document.write( "Question 1000815: If n(A)=10,n(A u B) =28 and n(A n B)=6; what is n(B)? \n" ); document.write( "
Algebra.Com's Answer #618151 by Edwin McCravy(20056)\"\" \"About 
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If n(A)=10,n(A u B) =28 and n(A n B)=6; what is n(B)?
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document.write( "There are three ways to do it:  Venn diagram, chart and formula.\r\n" );
document.write( "I'll show you all three ways:\r\n" );
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document.write( "Here's how to do it by Venn diagram:\r\n" );
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document.write( "The red circle represents the set A,\r\n" );
document.write( "The blue circle represents the set B,\r\n" );
document.write( "The overlapping part represents their intersection \r\n" );
document.write( "AnB, or A and B\r\n" );
document.write( "The left side of the red circle represents A only,\r\n" );
document.write( "or A and not B, written AnB'\r\n" );
document.write( "The right side of the blue circle represents B only,\r\n" );
document.write( "or B and not A, or in alphabetical order, not A and B,\r\n" );
document.write( "written AnB'\r\n" );
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document.write( "We are given the overlapping part, n(AnB)=6 so we replace\r\n" );
document.write( "n(AnB) by 6:\r\n" );
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document.write( "Now since we are given that n(A)=10, we know that the other part\r\n" );
document.write( "of the red circle must contain the other 4, so we replace n(AnB')\r\n" );
document.write( "by 4:\r\n" );
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document.write( "Finally we are given that AuB=28, which includes both circles,\r\n" );
document.write( "so since we only have one part to fill, A'nB, we know that it\r\n" );
document.write( "must be 28-10 = 18.  So we replace A'nB by 18:\r\n" );
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document.write( "Answer: n(B) = 6+18 = 24\r\n" );
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document.write( "Here's how to do it by chart:  n means \"and\", u means \"or\",\r\n" );
document.write( "prime (') after a set means \"everything but that set\" \r\n" );
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document.write( "           B          B'     totals      \r\n" );
document.write( "-------|----------|----------|--------\r\n" );
document.write( "A      |n(AnB)=6  | n(AnB')  |n(A)=10\r\n" );
document.write( "-------|----------|----------|--------\r\n" );
document.write( "A'     |n(A'nB)=? |n(A'nB')=?|n(A')=?  \r\n" );
document.write( "-------|----------|----------|--------\r\n" );
document.write( "totals | n(B)=?   | n(B')=?  |  28\r\n" );
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document.write( "We will assume that A'nB' is empty, so we fill in n(A'nB')=0\r\n" );
document.write( "We fill in n(A')=18 because 28-10 = 18\r\n" );
document.write( "We fill in n(A'nB)=18 because 18-0 = 18\r\n" );
document.write( "We fill in n(AnB')=4 because 10-6 = 4\r\n" );
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document.write( "           B          B'     totals      \r\n" );
document.write( "-------|----------|----------|--------\r\n" );
document.write( "A      |n(AnB)=6  | n(AnB')=4|n(A)=10\r\n" );
document.write( "-------|----------|----------|--------\r\n" );
document.write( "A'     |n(A'nB)=18|n(A'nB')=0|n(A')=18  \r\n" );
document.write( "-------|----------|----------|--------\r\n" );
document.write( "totals | n(B)=?   | n(B')=?  |  28\r\n" );
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document.write( "Now we can fill in n(B)=24 because 6+18=24,\r\n" );
document.write( "which is what we were asked for.  We can\r\n" );
document.write( "also fill in n(B') both because 4+0=4 and\r\n" );
document.write( "also because 28-24 = 4. \r\n" );
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document.write( "           B          B'     totals      \r\n" );
document.write( "-------|----------|----------|--------\r\n" );
document.write( "A      |n(AnB)=6  | n(AnB')=4|n(A)=10\r\n" );
document.write( "-------|----------|----------|--------\r\n" );
document.write( "A'     |n(A'nB)=18|n(A'nB')=0|n(A')=18  \r\n" );
document.write( "-------|----------|----------|--------\r\n" );
document.write( "totals | n(B)=24  | n(B')=4  |  28 \r\n" );
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document.write( "Answer n(B)=24.\r\n" );
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document.write( "Here's how to do it by formula.  No thinking required.\r\n" );
document.write( "Just plug in the magic formula, turn the crank, and out\r\n" );
document.write( "pops the answer. \r\n" );
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document.write( "If n(A)=10,n(A u B) =28 and n(A n B)=6; what is n(B)?\r\n" );
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document.write( "The magic formula is\r\n" );
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document.write( "n(AuB) = n(A) + n(B) - n(AnB)\r\n" );
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document.write( "   28  =  10  + n(B) -   6\r\n" );
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document.write( "    28 = 4 + n(B)\r\n" );
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document.write( "    24 = n(B)\r\n" );
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document.write( "Edwin
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