document.write( "Question 1073560: suppose A and B are sets such that A Union B has 20 elements, A intersection B has 7 elements and the number of elements in B is twice that of A.what is the number of elements in: A and B \n" ); document.write( "

Algebra.Com's Answer #688382 by ikleyn(52804) $\"\"$ $\"About$

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\n" ); document.write( "suppose A and B are sets such that A Union B has 20 elements, A intersection B has 7 elements and the number of elements in B
\n" ); document.write( "is twice that of A.what is the number of elements in $\"highlight%28each_of_the_sets%29\"$ A and B.
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document.write( "Let a = n(A)  (the number of elements in the set A);\r\n" );
document.write( "    b = n(B)   (same for B).\r\n" );
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document.write( "Then you have these equations\r\n" );
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document.write( "20 = a + b - 7,     (1)\r\n" );
document.write( "b = 2a.             (2)\r\n" );
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document.write( "Regarding equation (1), it is the particular case of the general equation\r\n" );
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document.write( "n(A U B) = n(A) + n(B) - n(A n B).   (*)  (See my comments at the END)\r\n" );
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document.write( "To solve the system (1), (2), simply substitute the expression (2) into (1). You will get\r\n" );
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document.write( "20 = a + 2a - 7  --->  3a = 27  --->  a =  = 9.\r\n" );
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document.write( "Then b = 2*a = 2*9 = 18.\r\n" );
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document.write( "Answer.  a = n(A) = 9,   b = n(B) = 19.\r\n" );
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document.write( "Check.   n(A) + n(B) - 7 = 9 + 18 - 7 = 20.\r\n" );
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\n" ); document.write( "\n" ); document.write( "Regarding the equation (*), its proof is OBVIOUS:\r
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\n" ); document.write( "\n" ); document.write( "count elements of A. Then add the elements of B. Elements in the intersection are counted twice.
\n" ); document.write( "Therefore, subtract n(a n B) from the sum n(A) + n(B), and you will get n(a U B).\r
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\n" ); document.write( "\n" ); document.write( "See the lesson \r
\n" ); document.write( "\n" ); document.write( " - Counting elements in sub-sets of a given finite set\r
\n" ); document.write( "\n" ); document.write( "in this site.\r
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\n" ); document.write( "\n" ); document.write( "Also, you have this free of charge online textbook in ALGEBRA-I in this site\r
\n" ); document.write( "\n" ); document.write( " - ALGEBRA-I - YOUR ONLINE TEXTBOOK.\r
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\n" ); document.write( "\n" ); document.write( "The referred lessons are the part of this online textbook under the topic \"Miscellaneous word problems\".\r
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