Q350: Inference Rules and Proofs - Handout 2: Sample Proofs for the Arguments in Handout 1

These are sample answers to the proofs in Handout 1. If your answer is different, it may be correct also. Check with one of the instructors to be sure.
  1. 1. |_ Cube(a) ^ Cube(b)    
    2. | Cube(a)     ^ Elim: 1
    3. | Cube(a) v Cube(c)     v Intro: 2

     
     
  2. 1. |_ Cube(a) ^ Cube(b)    
    2. | Cube(a)     ^ Elim: 1
    3. | Cube(b)     ^ Elim: 1
    4. | Cube(a) v Cube(c)     v Intro: 2
    5. | (Cube(a) v Cube(c)) ^ Cube(b)     ^ Intro: 4, 3

     
     
  3. 1. | Cube(a) → Cube(b)    
    2. |_ Cube(a) ^ Cube(c)    
    3. | Cube(a)     ^ Elim: 2
    4. | Cube(b)     → Elim: 1, 3
    5. | Cube(c)     ^ Elim: 2
    6. | Cube(c) ^ Cube(b)     ^ Intro: 5, 4

     
     
  4. 1. | Cube(a) ↔ Cube(b)    
    2. | Cube(b)    
    3. |_ Cube(b) → Cube(c)    
    4. | Cube(a)     ↔ Elim: 1, 2
    5. | Cube(c)     → Elim: 3, 2
    6. | Cube(a) ^ Cube(c)     ^ Intro: 4, 5

     
     
  5. 1. | (P v Q) ^ R    
    2. | R → S    
    3. |_ S → T    
    4. | P v Q     ^ Elim: 1
    5. | R     ^ Elim: 1
    6. | S     → Elim: 2, 5
    7. | T     → Elim: 3, 6
    8. | T ^ (P v Q)     ^ Intro: 7, 4

     
     
  6. 1. | P ^ Q ^ R    
    2. | (Q v S) → T    
    3. |_ T → U    
    4. | Q     ^ Elim: 1
    5. | Q v S     v Intro: 4
    6. | T     → Elim: 2, 5
    7. | U     → Elim: 3, 6
    8. | U v V     v Intro: 7

     
     
  7. 1. | P    
    2. | P ↔ (Q v R)    
    3. | (Q v R) → S    
    4. |_ S → T    
    5. | Q v R     ↔ Elim: 2, 1
    6. | S     → Elim: 3, 5
    7. | T     → Elim: 4, 6
    8. | P ^ S ^ T     ^ Intro: 1, 6, 7

     
     
  8. 1. | P → W    
    2. | (Q v R) ↔ (S ^ T)    
    3. | Q    
    4. | S → V    
    5. |_ V → P    
    6. | Q v R     v Intro: 3
    7. | S ^ T     ↔ Elim: 2, 6
    8. | S     ^ Elim: 7
    9. | V     → Elim: 4, 8
    10. | P     → Elim: 5, 9
    11. | W     → Elim: 1, 10