document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1195639>Question 1195639</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Let S={1,2,3,4,5,6,7,8,9,10} be the universal set.\r<BR>\n" );
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document.write( "Let sets A, B, and C be subsets of S , where:\r<BR>\n" );
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document.write( "Set A={1,6,7,10}\r<BR>\n" );
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document.write( "Set B={2,9,10}\r<BR>\n" );
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document.write( "Set C={3,4,5,7,8}\r<BR>\n" );
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document.write( "LIST the elements in the set A∩B∩C:A∩B∩C= {___}\r<BR>\n" );
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document.write( "LIST the elements in the set A∪B∪C:A∪B∪C= {__}<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 #828282 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=828282\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "∩ is the intersection symbol<BR>\n" );
document.write( "U is the union symbol\r<BR>\n" );
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document.write( "Set A∩B is where we look at elements in both sets A and B at the same time. This is the overlapped region in the Venn Diagram.\r<BR>\n" );
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document.write( "Similarly, A∩B∩C looks at numbers in all three sets at the same time. \r<BR>\n" );
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document.write( "Set A={1,6,7,10}<BR>\n" );
document.write( "Set B={2,9,10}<BR>\n" );
document.write( "Set C={3,4,5,7,8}\r<BR>\n" );
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document.write( "The value 1 is in set A, but not in any other set<BR>\n" );
document.write( "6 is in set A, but not in any other set<BR>\n" );
document.write( "7 is in A and C, but not set B, so this value is crossed off as well<BR>\n" );
document.write( "10 is in A and B, but not in set C\r<BR>\n" );
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document.write( "The values 1,6,7 and 10 are not found in all three sets.<BR>\n" );
document.write( "As you can see, we only need to check one set to determine A∩B∩C since this intersected set must consist of values from that particular single set mentioned.\r<BR>\n" );
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document.write( "In other words, there isn't any value that is in all three sets at the same time. \r<BR>\n" );
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document.write( "Therefore, A∩B∩C = { } is the empty set.<BR>\n" );
document.write( "We write two curly braces with nothing between them. <BR>\n" );
document.write( "Not even 0 is in this set.<BR>\n" );
document.write( "The set { } is different from { 0 }\r<BR>\n" );
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document.write( "For the next part, we'll be doing set union.<BR>\n" );
document.write( "A∪B means we combine sets A and B into one bigger set.<BR>\n" );
document.write( "Toss out any duplicates.\r<BR>\n" );
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document.write( "Example:<BR>\n" );
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document.write( "A = {1,2}\r\n" );
document.write( "B = {3,4}\r\n" );
document.write( "A∪B = {1,2     3,4}\r\n" );
document.write( "A∪B = {1,2,3,4}\r\n" );
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document.write( "The spacing is intentional to show how sets A and B combine together. When writing the actual union, you'd just write a list of numbers like normal without the massive gap.\r<BR>\n" );
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document.write( "Another example:<BR>\n" );
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document.write( "A = {1,2,3}\r\n" );
document.write( "B = {3,4,5}\r\n" );
document.write( "A∪B = {1,2,3     3,4,5}\r\n" );
document.write( "A∪B = {1,2,3     <font color=red><s>3</s></font>,4,5}\r\n" );
document.write( "A∪B = {1,2,3,4,5}\r\n" );
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document.write( "We combine the 1,2,3 with 3,4,5<BR>\n" );
document.write( "Cross off the second copy of 3 to avoid duplicates.\r<BR>\n" );
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document.write( "Going back to the original problem<BR>\n" );
document.write( "Set A={1,6,7,10}<BR>\n" );
document.write( "Set B={2,9,10}<BR>\n" );
document.write( "Set C={3,4,5,7,8}\r<BR>\n" );
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document.write( "We have<BR>\n" );
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document.write( "A∪B∪C = {1,6,7,10,        2,9,10,       3,4,5,7,8}\r\n" );
document.write( "A∪B∪C = {1,2,3,4,5,6,7,8,9,10}\r\n" );
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document.write( "Common practice is to sort the values in the set from smallest to largest.<BR>\n" );
document.write( "Don't forget to erase any duplicates.\r<BR>\n" );
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document.write( "It turns out that we have the universal set S = {1,2,3,4,5,6,7,8,9,10} as the result of the union of all three sets mentioned. <BR>\n" );
document.write( "All of the values in the universe are found in set A, set B, or set C.\r<BR>\n" );
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document.write( "We have this interesting contrast going on.<BR>\n" );
document.write( "A∩B∩C is the empty set, aka \"nothing\"<BR>\n" );
document.write( "A∪B∪C consists of everything in the universal set.<BR>\n" );
document.write( "This won't always happen. It is possible your teacher purposefully crafted these sets in this fashion to attain this dichotomy.\r<BR>\n" );
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document.write( "Answers: <BR>\n" );
document.write( "A∩B∩C = <font color=red>{  }</font> which is the empty set<BR>\n" );
document.write( "A∪B∪C = <font color=red>{1,2,3,4,5,6,7,8,9,10}</font> which in this case is the universal set<BR>\n" );
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