document.write( "Question 1207830: Construct a formal proof of validity for the following arguments by means of Natural Deduction. You are NOT allowed to use Conditional Proof or Indirect Proof
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\n" ); document.write( "1. (A ⊃ B) • (C ⊃ D) // (A • C) ⊃ (B • D) \n" ); document.write( "
Algebra.Com's Answer #845971 by mccravyedwin(407)\"\" \"About 
You can put this solution on YOUR website!
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document.write( "I will assume your teacher will allow you to use \r\n" );
document.write( "Material implication (P ⊃ Q) ≡ (~P ∨ Q), which is\r\n" );
document.write( "easily proved by with a simple 4-line truth table.\r\n" );
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document.write( " 1. (A ⊃ B) • (C ⊃ D) // (A • C) ⊃ (B • D)\r\n" );
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document.write( " 2. (~A ∨ B) • (~C ∨ D)                   1, Material implication\r\n" );
document.write( " 3. ~A ∨ B                                2, Simplification\r\n" );
document.write( " 4. (~C ∨ D) • (~A ∨ B)                   2, Commutation\r\n" );
document.write( " 5. ~C ∨ D)                               4, Simplification\r\n" );
document.write( " 6. (~C ∨ D) ∨ ~A                         5, Addition \r\n" );
document.write( " 7. ~C ∨ (D ∨ ~A)                         6, Association\r\n" );
document.write( " 8. (D ∨ ~A) ∨ ~C                         7, Commutation\r\n" );
document.write( " 9. (~A ∨ D) ∨ ~C                         8, Commutation\r\n" );
document.write( "10. (~A ∨ B) ∨ ~C                         3, Addition\r\n" );
document.write( "11. [(~A ∨ B) ∨ ~C] • [(~A ∨ D) ∨ ~C]  10,9, Conjunction\r\n" );
document.write( "12. [(~A ∨ B) • (~A ∨ D)] ∨ ~C           11, Distribution                            \r\n" );
document.write( "13. [~A ∨ (B • D)] ∨ ~C                  12, Distribution\r\n" );
document.write( "14. ~A ∨ [(B • D) ∨ ~C]                  13, Association\r\n" );
document.write( "15. ~A ∨ [~C ∨ (B • D)]                  14, Commutation\r\n" );
document.write( "16. [~A ∨ ~C] ∨ (B • D)                  15, Association\r\n" );
document.write( "17. ~(A • C) ∨ (B • D)                   16, DeMorgan's law\r\n" );
document.write( "18. (A • C) ⊃ (B • D)                    17, Material implication    \r\n" );
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document.write( "If he or she will not allow you to use Material Implication, \r\n" );
document.write( "then re-post saying so.\r\n" );
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document.write( "Edwin
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