document.write( "Question 1035036: A, B, and C are sets: n(A)=30; n(B)=32; n(C)=31; n(A ∩ B)=16; n(A ∩ C)=19; n(B ∩ C)=18; and n(A ∩ B ∩ C)=8. Find n(A U B U C) and calculate n(A-(B U C)).. Thanks! \n" ); document.write( "

Algebra.Com's Answer #649696 by ikleyn(52915) $\"\"$ $\"About$

You can put this solution on YOUR website!
.
\n" ); document.write( "

\r\n" );
document.write( "Let me introduce more convenient notation.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Let |A| be the number of elements of the set A;\r\n" );
document.write( "    |B| be the number of elements of the set B;\r\n" );
document.write( "    |C| be the number of elements of the set C.\r\n" );
document.write( "\r\n" );
document.write( "Let AB be the intersection of A and B;\r\n" );
document.write( "    AC be the intersection of A and C;\r\n" );
document.write( "    BC be the intersection of B and C.\r\n" );
document.write( "\r\n" );
document.write( "Let |AB| be the number of elements in AB;\r\n" );
document.write( "    |AC| be the number of elements in AC;\r\n" );
document.write( "    |BC| be the number of elements in BC.\r\n" );
document.write( "\r\n" );
document.write( "Let ABC be the intersection of the sets A, B, and C, and\r\n" );
document.write( "    |ABC| be the number of elements in ABC.\r\n" );
document.write( "\r\n" );
document.write( "Then there is a remarkable formula:\r\n" );
document.write( "\r\n" );
document.write( "n(A U B U C) = |A| + |B| + |C| - |AB| - |AC| - |BC| + |ABC|.\r\n" );
document.write( "\r\n" );
document.write( "For its proof see the lesson Advanced problems on counting elements in sub-sets of a given finite set in this site.\r\n" );
document.write( "\r\n" );
document.write( "Now apply the formula. You have everything to calculate the right side. It is \r\n" );
document.write( "\r\n" );
document.write( "30 + 32 + 31 - 16 - 19 - 18 + 8 = 48.\r\n" );
document.write( "\r\n" );
document.write( "Answer. n(A U B U C) = 48.\r\n" );
document.write( "

\n" ); document.write( " \n" ); document.write( "

\n" );