document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1194404>Question 1194404</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A survey of 130 pet owners yielded the following information: 47 own fish; 53 own a bird; 50 own a cat; 64 own a dog; 2 own all four; 11 own only fish; 14 own only a bird; 10 own fish and a bird; 21 own fish and a cat; 24 own a bird and a dog; 27 own a cat and a dog; 3 own fish, a bird, a cat, and no dog; 1 owns fish, a bird, a dog, and no cat; 9 own fish, a cat, a dog, and no bird; and 10 own a bird, a cat, a dog, and no fish. How many of the surveyed pet owners have no fish, no birds, no cats, and no dogs? (They own other types of pets.) </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #826627 by </B><A HREF=/tutors/aboutme.mpl?userid=ikleyn><B>ikleyn(52786)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=ikleyn><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=ikleyn><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=826627\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> .<BR>\n" );
document.write( "For convenience, I will represent the given part in modified form<BR>\n" );
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document.write( "A survey of 130 pet owners yielded the following information: \r\n" );
document.write( "  (1)  47 own fish; 53 own a bird; 50 own a cat; 64 own a dog; \r\n" );
document.write( "  (2)   2 own all four; \r\n" );
document.write( "  (3)  11 own only fish; 14 own only a bird; \r\n" );
document.write( "  (4)  10 own fish and a bird; 21 own fish and a cat; 24 own a bird and a dog; 27 own a cat and a dog; \r\n" );
document.write( "  (5)   3 own fish, a bird, a cat, and no dog; \r\n" );
document.write( "  (6)   1 owns fish, a bird, a dog, and no cat; \r\n" );
document.write( "  (7)   9 own fish, a cat, a dog, and no bird; \r\n" );
document.write( "  (8)  10 own a bird, a cat, a dog, and no fish. \r\n" );
document.write( "</PRE>How many of the surveyed pet owners have no fish, no birds, no cats, and no dogs? <BR>\n" );
document.write( "(They own other types of pets.)<BR>\n" );
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document.write( "&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;I present here another solution, which entirely is build on using Inclusion-Exclusion principle.\r<BR>\n" );
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document.write( "Simply saying, they want you find the union set of owners, who own either fish, or birds, or cats or dogs,\r\n" );
document.write( "and then calculate the COMPLEMENT of this set to 130.\r\n" );
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document.write( "It would be a standard Inclusion-Exclusion problem, if the numbers of elements in single sets \r\n" );
document.write( "F (fish owners), B(bird owners), C (cat owners) and D (dog owners) be given, along with \r\n" );
document.write( "their in-pairs intersections, triple intersections and quadruple intersection.\r\n" );
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document.write( "But in the problem, the given are other combinations of intersections.\r\n" );
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document.write( "THEREFORE, if you want to reduce the problem to standard Inclusion-Exclusion, you should restore\r\n" );
document.write( "information about standard intersection subsets, based on given data.\r\n" );
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document.write( "We just have single subsets F, B, C, D given in statement (1), and a quadruple intersetion given in statement (2).\r\n" );
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document.write( "From statements (5), (6), (7), and (8), we can easy restore all triple intersections by adding FBCD = 2\r\n" );
document.write( "to given values in (5), (6) (7) and (8)\r\n" );
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document.write( "    FBC = 3 + 2 =  5;     (11)\r\n" );
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document.write( "    FBD = 1 + 2 =  3;     (12)\r\n" );
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document.write( "    FCD = 9 + 2 = 11;     (13)\r\n" );
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document.write( "    BCD = 10 + 2 = 12.    (14)\r\n" );
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document.write( "Regarding in-pair intersections, we just have given FB = 10, FC = 21, BD = 24, CD = 27 in statement (4).\r\n" );
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document.write( "So, of in-pair intersections, we need to restore FD and BC, that are not given in the input.\r\n" );
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document.write( "It is obvious, that the set F - F_only is the union of three subsets FB, FC and FD:\r\n" );
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document.write( "    F - F_only = FB U FC U FD;  so  n(F - F_only) = FB + FC+ FD,  or (applying Inclusion-Exclusion)\r\n" );
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document.write( "    47 - 11 = FB + FC + FD - FBC - FBD - FCD + FBCD = 10 + 21 + FD - 5 - 3 - 11 + 2.\r\n" );
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document.write( "It gives an equation  36 = 14 + FD, from which  FD = 36-14 = 22.\r\n" );
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document.write( "Similarly, the set B - B_only is the union of three subsets FB, BC and BD:\r\n" );
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document.write( "    B - B_only = FB U BC U BD;  so  n(B - B_only) = FB + BC+ BD,  or (applying Inclusion-Exclusion)\r\n" );
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document.write( "    53 - 14 = FB + BC + BD - FBC - FBD - BCD + FBCD = 10 + BC + 24 - 5 - 3 - 12 + 2.\r\n" );
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document.write( "It gives an equation  39 = 16 + BC, from which  BC = 39-16 = 23.\r\n" );
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document.write( "Now the union (F U B U C U D)  contains, according to Inclusion-Exclusion principle, \r\n" );
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document.write( "      F +  B  +  C +  D - FB - FC - FD - BC - BD - CD + FBC + FBD + FCD + BCD - FBCD = \r\n" );
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document.write( "    = 47 + 53 + 50 + 64 - 10 - 21 - 22 - 23 - 24 - 27 +   5 +   3 +  11 +  12 -    2 = 116 owners.\r\n" );
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document.write( "The complement of 116 to 130 is 14 owners.\r\n" );
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document.write( "<U>ANSWER</U>. 14 owners have other types of pets.\r\n" );
document.write( "</PRE>\r<BR>\n" );
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document.write( "Solved.\r<BR>\n" );
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document.write( "Notice that in my solution, the meaning of symbols FB, FC, FD, BC, BD, CD, FBC, FBD, FCD, BCD, FBCD <BR>\n" );
document.write( "is different from that in the post of the other tutor.\r<BR>\n" );
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document.write( "In my post, they mean in-pair, triples and quadruple intersection of the base subsets F, B, C and D.\r<BR>\n" );
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document.write( "On inclusion-exclusion principle, &nbsp;see this Wikipedia article\r<BR>\n" );
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document.write( "https://en.wikipedia.org/wiki/Inclusion%E2%80%93exclusion_principle\r<BR>\n" );
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document.write( "To see many other similar &nbsp;(and different) &nbsp;solved problems, &nbsp;see the lessons\r<BR>\n" );
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document.write( "&nbsp;&nbsp;&nbsp;&nbsp;- <A  rel=nofollow HREF=http://www.algebra.com/algebra/homework/word/misc/Counting-elements-in-sub-sets-of-a-given-finite-set.lesson>Counting elements in sub-sets of a given finite set</A>\r<BR>\n" );
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document.write( "&nbsp;&nbsp;&nbsp;&nbsp;- <A  rel=nofollow HREF=https://www.algebra.com/algebra/homework/word/misc/Advanced-probs-counting-elements-in-sub-sets-of-a-given-finite-set.lesson>Advanced problems on counting elements in sub-sets of a given finite set</A>\r<BR>\n" );
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document.write( "&nbsp;&nbsp;&nbsp;&nbsp;- <A  rel=nofollow HREF=https://www.algebra.com/algebra/homework/word/misc/Challenging-problems-on-counting-elements-in-subsets-of-a-given-finite-set.lesson>Challenging problems on counting elements in subsets of a given finite set</A> \r<BR>\n" );
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document.write( "&nbsp;&nbsp;&nbsp;&nbsp;- <A  rel=nofollow HREF=https://www.algebra.com/algebra/homework/word/misc/Selected-problems-on-counting-elements-in-subsets-of-a-given-finite-set.lesson>Selected problems on counting elements in subsets of a given finite set</A> \r<BR>\n" );
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document.write( "&nbsp;&nbsp;&nbsp;&nbsp;- <A  rel=nofollow HREF=https://www.algebra.com/algebra/homework/Permutations/Inclusion-Exclusion-principle.lesson>Inclusion-Exclusion principle problems</A> \r<BR>\n" );
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document.write( "in this site.\r<BR>\n" );
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document.write( " </SPAN>\n" );
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