document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=622759>Question 622759</A>: <I><SPAN STYLE=\"word-break: break-all;\">  Consider the set Z of all Integers and an integer m > 1. For all integers x and y &#61646; Z, if x – y is divisible by m, then show that this<BR>\n" );
document.write( "defines an equivalence relation on Z. An equivalence relation is reflective, symmetric, and transitive. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #391617 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=391617\">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( "Let <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \mathbb{Z}\"> be the set of all integers and let <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE m\ \in\ \mathbb{Z},\,m\ >\ 1\">, and accept the notation <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \alpha\ \equiv\ \beta\,(\text{mod}\,n)\"> to mean that <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \alpha\ -\ \beta\"> is divisible by <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE n\">.\r<BR>\n" );
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document.write( "Prove that the set <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE R\ =\ \{(x,y)\,|\,x,\,y\,\in\,\mathbb{Z},\,x\ \equiv\ y\,(\text{mod}\,m)\}\"> defines an equivalence relation:\r<BR>\n" );
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document.write( "1.  Since <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE x\ -\ x\ =\ 0\">, <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE m\ \Rightarrow\ x\ \equiv\ x\,(\text{mod}\,m)\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \Rightarrow\ (x,x)\ \in\ R\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \Rightarrow\ R\"> is reflexive.\r<BR>\n" );
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document.write( "2.  Let <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE (x,y)\ \in\ R\ \Rightarrow\ x\ \equiv\ y\,(\text{mod}\,m)\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \Rightarrow\ x\ -\ y\ =\ mq\ \exists\ q\ \in\ \mathbb{Z}\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \Rightarrow\ y\ -\ x\ =\ m(-q)\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \Rightarrow\ y\ \equiv\ x\,(\text{mod}\,m)\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \Rightarrow\ (y,x)\ \in\ R\">\r<BR>\n" );
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document.write( "Thus <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE (x,y)\ \in\ R\ \Rightarrow\ (y,x)\ \in\ R\ \Rightarrow R\ \">is symmetric.\r<BR>\n" );
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document.write( "3. Let <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE (x,y)\ \in\ R\"> and <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE (y,z)\ \in\ R\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \Rightarrow\ x\ \equiv\ y\,(\text{mod}\,m)\ \ \">and <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE\ y\ \equiv\ z\,(\text{mod}\,m)\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \Rightarrow\ x\ -\ y\ =\ mq\ \ \">and<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE\ \ y\ -\ z\ =\ mq'\ \exists\ q,q'\ \in\ \mathbb{Z}\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \Rightarrow\ (x\ -\ y)\ +\ (y\ -\ z)\ =\ m(q\ +\ q')\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \Rightarrow\ (x\ -\ z)\ =\ m(q\ +\ q'),\ q\ +\ q'\ \in\ \mathbb{Z}\">\r<BR>\n" );
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document.write( "Thus <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE (x,y)\ \in\ R\ \text{and}\ (y,z)\ in\ R\ \Rightarrow\ (x,z)\ \in\ R\ \Rightarrow R\ \">is transitive.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE R\ \ \"> is reflexive, symmetric, and transitive <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \Rightarrow\ R\"> is an equivalence relation.\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<BR>\n" );
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