document.write( "Question 952302: Premise: ~(A->(B&C))\r
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\n" ); document.write( "\n" ); document.write( "I've tried to Provisionally assume A<->(B&C), but I don't understand double arrow rules that well. \n" ); document.write( "
Algebra.Com's Answer #581586 by Edwin McCravy(20055)\"\" \"About 
You can put this solution on YOUR website!
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document.write( "This argument is NOT VALID!\r\n" );
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document.write( "The double arrow is the biconditional\r\n" );
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document.write( "P<->Q by definition is (P-Q)&(Q->P)\r\n" );
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document.write( "It is sometimes called \"equivalence\",\r\n" );
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document.write( "which means they have the same truth value.\r\n" );
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document.write( "But your conclusion is not valid.\r\n" );
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document.write( "We can show that it is not valid by a truth table:\r\n" );
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document.write( "            ?\r\n" );
document.write( "~[A->(B&C)] => ~[A<->(B&C)]\r\n" );
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document.write( "Since (B&C) appears in both, we can let D = (B&C)\r\n" );
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document.write( "           ?\r\n" );
document.write( "   ~(A->D) => ~(A<->~D)\r\n" );
document.write( "-----------------------\r\n" );
document.write( "     T  T       T    T     \r\n" );
document.write( "     T  F       T    F\r\n" );
document.write( "     F  T       F    T\r\n" );
document.write( "     F  F       F    F\r\n" );
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document.write( "becomes:\r\n" );
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document.write( "           ?\r\n" );
document.write( "   ~(A->D) => ~(A<->~D)\r\n" );
document.write( "-----------------------\r\n" );
document.write( "       T          F      \r\n" );
document.write( "       F          T \r\n" );
document.write( "       T          T \r\n" );
document.write( "       T          T  \r\n" );
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document.write( "which becomes:\r\n" );
document.write( "           \r\n" );
document.write( "           ?\r\n" );
document.write( "   ~(A->D) => ~(A<->~D)\r\n" );
document.write( "-----------------------\r\n" );
document.write( "   F          T          \r\n" );
document.write( "   T          F     \r\n" );
document.write( "   F          T     \r\n" );
document.write( "   F          T    \r\n" );
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document.write( "which becomes:\r\n" );
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document.write( "           ?\r\n" );
document.write( "   ~(A->D) => ~(A<->~D)\r\n" );
document.write( "-----------------------\r\n" );
document.write( "           T             \r\n" );
document.write( "           F        \r\n" );
document.write( "           T        \r\n" );
document.write( "           T    \r\n" );
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document.write( "It is not valid because it fails in case 2.\r\n" );
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document.write( "Edwin
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