document.write( "Question 1193969: What are the proof steps for A ≡ (B ≡ C) // (A ≡ B) ≡ C ? \n" ); document.write( "
Algebra.Com's Answer #826345 by Edwin McCravy(20055)\"\" \"About 
You can put this solution on YOUR website!
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document.write( "It's a cinch to prove it with truth tables, and murder to prove it by \r\n" );
document.write( "argument forms.  Since all the argument forms are proved by truth tables,\r\n" );
document.write( "your teacher should allow a truth table proof.\r\n" );
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document.write( "[A ≡ (B ≡ C)] ≡ [(A ≡ B) ≡ C]\r\n" );
document.write( "---------------------------\r\n" );
document.write( " T ≡ (T ≡ T)  ≡  (T ≡ T) ≡ T\r\n" );
document.write( " T ≡ (T ≡ F)  ≡  (T ≡ T) ≡ F\r\n" );
document.write( " T ≡ (F ≡ T)  ≡  (T ≡ F) ≡ T\r\n" );
document.write( " T ≡ (F ≡ F)  ≡  (T ≡ F) ≡ F\r\n" );
document.write( " F ≡ (T ≡ T)  ≡  (F ≡ T) ≡ T\r\n" );
document.write( " F ≡ (T ≡ F)  ≡  (F ≡ T) ≡ F\r\n" );
document.write( " F ≡ (F ≡ T)  ≡  (F ≡ F) ≡ T\r\n" );
document.write( " F ≡ (F ≡ F)  ≡  (F ≡ F) ≡ F\r\n" );
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document.write( "Do equivalence inside the parentheses\r\n" );
document.write( "If the values are the same on both sides\r\n" );
document.write( "of the ≡, put T, otherwise put F.\r\n" );
document.write( "Then erase what you used to get what\r\n" );
document.write( "you just put down.\r\n" );
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document.write( "[A ≡ (B ≡ C)] ≡ [(A ≡ B) ≡ C]\r\n" );
document.write( "-----------------------------\r\n" );
document.write( "[T ≡    T   ] ≡ [   T    ≡ T]\r\n" );
document.write( "[T ≡    F   ] ≡ [   T    ≡ F]\r\n" );
document.write( "[T ≡    F   ] ≡ [   F    ≡ T]\r\n" );
document.write( "[T ≡    T   ] ≡ [   F    ≡ F]\r\n" );
document.write( "[F ≡    T   ] ≡ [   F    ≡ T]\r\n" );
document.write( "[F ≡    F   ] ≡ [   F    ≡ F]\r\n" );
document.write( "[F ≡    F   ] ≡ [   T    ≡ T]\r\n" );
document.write( "[F ≡    T   ] ≡ [   T    ≡ F]\r\n" );
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document.write( "Now do equivalence inside the brackets\r\n" );
document.write( "As before, if the values are the same \r\n" );
document.write( "on both sides of the ≡, put T, otherwise \r\n" );
document.write( "put F. Then erase what you used to get \r\n" );
document.write( "what you just put down.\r\n" );
document.write( " \r\n" );
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document.write( "[A ≡ (B ≡ C)] ≡ [(A ≡ B) ≡ C]\r\n" );
document.write( "-----------------------------\r\n" );
document.write( "   T          ≡     T    \r\n" );
document.write( "   F          ≡     F    \r\n" );
document.write( "   F          ≡     F    \r\n" );
document.write( "   T          ≡     T    \r\n" );
document.write( "   F          ≡     F    \r\n" );
document.write( "   T          ≡     T    \r\n" );
document.write( "   T          ≡     T    \r\n" );
document.write( "   F          ≡     F    \r\n" );
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document.write( "Now do the final equivalence inside the \r\n" );
document.write( "brackets As before, if the values are the \r\n" );
document.write( "same on both sides of the ≡, put T, otherwise \r\n" );
document.write( "put F. Then erase what you used to get \r\n" );
document.write( "what you just put down.\r\n" );
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document.write( "[A ≡ (B ≡ C)] ≡ [(A ≡ B) ≡ C]\r\n" );
document.write( "-----------------------------\r\n" );
document.write( "              T         \r\n" );
document.write( "              T         \r\n" );
document.write( "              T         \r\n" );
document.write( "              T         \r\n" );
document.write( "              T         \r\n" );
document.write( "              T         \r\n" );
document.write( "              T         \r\n" );
document.write( "              T\r\n" );
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document.write( "Since we end up with all T's, that proves\r\n" );
document.write( "that  [A ≡ (B ≡ C)] ≡ [(A ≡ B) ≡ C] is a\r\n" );
document.write( "tautology.        \r\n" );
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document.write( "Edwin
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