document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=716179>Question 716179</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Hi i'm stuck on the following question:\r<BR>\n" );
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document.write( "\"For the set of positive rational numbers, show that division as the operation of combination is not associative. For this set is there any operation that is associative?\" </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #439769 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=439769\">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( "Given a set of positive integers <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \mathbb{Z}_\alpha\ =\ \{p,\, q,\, r,\, s,\, u,\, v\}\">, consider the combination:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \left(\frac{p}{q}\ \div\ \frac{r}{s}\right)\ \div\ \frac{u}{v}\ =\ \frac{ps}{qr}\ \div\ \frac{u}{v}\ =\ \frac{psv}{qru}\">\r<BR>\n" );
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document.write( "and consider:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \frac{p}{q}\ \div\ \left(\frac{r}{s}\ \div\ \frac{u}{v}\right)\ =\ \frac{p}{q}\ \div\ \frac{rv}{su}\ =\ \frac{psu}{qrv}\ \not\equiv\ \frac{psv}{qru}\">\r<BR>\n" );
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document.write( "In general, subtraction and division are neither associative nor commutative over the Real numbers.  Addition and Multiplication are both associative and commutative over all Real numbers.  The set of positive rational numbers is a subset of the Reals.\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( "<font face=\"Math1\" size=\"+2\">Egw to Beta kai to Sigma</font><BR>\n" );
document.write( "My calculator said it, I believe it, that settles it<BR>\n" );
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