document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1201401>Question 1201401</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Which of the following are true for all integers a,b, and c?<BR>\n" );
document.write( "I.(a+b)+c=a+(b+c)<BR>\n" );
document.write( "II.(a/b)/c=a/(b/c)<BR>\n" );
document.write( "III.(a+b)/c=a+(b/c) </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #835751 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=835751\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "<font color=red>Statement I is true</font> for all integers a,b,c<BR>\n" );
document.write( "It's also true for any real numbers a,b,c<BR>\n" );
document.write( "For more information, search out \"associative law of addition\".<BR>\n" );
document.write( "The basic idea is that we can add numbers in any grouping order we want. The order doesn't matter.<BR>\n" );
document.write( "Examples:<BR>\n" );
document.write( "(1+2)+3 = 1+(2+3) since both sides evaluate to 6<BR>\n" );
document.write( "(4+7)+2 = 4+(7+2) since both sides evaluate to 13<BR>\n" );
document.write( "An almost identical property would be the commutative property of addition. That property is a+b = b+a.\r<BR>\n" );
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document.write( "Statement II is NOT true for all integers a,b,c<BR>\n" );
document.write( "We can construct a counterexample to see why.<BR>\n" );
document.write( "a = 1<BR>\n" );
document.write( "b = 2<BR>\n" );
document.write( "c = 4<BR>\n" );
document.write( "LHS = left hand side = (a/b)/c = (1/2)/4 = 0.5/4 = 0.125 exactly<BR>\n" );
document.write( "RHS = right hand side = a/(b/c) = 1/(2/4) = 1/(0.5) = 2<BR>\n" );
document.write( "LHS = RHS is not the case since 0.125 = 2 is false\r<BR>\n" );
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document.write( "Statement III is also not true in general<BR>\n" );
document.write( "Counterexample:<BR>\n" );
document.write( "a = 1<BR>\n" );
document.write( "b = 2<BR>\n" );
document.write( "c = 3<BR>\n" );
document.write( "LHS = (a+b)/c = (1+2)/3 = 3/3 = 1<BR>\n" );
document.write( "RHS = a + (b/c) = 1 + (2/3) = 1.667 approximately<BR>\n" );
document.write( "LHS = RHS is false since 1 = 1.667 is false.\r<BR>\n" );
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document.write( "Conclusion: <font color=red size=4>Statement I</font> is the only true statement of the trio.<BR>\n" );
document.write( "</font> </SPAN>\n" );
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