document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1002316>Question 1002316</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Is {0,2,4,6} closed under multiplication? Or Is {1,3,5,7} closed under multiplication? <BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #619253 by </B><A HREF=/tutors/aboutme.mpl?userid=Theo><B>Theo(13342)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Theo><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=Theo><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=619253\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> the statement is a little bit confusing.\r<BR>\n" );
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document.write( "the definition of closure is that the operation on the set yields a member of the same set.\r<BR>\n" );
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document.write( "based on that definition, the set {0,2,4,6} is not closed under multiplication because 2*4 = 8 and 2*6 = 12 and 4*6 = 24 are not members of that set.\r<BR>\n" );
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document.write( "similarly, the set of {1,3,5,7} is not closed under multiplication because 3*5 = 15 and 3 * 7 = 21 and 5 * 7 = 35 are not members of that set.\r<BR>\n" );
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document.write( "now, if you define {0,2,4,6} as being members of the set of even numbers, then it is closed under multiplication because the results of multiplying an even number by an even number will always yield an even number.\r<BR>\n" );
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document.write( "same with {1,3,5,7} being defined as members of the set of odd numbers.  it is then closed under multiplication because any odd number multiplied by any odd number will will always yield an odd number.\r<BR>\n" );
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document.write( "so, if your sets are defined as being ONLY those elements shown, neither set is closed under multiplication.\r<BR>\n" );
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document.write( "here's some references that may help you.\r<BR>\n" );
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document.write( "<a  rel=nofollow HREF= \"http://mathbitsnotebook.com/Algebra1/RealNumbers/RNClosure.html\" target = \"_blank\">http://mathbitsnotebook.com/Algebra1/RealNumbers/RNClosure.html</a>\r<BR>\n" );
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document.write( "<a  rel=nofollow HREF= \"http://www.regentsprep.org/regents/math/algebra/an1/closure.htm\" target = \"_blank\">http://www.regentsprep.org/regents/math/algebra/an1/closure.htm</a>\r<BR>\n" );
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document.write( "<a  rel=nofollow HREF= \"https://answers.yahoo.com/question/index?qid=20090901203653AALmEAw\" target = \"_blank\">https://answers.yahoo.com/question/index?qid=20090901203653AALmEAw</a>\r<BR>\n" );
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document.write( "<a  rel=nofollow HREF= \"http://www.cwladis.com/math100/Lecture2Groups.htm\" target = \"_blank\">http://www.cwladis.com/math100/Lecture2Groups.htm</a>\r<BR>\n" );
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document.write( "since your problem didn't defined what the sets are closed under (like set of integers or set of even numbers, etc), then you are dealing with only members of those sets shown and therefore both are not closed under multiplication or additon for that matter.\r<BR>\n" );
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document.write( "that's my take.\r<BR>\n" );
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