TD >T</TD><TD >T</TD><TD >F</TD><TD >F</TD><TD >F</TD></tr><tr><TD >T</TD><TD >F</TD><TD >T</TD><TD >T</TD><TD >T</TD></tr><tr><TD >T</TD><TD >F</TD><TD >F</TD><TD >T</TD><TD >T</TD></tr><tr><TD >F</TD><TD >T</TD><TD >T</TD><TD >T</TD><TD >F</TD></tr><tr><TD >F</TD><TD >T</TD><TD >F</TD><TD >F</TD><TD >F</TD></tr><tr><TD >F</TD><TD >F</TD><TD >T</TD><TD >T</TD><TD >F</TD></tr><tr><TD >F</TD><TD >F</TD><TD >F</TD><TD >T</TD><TD >F</TD></tr>\n" );
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document.write( "Truth Table for <font color=blue>(P & Q) > (Q & R)</font>\r<BR>\n" );
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document.write( "<tr><th>P</th><th>Q</th><th>R</th><th>P &amp; Q</th><th>P &amp; R</th><th>(P &amp; Q) &gt; (P &amp; R)</th></tr><tr><TD >T</TD><TD >T</TD><TD >T</TD><TD >T</TD><TD >T</TD><TD >T</TD></tr><tr><TD >T</TD><TD >T</TD><TD >F</TD><TD >T</TD><TD >F</TD><TD >F</TD></tr><tr><TD >T</TD><TD >F</TD><TD >T</TD><TD >F</TD><TD >T</TD><TD >T</TD></tr><tr><TD >T</TD><TD >F</TD><TD >F</TD><TD >F</TD><TD >F</TD><TD >T</TD></tr><tr><TD >F</TD><TD >T</TD><TD >T</TD><TD >F</TD><TD >F</TD><TD >T</TD></tr><tr><TD >F</TD><TD >T</TD><TD >F</TD><TD >F</TD><TD >F</TD><TD >T</TD></tr><tr><TD >F</TD><TD >F</TD><TD >T</TD><TD >F</TD><TD >F</TD><TD >T</TD></tr><tr><TD >F</TD><TD >F</TD><TD >F</TD><TD >F</TD><TD >F</TD><TD >T</TD></tr>\n" );
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document.write( "Carefully compare (row by row) the last columns of each table. Here's a side by side comparison of the last two columns of each table.\r<BR>\n" );
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document.write( "<tr><th>P &amp; (Q &gt; R)</th><th>(P &amp; Q) &gt; (P &amp; R)</th></tr><tr><TD >T</TD><TD >T</TD></tr><tr><TD >F</TD><TD >T</TD></tr><tr><TD >T</TD><TD >T</TD></tr><tr><TD >T</TD><TD >T</TD></tr><tr><TD >T</TD><TD >T</TD></tr><tr><TD >F</TD><TD >T</TD></tr><tr><TD >T</TD><TD >T</TD></tr><tr><TD >T</TD><TD >T</TD></tr>\n" );
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document.write( "as you can see the columns do not match up. Therefore, the two logical expressions aren't equivalent. \r<BR>\n" );
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document.write( "They aren't contradictory to one another because sometimes they do match up (like in row 1 we have two T's) but other times they don't (row 2 with F and then T). For a contradiction, the values need to be opposite one another (one must be true and the other false or vice versa) and this needs to apply to every row.\r<BR>\n" );
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document.write( "The two expressions are consistent because it is possible for them both to be true under the same truth values (for P,Q,R). One example of such is in row 1 of the third table shown above. Not all rows need to have T's in them. All we need is one row with nothing but T's.\r<BR>\n" );
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document.write( "For further reading, check out this page<BR>\n" );
document.write( "<a  rel=nofollow HREF=\"http://www.butte.edu/resources/interim/wmwu/iLogic/3.2/iLogic_3_2.html\">http://www.butte.edu/resources/interim/wmwu/iLogic/3.2/iLogic_3_2.html</a><BR>\n" );
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