document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=207968>Question 207968</A>: <I><SPAN STYLE=\"word-break: break-all;\">  I have four problems to use this with if you could help me with one it would be great thanks.An operation $ has been defined over the set of natural numbers. in each case, find 2$3, determine whether or not the set of natural numbers is closed under $, determine if $ is commutative and determine if is associative.\r<BR>\n" );
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document.write( "x$y=(1+x)+y </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #157314 by </B><A HREF=/tutors/aboutme.mpl?userid=stanbon><B>stanbon(75887)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=stanbon><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/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=157314\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> An operation $ has been defined over the set of natural numbers. in each case, find 2$3, determine whether or not the set of natural numbers is closed under $, determine if $ is commutative and determine if is associative.<BR>\n" );
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document.write( "x$y=(1+x)+y<BR>\n" );
document.write( "Then 2$3 = (1+2)+3 = 6<BR>\n" );
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document.write( "Naturals closed under $ ?<BR>\n" );
document.write( "Let a and b be natural numbers<BR>\n" );
document.write( "Then a$b = (1+a)+b = 1+(a+b)<BR>\n" );
document.write( "Since the natural numbers are closed under addition,<BR>\n" );
document.write( "1 + (a+b) is is a natural number. So the set of <BR>\n" );
document.write( "natural numbers is closed under $.<BR>\n" );
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document.write( "Is # communtative?<BR>\n" );
document.write( "Show that a$b = b$a<BR>\n" );
document.write( "(1+a)+b = (1+b)+a<BR>\n" );
document.write( "1 + a + b = 1+ b + a<BR>\n" );
document.write( "Since the naturals are commutative,<BR>\n" );
document.write( "1 + a + b = 1 + a+b<BR>\n" );
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document.write( "Is $ associative?<BR>\n" );
document.write( "Show that (a$b)$c = a$(b$c)<BR>\n" );
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document.write( "Can you do that?<BR>\n" );
document.write( "Cheers,<BR>\n" );
document.write( "Stan H. </SPAN>\n" );
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