document.write( "Question 1015522: out of 136 students 67 read tribune, 56 read punch,40read the nation. 5 read all the three,12 read punch nation,9read tribune and nation.how many students read tribune and punch. \n" ); document.write( "
Algebra.Com's Answer #631941 by ikleyn(52864)\"\" \"About 
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\n" ); document.write( "out of 136 students 67 read tribune, 56 read punch,40 read the nation. 5 read all the three,12 read punch nation, 9 read tribune and nation.how many students read tribune and punch.
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\n" ); document.write( "\n" ); document.write( "Can we be sure that of 136 students each reads at least one of T, P or N?\r
\n" ); document.write( "\n" ); document.write( "Without this info the problem can not be solved.\r
\n" ); document.write( "\n" ); document.write( "And why you didn't say it from the very beginning?\r
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\n" ); document.write( "\n" ); document.write( "OK. I decided to fix the condition myself as it should be. See below.\r
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\n" ); document.write( "\n" ); document.write( "In a group of 136 students each reads at least one of three papers, Tribune, Punch and the Nation.
\n" ); document.write( "67 read Tribune, 56 read Punch, 40 read the Nation. 5 read all the three, 12 read Punch and Nation, 9 read Tribune and the Nation.
\n" ); document.write( "How many students read Tribune and Punch? \r
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document.write( "We have the set A of 136 students and three its subsets T (67 elements), P (56 elements), and N (40 elements).\r\n" );
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document.write( "Let us denote TP the intersection of T and P for brevity; denote TN the intersection of T and N; and denote\r\n" );
document.write( "PN the intersection of P and N.\r\n" );
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document.write( "Also let us denote TPN the intersection of all the three subsets T, P and N.\r\n" );
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document.write( "We know that the subset T consists of 67 elements, the subset P consists of 56 elements and the subset N consists of 40 elements.\r\n" );
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document.write( "We are also given that PN consists of 12 elements, TN consists of 9 elements, and TPN consists of 5 elements.\r\n" );
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document.write( "The question is: how many elements are in TP?\r\n" );
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document.write( "For any given subset X, let us use the symbol |X| for the number of elements in X.\r\n" );
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document.write( "Then there is a remarkable equality, which connects the number of elements in the set A; in its subsets T, P and N;\r\n" );
document.write( "in their intersections TP, TN and PN; and in the subset TPN:\r\n" );
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document.write( "|A| = |T| + |P| + |N| - |TP| - |TN| - |PN| + |TPN|.     (1)\r\n" );
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document.write( "Now look how this equality will help us to solve our problem.\r\n" );
document.write( "Simply substitute the known values into this equality. You will get\r\n" );
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document.write( "136 = 67 + 56 + 40 - |TP| - 9 - 12 + 5.                 (2)\r\n" );
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document.write( "|TP| is the only unknown in this equation, and you can easily find it by isolating.\r\n" );
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document.write( "|TP| = 11.\r\n" );
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document.write( "Actually, the equality (1) is valid for any finite set A and for any three its subsets T, P and N that jointly cover the set A. \r\n" );
document.write( "The proof is in this dialog:\r\n" );
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document.write( "Physicist: Having the goal to calculate the number of elements in A, let us sum up the amounts of elements in subsets T, P and N:\r\n" );
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document.write( "   |A| = |T| + |P| + |N|.                               (3)\r\n" );
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document.write( "Mathematician: Hmmm, interesting . . . But it is not exactly right, because the subsets T, P and N have intersections.\r\n" );
document.write( "               By adding up the amounts of elements according (3), we count twice the elements of each intersection TP, TN and PN.\r\n" );
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document.write( "               So, it works only as the \"first approximation\" of the number |A|.\r\n" );
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document.write( "Physicist: Agree. But we can easily fix the situation. Simply, take off the amounts of elements in the intersections, like this\r\n" );
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document.write( "   |A| = |T| + |P| + |N| - |TP| - |TN| - |PN|.           (4)\r\n" );
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document.write( "Mathematician: Hmmm, a good idea . . . But it is still not exactly right, unfortunately. \r\n" );
document.write( "               Notice that when we sum up (3), we count the elements in the tripled intersection TPN three times.\r\n" );
document.write( "               In the next step (4), we take off these elements of the intersection TPN three times. \r\n" );
document.write( "               It means that the elements of the intersection TPN do not contribute at all to the value of |A| in formula (4).\r\n" );
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document.write( "Physicist: Yes, you are right.\r\n" );
document.write( "               But we can make one step forward and add the amount |TPN| to the right side of (4).\r\n" );
document.write( "               Then we will have \r\n" );
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document.write( "   |A| = |T| + |P| + |N| - |TP| - |TN| - |PN| + |TPN|,\r\n" );
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document.write( "               exactly as the formula (1) says.\r\n" );
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document.write( "After completing the proof (it is really completing and it is really the proof) Mathematician and Physicist went for lunch.\r\n" );
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document.write( "(Alternative version: they wrote this text and submitted it to the forum www.algebra.com)\r\n" );
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