Relationships in Truth Statements

Answers

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