document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=972808>Question 972808</A>: <I><SPAN STYLE=\"word-break: break-all;\"> I'm stuck on the following question: Use truth tables to establish whether the following arguments are valid. If any arguments are not valid, give counterexamples to them. If any arguments are valid, explain carefully why they are valid. \r<BR>\n" );
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document.write( "e.g. P or Q, not P, Q \r<BR>\n" );
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document.write( "or not (P or not Q) P <> Q\r<BR>\n" );
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document.write( "I'm not sure how to provide counterexamples or explain how an argument is valid.  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #595078 by </B><A HREF=/tutors/aboutme.mpl?userid=solver91311><B>solver91311(24713)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=solver91311><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=solver91311><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=595078\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font face=\"Times New Roman\" size=\"+2\">\r<BR>\n" );
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document.write( "I assume that your first example is:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ P\ \large \vee \LARGE\ Q\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \neg P\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \text{Therefore } Q\">\r<BR>\n" );
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document.write( "This is valid by disjunctive syllogism.\r<BR>\n" );
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document.write( "I'm not sure what you mean by your second example.  If by \"<>\" you mean to indicate the biconditional (you should represent this by using \"<->\" or \"iff\"), then the argument is invalid.  Assume both P and Q are true, then the bi-conditional is true, but <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \neg(T\ \large\vee\Large \neg T)\"> is false.  Hence <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large P\ \large\wedge\Large\ Q\"> is a possible counterexample.  If you meant something different than <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large P\ \Leftrightarrow\ Q\"> when you wrote P<>Q, then you are on your own.\r<BR>\n" );
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document.write( "John<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE e^{i\pi}\ +\ 1\ =\ 0\"><BR>\n" );
document.write( "My calculator said it, I believe it, that settles it\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \ \\r\"> </SPAN>\n" );
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