document.write( "Question 915332: Use an ordinary truth table to answer the following problems. Construct the truth table as per the instructions in the textbook.\r
\n" ); document.write( "\n" ); document.write( "Given the argument:\r
\n" ); document.write( "\n" ); document.write( "S ⊃ (K ∨ ∼ S) / K ⊃ S // S ≡ K\r
\n" ); document.write( "\n" ); document.write( "This argument is:
\n" ); document.write( "
\n" ); document.write( "a. Invalid; fails in 2nd line.
\n" ); document.write( "b. Invalid; fails in 3rd line.
\n" ); document.write( "c. Invalid; fails in 1st line.
\n" ); document.write( "d. Valid.
\n" ); document.write( "e. Invalid; fails in 4th line. \n" ); document.write( "
Algebra.Com's Answer #555548 by Edwin McCravy(20055)\"\" \"About 
You can put this solution on YOUR website!
Question 915332
\n" ); document.write( "
\r\n" );
document.write( "\r\n" );
document.write( "There was a slight mistake before.  The argument is valid. Here is the corrected version.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "I do truth tables different, but it's the same way. I begin\r\n" );
document.write( "by rewriting the argument with T's and F's under the S's and K's:\r\n" );
document.write( "\r\n" );
document.write( "ORIGINAL TRUTH TABLE\r\n" );
document.write( "\r\n" );
document.write( "S ⊃ (K ∨ ∼ S) / K ⊃ S // S ≡ K\r\n" );
document.write( "-------------------------------\r\n" );
document.write( "T ⊃ (T ∨ ∼ T) / T ⊃ T // T ≡ T\r\n" );
document.write( "T ⊃ (F ∨ ∼ T) / F ⊃ T // T ≡ F\r\n" );
document.write( "F ⊃ (T ∨ ∼ F) / T ⊃ F // F ≡ T\r\n" );
document.write( "F ⊃ (F ∨ ∼ F) / F ⊃ F // F ≡ F\r\n" );
document.write( "\r\n" );
document.write( "Then I place the truth values under the symbols ∼, ⊃ and ≡\r\n" );
document.write( "\r\n" );
document.write( "S ⊃ (K ∨ ∼ S) / K ⊃ S // S ≡ K\r\n" );
document.write( "-------------------------------\r\n" );
document.write( "T ⊃ (T ∨ F T) / T T T // T T T\r\n" );
document.write( "T ⊃ (F ∨ F T) / F T T // T F F\r\n" );
document.write( "F ⊃ (T ∨ T F) / T F F // F F T\r\n" );
document.write( "F ⊃ (F ∨ T F) / F T F // F T F\r\n" );
document.write( "\r\n" );
document.write( "Then I erase all the used truth values:\r\n" );
document.write( "\r\n" );
document.write( "S ⊃ (K ∨ ∼ S) / K ⊃ S // S ≡ K\r\n" );
document.write( "-------------------------------\r\n" );
document.write( "T ⊃ (T ∨ F  ) /   T   //   T \r\n" );
document.write( "T ⊃ (F ∨ F  ) /   T   //   F \r\n" );
document.write( "F ⊃ (T ∨ T  ) /   F   //   F \r\n" );
document.write( "F ⊃ (F ∨ T  ) /   T   //   T \r\n" );
document.write( "\r\n" );
document.write( "Place truth values under symbol ∨:\r\n" );
document.write( "\r\n" );
document.write( "S ⊃ (K ∨ ∼ S) / K ⊃ S // S ≡ K\r\n" );
document.write( "-------------------------------\r\n" );
document.write( "T ⊃ (T T F  ) /   T   //   T \r\n" );
document.write( "T ⊃ (F F F  ) /   T   //   F \r\n" );
document.write( "F ⊃ (T T T  ) /   F   //   F \r\n" );
document.write( "F ⊃ (F T T  ) /   T   //   T\r\n" );
document.write( "\r\n" );
document.write( "Erase used truth values:\r\n" );
document.write( "\r\n" );
document.write( "S ⊃ (K ∨ ∼ S) / K ⊃ S // S ≡ K\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   //   F \r\n" );
document.write( "F ⊃    T      /   T   //   T \r\n" );
document.write( "\r\n" );
document.write( "Place truth values under ⊃ \r\n" );
document.write( "\r\n" );
document.write( "S ⊃ (K ∨ ∼ S) / K ⊃ S // S ≡ K\r\n" );
document.write( "-------------------------------\r\n" );
document.write( "T T    T      /   T   //   T \r\n" );
document.write( "T F    F      /   T   //   F \r\n" );
document.write( "F T    T      /   F   //   F \r\n" );
document.write( "F T    T      /   T   //   T \r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Erase used truth values:\r\n" );
document.write( "\r\n" );
document.write( "S ⊃ (K ∨ ∼ S) / K ⊃ S // S ≡ K\r\n" );
document.write( "-------------------------------\r\n" );
document.write( "  T           /   T   //   T \r\n" );
document.write( "  F           /   T   //   F \r\n" );
document.write( "  T           /   F   //   F \r\n" );
document.write( "  T           /   T   //   T\r\n" );
document.write( "\r\n" );
document.write( "Place truth values under /, which is the same same \r\n" );
document.write( "as ∧ (and, conjunction)\r\n" );
document.write( "\r\n" );
document.write( "S ⊃ (K ∨ ∼ S) / K ⊃ S // S ≡ K\r\n" );
document.write( "-------------------------------\r\n" );
document.write( "  T           T   T   //   T \r\n" );
document.write( "  F           F   T   //   F \r\n" );
document.write( "  T           F   F   //   F \r\n" );
document.write( "  T           T   T   //   T\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Erase the used truth values\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "S ⊃ (K ∨ ∼ S) / K ⊃ S // S ≡ K\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( "\r\n" );
document.write( "Place truth values under the // which is the same as ⊃\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( " S ⊃ (K ∨ ∼ S) / K ⊃ S // S ≡ K\r\n" );
document.write( "-------------------------------\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( "               T        T   T\r\n" );
document.write( "\r\n" );
document.write( "Valid.  There are only T's under the //\r\n" );
document.write( "\r\n" );
document.write( "Edwin
\n" ); document.write( "
\n" );