document.write( "Question 978478: What are the proofs?
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Algebra.Com's Answer #599863 by Edwin McCravy(20060)\"\" \"About 
You can put this solution on YOUR website!
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document.write( "That's the law of composition:\r\n" );
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document.write( "That's a basic law, so we prove it with a truth table:\r\n" );
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document.write( "(A → B) & (A → C) ├ A→ (B & C)\r\n" );
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document.write( "Place TTTTFFFF under the A's, TTFFTTFF under the B's and TFTFTFTF under\r\n" );
document.write( "the C's:\r\n" );
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document.write( "(A → B) & (A → C) ├ A→ (B & C)\r\n" );
document.write( " T   T     T   T    T   T   T\r\n" );
document.write( " T   T     T   F    T   T   F\r\n" );
document.write( " T   F     T   T    T   F   T\r\n" );
document.write( " T   F     T   F    T   F   F\r\n" );
document.write( " F   T     F   T    F   T   T\r\n" );
document.write( " F   T     F   F    F   T   F\r\n" );
document.write( " F   F     F   T    F   F   T\r\n" );
document.write( " F   F     F   F    F   F   F\r\n" );
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document.write( "Under the first → put F only when it has T on the left and F on the\r\n" );
document.write( "right.  Otherwise put T.  Then you erase what's under the A and the B.\r\n" );
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document.write( "(A → B) & (A → C) ├ A→ (B & C)\r\n" );
document.write( "   T       T   T    T   T   T\r\n" );
document.write( "   T       T   F    T   T   F\r\n" );
document.write( "   F       T   T    T   F   T\r\n" );
document.write( "   F       T   F    T   F   F\r\n" );
document.write( "   T       F   T    F   T   T\r\n" );
document.write( "   T       F   F    F   T   F\r\n" );
document.write( "   T       F   T    F   F   T\r\n" );
document.write( "   T       F   F    F   F   F \r\n" );
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document.write( "Under the second →, do the same. Put F only when it has T on the left \r\n" );
document.write( "and F on the right.  Otherwise put T.  Then you erase what's under the \r\n" );
document.write( "A and the C.\r\n" );
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document.write( "(A → B) & (A → C) ├ A→ (B & C)\r\n" );
document.write( "   T         T      T   T   T\r\n" );
document.write( "   T         F      T   T   F\r\n" );
document.write( "   F         T      T   F   T\r\n" );
document.write( "   F         F      T   F   F\r\n" );
document.write( "   T         T      F   T   T\r\n" );
document.write( "   T         T      F   T   F\r\n" );
document.write( "   T         T      F   F   T\r\n" );
document.write( "   T         T      F   F   F  \r\n" );
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document.write( "Under the first & put T only when it has T's on both sides of the &, otherwise\r\n" );
document.write( "put F, then erase what's under the first two →'s\r\n" );
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document.write( "(A → B) & (A → C) ├ A→ (B & C)\r\n" );
document.write( "        T           T   T   T\r\n" );
document.write( "        F           T   T   F\r\n" );
document.write( "        F           T   F   T\r\n" );
document.write( "        F           T   F   F\r\n" );
document.write( "        T           F   T   T\r\n" );
document.write( "        T           F   T   F\r\n" );
document.write( "        T           F   F   T\r\n" );
document.write( "        T           F   F   F \r\n" );
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document.write( "Under the second & put T only when it has T's on both sides of the &, otherwise\r\n" );
document.write( "put F, then erase what's under the B and C:\r\n" );
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document.write( "(A → B) & (A → C) ├ A → (B & C)\r\n" );
document.write( "        T           T      T\r\n" );
document.write( "        F           T      F\r\n" );
document.write( "        F           T      F   \r\n" );
document.write( "        F           T      F  \r\n" );
document.write( "        T           F      T \r\n" );
document.write( "        T           F      F\r\n" );
document.write( "        T           F      F   \r\n" );
document.write( "        T           F      F   \r\n" );
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document.write( "Under the third →, do the same as before. Put F only when it has T on \r\n" );
document.write( "the left and F on the right.  Otherwise put T.  Then you erase what's \r\n" );
document.write( "under the A and the (B & C).\r\n" );
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document.write( "(A → B) & (A → C) ├ A → (B & C)\r\n" );
document.write( "        T             T     \r\n" );
document.write( "        F             F     \r\n" );
document.write( "        F             F        \r\n" );
document.write( "        F             F       \r\n" );
document.write( "        T             T      \r\n" );
document.write( "        T             T     \r\n" );
document.write( "        T             T        \r\n" );
document.write( "        T             T     \r\n" );
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document.write( "Finally under the ├, place T is both sides are the same, and put F\r\n" );
document.write( "otherwise. Then erase the columns on each side of it.\r\n" );
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document.write( "(A → B) & (A → C) ├ A → (B & C)\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 they are all T, the law of composition is proved.\r\n" );
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document.write( "Edwin
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