document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=993909>Question 993909</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Determine whether the relation on the set of all real numbers is reflexive, symmetric ,antisymmetric , and transitive when (x ,y) €R if and only if<BR>\n" );
document.write( "1) x + y= 0<BR>\n" );
document.write( "2) x = +,- y<BR>\n" );
document.write( "3)x -y is a rational number<BR>\n" );
document.write( "4)x =2y<BR>\n" );
document.write( "5)xy is greater than equal to 0<BR>\n" );
document.write( "6)xy= 0<BR>\n" );
document.write( "7)x =1<BR>\n" );
document.write( "8)x =1 or y =1 <BR>\n" );
document.write( "Please explain the answer </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #613327 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=613327\">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( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ R\ =\ \{x, y\ \in \mathbb{R}\ |\ x\ +\ y\ =\ 0\}\">\r<BR>\n" );
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document.write( "Which is the same thing as\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \{x, y\ \in \mathbb{R}\ |\ x\ =\ a,\ y\ =\ -a\}\">\r<BR>\n" );
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document.write( "So if <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large <a, b>\ \in\ R\"> then <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large b\ =\ -a\">, and therefore <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large <a, a>\ \not\in R\"> unless <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large a\ =\ 0\">.\r<BR>\n" );
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document.write( "Therefore <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large R\"> is neither reflexive or irreflexive.\r<BR>\n" );
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document.write( "But since <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large a\"> can be negative, <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large <a, -a> \in\ R\ \right\ <-a,a>\ \in\ R\">\r<BR>\n" );
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document.write( "Therefore, R is symmetric.\r<BR>\n" );
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document.write( "Since <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large a\ \not=\ -a\"> except when <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large a\ =\ 0\">, <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large R\"> is not asymmetric.\r<BR>\n" );
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document.write( "Since if <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large <a, b> \in\ R\"> then <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large b\ =\ -a\">, so if <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large <b, c> \in\ R\"> then <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large c\ =\ a\">.\r<BR>\n" );
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document.write( "Therefore R is transitive.\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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