document.write( "Question 459075: I am working on cardinality of sets and the problem is:\r
\n" ); document.write( "\n" ); document.write( "n(AᑎB)=6
\n" ); document.write( "n(BᑎC)=7
\n" ); document.write( "n(AᑎC)=8
\n" ); document.write( "n(AᑌBᑌC)=25
\n" ); document.write( "n(AᑎBᑎC)=4
\n" ); document.write( "n(B-A)=5
\n" ); document.write( "n(BᑌC)=20\r
\n" ); document.write( "\n" ); document.write( "Find the cardinalities n(A), n(B), n(C) \n" ); document.write( "
Algebra.Com's Answer #315252 by Edwin McCravy(20054)\"\" \"About 
You can put this solution on YOUR website!
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document.write( "We will draw a Venn diagram consisting of 3 overlapping\r\n" );
document.write( "circles A, B, C and let the letters t,u,v,w,x,y,z indicate the\r\n" );
document.write( "number of elements that are in each of the 7 regions created\r\n" );
document.write( "by the overlapping circles.\r\n" );
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document.write( "n(A) = t+u+w+x\r\n" );
document.write( "n(B) = u+v+x+y\r\n" );
document.write( "n(C) = w+x+y+z\r\n" );
document.write( "n(AᑎB) = u+x = 6\r\n" );
document.write( "n(AᑎC) = w+x = 8\r\n" );
document.write( "n(BᑎC) = x+y = 7\r\n" );
document.write( "n(AᑎBᑎC) = x = 4\r\n" );
document.write( "n(AᑌB) = t+u+v+w+x+y\r\n" );
document.write( "n(AᑌC) = t+u+w+x+y+z\r\n" );
document.write( "n(BᑌC) = u+v+w+x+y+z = 20\r\n" );
document.write( "n(AᑌBᑌC) = t+u+v+w+x+y+z = 25\r\n" );
document.write( "n(B-A) = v+y = 5 \r\n" );
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document.write( "Only one of those given pieces of data\r\n" );
document.write( "consists of only one of the 7 regions.\r\n" );
document.write( "That one is \r\n" );
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document.write( "n(AᑎBᑎC)= x = 4\r\n" );
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document.write( "so we will replace x by 4 in the region right in the\r\n" );
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document.write( "n(AᑎB) = u+x = 6\r\n" );
document.write( "n(AᑎC) = w+x = 8\r\n" );
document.write( "n(BᑎC) = x+y = 7\r\n" );
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document.write( "Since we know that n(AᑎB) = u+x = u+4 = 6\r\n" );
document.write( "we know that u=2 so we replace u by 2\r\n" );
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document.write( "Also since we know that n(AᑎC) = w+x = w+4 = 8\r\n" );
document.write( "we know that w=4 so we replace w by 4\r\n" );
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document.write( "Also since we know that n(BᑎC) = x+y = 4+y = 7\r\n" );
document.write( "we know that y=3 so we replace y by 3\r\n" );
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document.write( "Now we are given that \r\n" );
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document.write( "n(B-A) = v+y = 5\r\n" );
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document.write( "and since we know that v+y = 5, and y=3,\r\n" );
document.write( "v+3 = 5 and so v=2.  So we replace v by 2 \r\n" );
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document.write( "Now we know how many elements there are in B,\r\n" );
document.write( "n(B) = u+v+x+y = 2+2+4+3 = 11\r\n" );
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document.write( "So the cardinality of B is 11.\r\n" );
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document.write( "Now since n(BᑌC) = u+v+w+x+y+z = 20,\r\n" );
document.write( "2+2+4+4+3+z = 20,\r\n" );
document.write( "       15+z = 20\r\n" );
document.write( "          z = 5\r\n" );
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document.write( "So now we replace z by 5\r\n" );
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document.write( "and now we know the cardinality of C, because\r\n" );
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document.write( "n(C) = w+x+y+z = 4+4+3+5 = 16\r\n" );
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document.write( "Since n(AᑌBᑌC) = t+u+v+w+x+y+z = 25,\r\n" );
document.write( "t+2+2+4+4+3+5 = 25\r\n" );
document.write( "         t+20 = 25\r\n" );
document.write( "            t = 5\r\n" );
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document.write( "So we replace t by 5\r\n" );
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document.write( "So now we know that the cardinality of A is\r\n" );
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document.write( "n(A) = t+u+w+x = 5+2+4+4 = 15\r\n" );
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document.write( "Edwin
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