document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=169812>Question 169812</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Of all theses sets which ones are proper subsets {}- {a} - {b} - {c} - {d} - {a,b} - {a,c} - {a,d} - {b,c} - {b,d} - {c,d} - {a,b,c} - {a,b,d} - {a,c,d} - {b,c,d} - {a,b,c,d} I know that the empty set is a proper subset and I think that the next 4 after that are as well but I am not sure about the other ones, any help would be appreciated. <BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #125206 by </B><A HREF=/tutors/aboutme.mpl?userid=solver91311><B>solver91311(24713)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=solver91311><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=solver91311><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=125206\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Proper subsets of what?  You did not specify the original set, so the question as stated cannot be answered.  However...\r<BR>\n" );
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document.write( "A proper subset must be strictly contained, therefore must exclude at least one element of the original set.\r<BR>\n" );
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document.write( "The empty set is a proper subset of any non-empty set.\r<BR>\n" );
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document.write( "So:\r<BR>\n" );
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document.write( "If the original set was the set of all lower case letters of the English alphabet, then all of the listed subsets would be proper.\r<BR>\n" );
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document.write( "If the original set was the set {a, b, c, d} then all of the listed subsets with the exeception of the very last one would be proper subsets.\r<BR>\n" );
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document.write( "If the original set was the set {a, b, d}, then any of the listed sets containing the element 'c' or any of the listed sets with 3 or more elements would NOT be proper subsets, while all of the rest would be.\r<BR>\n" );
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document.write( "Hope that helps.  And, in the future, give us the WHOLE problem to work with.  Thanks. \r<BR>\n" );
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