document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=467766>Question 467766</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Given the following sets, select the statement below that is true.<BR>\n" );
document.write( "A= {l,a,t,e,r}  B= {l,a,t,e} C= {t,a,l,e}  D= {e,a,t}<BR>\n" );
document.write( " B is a proper subset of C and C is a subset of A<BR>\n" );
document.write( " C is a subset of B and D is aproper subset of B<BR>\n" );
document.write( " D is a proper subset of A and A is a proper subset of D <BR>\n" );
document.write( " B is a proper subset of A and C is a proper subset of D<BR>\n" );
document.write( " D is a subset of A and A is a proper subset of C </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #321358 by </B><A HREF=/tutors/aboutme.mpl?userid=moshiz08><B>moshiz08(60)</B></A><img src=\"/images/stars/stars-08.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=moshiz08><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=moshiz08><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=321358\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Recall that two sets are equal if they have exactly the same elements. In your example, B and C are equal because the elements are the same letters a,e,l,t. <BR>\n" );
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document.write( "We say that X is a subset of Y if everything that is in X is also in Y. We say that X is a proper subset of Y if X is a subset of Y which is not equal to Y. <BR>\n" );
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document.write( "Note that if X equals Y, then X is a subset of Y because everything in X must be in Y. Similarly Y is a subset of X because everything in Y is also in X. In other words, X = Y if and only if X is a subset of Y and Y is a subset of X. <BR>\n" );
document.write( "<br>With these definitions in mind, let's go through each answer choice. <BR>\n" );
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document.write( " B is a proper subset of C and C is a subset of A<BR>\n" );
document.write( "-FALSE. It is true that C is a subset of A. But B is equal to C, so it is not a PROPER subset. \r<BR>\n" );
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document.write( " C is a subset of B and D is aproper subset of B<BR>\n" );
document.write( "-TRUE. C is a subset of B because C=B, so everything in C is also in B. D is a proper subset of B because everything in D is also in B, but they are not equal. \r<BR>\n" );
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document.write( " D is a proper subset of A and A is a proper subset of D <BR>\n" );
document.write( "-FALSE. D is a proper subset of A, but A is NOT a proper subset of D. (In fact, A is not even a subset of D because it has elements l and r which are not in D. In general if X is a proper subset of Y, then Y cannot be a proper subset of X. Can you see why?) \r<BR>\n" );
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document.write( " B is a proper subset of A and C is a proper subset of D<BR>\n" );
document.write( "-FALSE. B is a proper subset of A, but C is NOT a proper subset of D because there is an element l in C which is not in D. \r<BR>\n" );
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document.write( " D is a subset of A and A is a proper subset of C<BR>\n" );
document.write( "-FALSE. D is a subset of A, but A is NOT a proper subset of C. As in the third choice, A is not even a subset of C because there is an element r in A which is not in C.  </SPAN>\n" );
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