Relationships in Truth Statements

Exercise

Construct proofs for the following valid arguments. The first fifteen proofs can be complete in three or less additional lines. The next five proofs will be a bit longer. It is important to note that there is always more than one way to construct a proof. If your proof differs from the answer key, that doesn't mean it is wrong. #1

1. 
   
   1. A ⋅ B
      
   2. (A v C) ⊃ D /∴ A ⋅ D
      
2.  
   
   1. A
      
   2. B /∴ (A v C) ⋅ B
      
3. 
   
   1. D ⊃ E
   2. D ⋅ F /∴ E
      
4. 
   
   1. J ⊃ K
      
   2.  J /∴ K v L
      
5. 
   
   1. A v B
      
   2. ~A ⋅ ~C /∴ B
      
6. 
   
   1. A ⊃ B
      
   2. ~B ⋅ ~C /∴ ~A
      
7. 
   
   1. D ⊃ E
      
   2. (E ⊃ F) ⋅ (F⊃ D) /∴D ⊃ F
      
8. 
   
   1. (T ⊃ U) ⋅ (T ⊃ V)
   2. T /∴ U v V
      
9. 
   
   1. (E ⋅ F) v (G ⊃ H)
      
   2. I ⊃ G
      
   3. ~(E ⋅ F) /∴ I ⊃ H
      
10. 
    
    1. M ⊃ N
       
    2. O ⊃ P
       
    3. N ⊃ P
       
    4. (N ⊃ P) ⊃ (M v O) /∴N v P
       
11. 
    
    1. A v (B ⊃ A)
       
    2. ~A ⋅ C /∴ ~B
       
12. 
    
    1. (D v E) ⊃ (F ⋅ G)
       
    2. D /∴ F
       
13. 
    
    1. T ⊃ U
       
    2. V v ~U
       
    3. ~V ⋅ ~W /∴ ~T
       
14. 
    
    1. (A v B) ⊃ ~C
       
    2. C v D
       
    3. A /∴ D
       
15. 
    
    1. L v (M ⊃ N)
    2. ~L ⊃ (N ⊃ O)
       
    3. ~L /∴ M ⊃ O
       
16. 
    
    1. A ⊃ B
    2. A v (C ⋅ D)
       
    3. ~B ⋅ ~E /∴ C
       
17. 
    
    1. (F ⊃ G) ⋅ (H ⊃ I)
    2. J ⊃ K
    3. (F v J) ⋅ (H v L) /∴ G v K
       
18. 
    
    1. (E v F) ⊃ (G ⋅ H)
       
    2. (G v H) ⊃ I
       
    3. E /∴ I
       
19. 
    
    1. (N v O) ⊃ P
       
    2. (P v Q) ⊃ R
       
    3. Q v N
       
    4. ~Q /∴ R
       
20. 
    
    1. J ⊃ K
       
    2. K v L
       
    3. (L ⋅ ~J) ⊃ (M ⋅ ~J)
       
    4. ~K    /∴ M