document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1203441>Question 1203441</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Which of the ff are associative binary operations? Show your solutions? <BR>\n" );
document.write( "i. (ℤ,*), where x*y=(x+y)-(x·y) for all x,y,∈ℤ. <BR>\n" );
document.write( "ii. (ℝ,*), where x*y=max (x,y) for all x,y,∈ℝ. <BR>\n" );
document.write( "iii. (ℝ,*), where x*y= |x+y|  for all x,y,∈ℝ. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #839007 by </B><A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><B>math_tutor2020(3817)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><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=839007\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "I'll work on part (i) only, and leave parts (ii) and (iii) for the student to do.\r<BR>\n" );
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document.write( "If the operator star was associative, then a*(b*c) = (a*b)*c must be true. \r<BR>\n" );
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document.write( "Let's do a bit of scratch work<BR>\n" );
document.write( "x*y=(x+y)-(x·y)<BR>\n" );
document.write( "a*b=(a+b)-(a·b)<BR>\n" );
document.write( "b*c=(b+c)-(b·c)\r<BR>\n" );
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document.write( "Then we can say:<BR>\n" );
document.write( "a*(b*c)=a*((b+c)-(b·c))<BR>\n" );
document.write( "a*(b*c)=(a+(b+c)-bc)-a((b+c)-(b·c))<BR>\n" );
document.write( "a*(b*c)=(a+(b+c)-bc)+(-a(b+c)+abc)<BR>\n" );
document.write( "a*(b*c)=(a+b+c-bc)+(-ab-ac+abc)<BR>\n" );
document.write( "a*(b*c)=(a+b+c)+(-bc-ab-ac+abc)<BR>\n" );
document.write( "Be careful not to mix up the star operator with the multiplication symbol.\r<BR>\n" );
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document.write( "And furthermore,<BR>\n" );
document.write( "(a*b)*c = ((a+b)-(a·b))*c<BR>\n" );
document.write( "(a*b)*c = ((a+b)-(a·b)+c)  - ((a+b)-(a·b))*c <BR>\n" );
document.write( "(a*b)*c = ((a+b)-(a·b)+c)  - (ac+bc)+(abc)<BR>\n" );
document.write( "(a*b)*c = ((a+b)-(a·b)+c)  + (-ac-bc)+(abc)<BR>\n" );
document.write( "(a*b)*c = (a+b+c-ab) + (-ac-bc+abc)<BR>\n" );
document.write( "(a*b)*c = (a+b+c) + (-ab-ac-bc+abc)\r<BR>\n" );
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document.write( "In summary we found these equations<BR>\n" );
document.write( "a*(b*c)=(a+b+c)+(-bc-ab-ac+abc)<BR>\n" );
document.write( "(a*b)*c = (a+b+c) + (-ab-ac-bc+abc)\r<BR>\n" );
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document.write( "Compare the right hand sides of a*(b*c) and (a*b)*c <BR>\n" );
document.write( "Both have the same exact terms. Compare the terms carefully.\r<BR>\n" );
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document.write( "Therefore, we have proven that a*(b*c) = (a*b)*c<BR>\n" );
document.write( "This shows the operator star is associative when we have defined the operator star to be x*y = (x+y) - xy<BR>\n" );
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