Symbolic Logic 5E: 3.1, II

“Each of the following is a formal proof of validity for the indicated argument. State the ‘justification’ for each line that is not a premiss”

Note: The justification for each premise is placed at the end of each line and is  colored green.

1.

  1. (A ∧ B)→[A→(D ∧ E)]
  2. (A ∧ B) ∧ C …Therefore, D ∨ E
  3. A ∧ B (2,SIMP)
  4. A→(D ∨ E) (3,1,MP)
  5. A (3,SIMP)
  6. D ∧ E (4,MP)
  7. D (6,SIMP)
  8. D ∨ E (7,ADD)

2.

  1. F ∨ (G ∨ H)
  2. (G→I) ∧ ( H→J)
  3. (I ∨ J)→(F ∨ H)
  4. ¬F …Therefore, H
  5. G ∨ H (4,1,DS)
  6. I ∨ J (5,2,DS)
  7. F ∨ H (6,3,MP)
  8. H (4,7DS)

3.

  1. K→L
  2. M→N
  3. (O→N) ∧ (P→L)
  4. (¬N ∨ ¬L) ∧ (¬M ∨ ¬K) …Therefore, (¬O ∨ ¬P) ∧ (¬M ∨ ¬K)
  5. (K→L) ∧ (M→N) (1,2,CONJ)
  6. (¬N ∨ ¬L) (4,SIMP)
  7. (¬M ∨ ¬K) (4,SIMP)
  8. (¬O ∨ ¬P) (8,3,DD)
  9. (¬O ∨ ¬P) ∧ (¬M ∨ ¬K) (8,7,CONJ)

4.

  1. Q→(R→S)
  2. (R→S)→T
  3. (S ∧ U)→¬V
  4. ¬V→(R≡¬W)
  5. ¬T ∨ ¬(R≡¬W) …Therefore, ¬Q ∨ ¬(S ∧ U)
  6. Q→T (1,2,HS)
  7. (S ∧ U)→( R≡¬W) (3,4,HS)
  8. (Q→T) ∧ [(S ∧ U)→( R≡¬W)] (6,7,CONJ)
  9. ¬Q ∨ ¬(S ∧ U) (8,5,DD)

5.

  1. (¬X ∨ ¬Y)→[A→(P ∧ ¬Q)]
  2. (¬X ∧ ¬R)→[(P ∧ ¬Q)→Z]
  3. (¬X ∧ ¬R) ∧ (¬Z ∨ A) …Therefore, A→Z
  4. (¬X ∧ ¬R) (3,SIMP)
  5. (P ∧ ¬Q)→Z (4,2,MP)
  6. ¬X (4,SIMP)
  7. ¬X ∨ ¬Y (6,ADD)
  8. A→(P ∧ ¬Q) (7,1,ADD)
  9. A→Z (8,5,HS)

6.

  1. A→B
  2. C→D
  3. ¬B ∨ ¬D
  4. ¬¬A
  5. (E ∧ F)→C …Therefore, ¬(E ∧ F)
  6. (A→B) ∧ (C→D) (1,2,CONJ)
  7. ¬A ∨ ¬C (3,6,DD)
  8. ¬C (4,7,DS)
  9. ¬(E ∧ F) (8,5,MT)

7.

  1. (G→H)→(I≡J)
  2. K ∨ ¬(L→M)
  3. (G→H) ∨ ¬K
  4. N→( L→M)
  5. ¬(I≡J) …Therefore, ¬N
  6. ¬(G→H) (5,1,MT)
  7. ¬K (6,3,DS)
  8. ¬(L→M) (7,2,DS)
  9. ¬N(8,4,MT)

8.

  1. (O→¬P) ∧ (¬Q→R)
  2. (S→T) ∧ (¬U→¬V)
  3. (¬P→S) ∧ (R→¬U)
  4. (T ∨ ¬V)→(W ∧ X)
  5. O ∨ ¬Q …Therefore, W ∧ X
  6. ¬P ∨ R(5,1,CD)
  7. S ∨ ¬U(6,3,CD)
  8. T ∨ ¬U(7,2,CD)
  9. W ∧ X (8,4,MP)

9.

  1. [(A ∨ ¬B) ∨ C]→[D→(E≡F)]
  2. (A ∨ ¬B)→ [(F≡G)→H]
  3. A→[(E≡F)→(F≡G)]
  4. A …Therefore, D→H
  5. A ∨ ¬B (4,ADD)
  6. (A ∨ ¬B) ∨ C (5,ADD)
  7. D→(E≡F) (6,1,MP)
  8. (E≡F)→(F≡G) (4,3,MP)
  9. D→(F≡G) (7,8,HS)
  10. (F≡G)→H (5,2,MP)
  11. D→H(9,10,HS)

10.

  1. H→(I→J)
  2. K→(I→J)
  3. (¬H ∧ ¬K)→(¬L ∧ ¬M)
  4. (¬L→¬N) ∧ (¬M→¬O)
  5. (P→N) ∧ (Q→O)
  6. ¬(I→J) …Therefore, ¬P ∨ ¬Q
  7. ¬H (6,1,MT)
  8. ¬K (6,2,MT)
  9. ¬H ∧ ¬K (8,7,CONJ)
  10. ¬L ∧ ¬M (9,3,MP)
  11. ¬N ∨ ¬O (10,4,CD)
  12. ¬P ∨ ¬Q (11,5,DD)
Advertisements

Leave a comment

Filed under Solution Sets

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s