SOLUTION: Use natural deduction to derive the conclusion in each problem. Use natural deduction to prove the following logical truth: (P ⊃ Q) ≡ [P ⊃ (Q ∨ ∼P)]

1.  (P --> Q) ==  P --> (Q v ~P) 



// Note to prove A == B, need to show A --> B and  B --> A

// Also note I'm using simplified notation as it is easier to enter



2. :: P                                Conditional Proof (CP) assumption #1

3. :: Q                                2,1   Modus Ponens (MP)

4. :: Q v ~P                           3     Addition (ADD) 

5. :: P --> (Q v ~P)                   2-4  CP

6. :: (P --> Q) --> (P --> (Q v ~P))   1-5  CP



// Now go the other way

7. :: P --> (Q v ~P)                   CP assumption #2

8. :: P                                CP assumption #3

9. :: (Q v ~P)                         8,7 MP

10.:: P --> Q                          9   Material Implication (MI)

11.:: (P --> (Q v ~P)) -->  (P --> Q)  7-10  CP

12.:: (P --> Q) == (P --> (Q v ~P))    6,11  Material Equivalence (ME)

13.(P --> Q) == (P --> (Q v ~P))       2-12  CP



There may be shorter ways to go, but this is what came to mind.