document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1182916>Question 1182916</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A relation ⋆ is defined on z by  x ⋆ y if and only if there exists k ϵ z such that  y=x+5k .<BR>\n" );
document.write( "	Is ⋆ reflexive?<BR>\n" );
document.write( "	Is ⋆ symmetric?<BR>\n" );
document.write( "	Is ⋆ anti-symmetric?<BR>\n" );
document.write( "	Is ⋆ transitive?<BR>\n" );
document.write( "	Is ⋆ an equivalence relation, a partial order, both, or neither?\r<BR>\n" );
document.write( "\n" );
document.write( "Thanks in advance!  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #813112 by </B><A HREF=/tutors/aboutme.mpl?userid=ikleyn><B>ikleyn(52799)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=ikleyn><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=ikleyn><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=813112\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> .<BR>\n" );
document.write( "A relation ⋆ is defined on z by x ⋆ y if and only if there exists k ϵ z such that y=x+5k .<BR>\n" );
document.write( "Is ⋆ reflexive?<BR>\n" );
document.write( "Is ⋆ symmetric?<BR>\n" );
document.write( "Is ⋆ anti-symmetric?<BR>\n" );
document.write( "Is ⋆ transitive?<BR>\n" );
document.write( "Is ⋆ an equivalence relation, a partial order, both, or neither?<BR>\n" );
document.write( "Thanks in advance!<BR>\n" );
document.write( "~~~~~~~~~~~~~~~~~~~~~~~\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<PRE STYLE=\"white-space: pre-wrap; white-space: -moz-pre-wrap; white-space: -pre-wrap; white-space: -o-pre-wrap; word-wrap: break-word;\">\r\n" );
document.write( "As defined in the post, two integer numbers x and y are in relation  \" ⋆ \"  if and only if their difference  x - y is a multiple of 5.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Therefore, if you are familiar with the definition of the terms, this relation is\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "    - reflexive :  (x⋆x)  is TRUE for any integer  number x, since  x-x = 0 is a multiple of 5;\r\n" );
document.write( "\r\n" );
document.write( "    - symmetric :  (x⋆y) implies (y⋆x),  since if x-y is a multiple of 5, it implies that y-x is a multiple of 5;\r\n" );
document.write( "\r\n" );
document.write( "    - tranzitive : (x⋆y) and (y⋆z) implies (x⋆z), since if x-y is multiple of 5 and y-z is a multiple of 5,\r\n" );
document.write( "\r\n" );
document.write( "                                                  then  x-z is a multiple of 5, too.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Finally, since the relation  \" ⋆ \" is reflexive, symmetric and transitive (as we proved it above), it is equivalence relation, by the definition.\r\n" );
document.write( "</PRE>\r<BR>\n" );
document.write( "\n" );
document.write( "Solved.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );