document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1062817>Question 1062817</A>: <I><SPAN STYLE=\"word-break: break-all;\"> I apologise beforehand if this question does not fall under the appropriate topics. \r<BR>\n" );
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document.write( "Let C = {5r + 2 | r is an element of Z}, and<BR>\n" );
document.write( "D = {10s - 3 | s is an element of Z}.\r<BR>\n" );
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document.write( "Is C a subset of D or D is a subset of C? Explain your answer.\r<BR>\n" );
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document.write( "Thank you for noticing and answering this question. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #677815 by </B><A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><B>jim_thompson5910(35256)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><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=677815\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let's replace r with something you'd find in the set Z. The set Z stands for the set of integers. An easy integer to deal with is r = 0\r<BR>\n" );
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document.write( "5r+2 = 5*0+2 = 2\r<BR>\n" );
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document.write( "So 2 is in the set C. If C was a subset of D, then 2 must also be in set D for it to work. Let's find out. Set 10s-3 equal to 2 and solve for s\r<BR>\n" );
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document.write( "10s-3 = 2<BR>\n" );
document.write( "10s = 5<BR>\n" );
document.write( "s = 5/10<BR>\n" );
document.write( "s = 1/2<BR>\n" );
document.write( "s = 0.5 <--- this is NOT an integer\r<BR>\n" );
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document.write( "since s is NOT an integer, this means that 2 is NOT in set D. So set C is definitely not a subset of D\r<BR>\n" );
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document.write( "Let's repeat the steps above, but now start with set D. Plug in s = 0\r<BR>\n" );
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document.write( "10s-3 = 10*0 - 3 = -3\r<BR>\n" );
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document.write( "So -3 is definitely in set D. Is it in set C? Set -3 equal to 5r+2 and solve for r to find out\r<BR>\n" );
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document.write( "5r+2 = -3<BR>\n" );
document.write( "5r = -5<BR>\n" );
document.write( "r = -1\r<BR>\n" );
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document.write( "If r = -1, then it leads to -3. This tells us that -3 is definitely in set C\r<BR>\n" );
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document.write( "We can repeat the last section with other values of s to find corresponding values of r.\r<BR>\n" );
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document.write( "It turns out that with a bit of trial and error, we can manipulate the form for set D to match what we see in set C, like so...\r<BR>\n" );
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document.write( "10s - 3<BR>\n" );
document.write( "2*5s - 3<BR>\n" );
document.write( "2*5s + 2 - 5<BR>\n" );
document.write( "2*5s - 5 + 2<BR>\n" );
document.write( "5*2s - 5 + 2<BR>\n" );
document.write( "5*(2s - 1) + 2\r<BR>\n" );
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document.write( "The (2s-1) portion is an integer so we basically have this format 5*(integer) + 2 which is what set C is saying.\r<BR>\n" );
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document.write( "Therefore, any number in set D is also in set C.\r<BR>\n" );
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document.write( "This means set D is a subset of set C. </SPAN>\n" );
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