Symbolic Logic 5E: 3.2, III, Second Half

Note: I cheated and used conditional proof (CP) for problems 27-29. I don’t think this is such an egregious offense, as CP is introduced a mere four pages later. The point of doing these problems pre-CP is to see how helpful CP is. So save yourself some time and trust me: CP makes solving proofs much, much easier!

16.

  1. A→(B→C)
  2. C→(D ∧ E) …Therefore, A→(B→D)
  3. ¬A ∨ (B→C) (1,IMP)
  4. ¬A ∨ (¬B ∨ C) (3,IMP)
  5. (¬A ∨ ¬B) ∧ (¬A ∨ C) (4,DIST)
  6. ¬A ∨ ¬B (5,SIMP)
  7. ¬(A ∧ B) (6,DeM)
  8. ¬(A ∧ B) ∨ D (7,ADD)
  9. (A ∧ B)→D (8,IMP)
  10. A→(B→D) (9,EXP)

17.

  1. E→F
  2. G→F …Therefore, (E ∨ G)→F
  3. ¬E ∨ F (1,IMP)
  4. ¬G ∨ F (2,IMP)
  5. (¬E ∨ F) ∧ (¬G ∨ F) (3,4,CONJ)
  6. F ∨ (¬E ∧ ¬G) (5,DIST)
  7. (¬E ∧ ¬G) ∨ F (6,COMM)
  8. ¬(¬E ∧ ¬G)→F (7,IMP)
  9. (E ∨ G)→F (8,DeM)

18.

  1. [(H ∧ I)→J] ∧ [¬K→(I ∧ ¬J)] …Therefore, H→K
  2. (H ∧ I)→J (1,SIMP)
  3. ¬(H ∧ I) ∨ J (2,IMP)
  4. (¬H ∨ ¬I) ∨ J (3,DeM)
  5. ¬H ∨(¬I ∨ J) (4,ASSOC)
  6. H→(¬I ∨ J) (5,IMP)
  7. ¬K→(I ∧ ¬J) (1,SIMP)
  8. ¬(I ∧ ¬J)→K (7,TRANS)
  9. (¬I ∨ J)→K (8,DeM)
  10. H→K (6,9,HS)

19.

  1. [L ∧ (M ∨ N)]→(M ∧ N) …Therefore, L→(M→N)
  2. L→[(M ∨ N)→(M ∧ N)] (1,EXP)
  3. L→[¬(M ∨ N) ∨ (M ∧ N)] (2,IMP)
  4. L→[(¬M ∧ ¬N) ∨ (M ∧ N)] (3,DeM)
  5. L→(M≡N) (4,EQUIV)
  6. L→[(M→N) ∧ (N→M)] (5,EQUIV)
  7. ¬L ∨ [(M→N) ∧ (N→M)] (6,IMP)
  8. [¬L ∨ (M→N)] ∧ [¬L ∨ (N→M)] (7,DIST)
  9. ¬L ∨ (M→N) (8,SIMP)
  10. L→(M→N) (9,IMP)

20. In back of book.

21.

  1. S→(T ∧U)
  2. (T ∨ U)→V …Therefore, S→V
  3. ¬S ∨ (T ∧U) (1,IMP)
  4. (¬S ∨ T) ∧ (¬S ∨ U) (3,DIST)
  5. ¬(T ∨ U) ∨ V (2,IMP)
  6. (¬T ∧ ¬U) ∨ V (5,DeM)
  7. (V ∨ ¬T) ∧ (V ∨ ¬U) (6,DIST)
  8. ¬S ∨ T (4,SIMP)
  9. S→T (8,IMP)
  10. V ∨ ¬T (7,SIMP)
  11. ¬T ∨ V(10,COMM)
  12. T→V (11,IMP)
  13. S→V (9,12,HS)

22.

  1. ¬W ∨ [(X→Y) ∧ (Z→Y)]
  2. W ∧ (X ∨ V) …Therefore, Y
  3. W (2,SIMP)
  4. (X→Y) ∧ (Z→Y) (3,1,DS)
  5. X ∨ V (2,SIMP)
  6. Y ∨ Y (5,4,CD)
  7. Y (6,TAUT)

23.

  1. (A ∨ B)→(C ∧ D)
  2. ¬A→(E→¬E)
  3. ¬C …Therefore, ¬E
  4. A ∨ (E→¬E) (2,IMP)
  5. A ∨ (¬E ∨ ¬E) (4,IMP)
  6. A ∨ ¬E (5,TAUT)
  7. ¬C ∨ ¬D (3,ADD)
  8. ¬(C ∧ D) (7,DeM)
  9. ¬(A ∨ B) (8,1,MT)
  10. ¬A ∧ ¬B (9,DeM)
  11. ¬A (10,IMP)
  12. ¬E (11,6,DS)

24.

  1. (F→G) ∧ (H→I)
  2. F ∨ H
  3. (F→¬I) ∧ (H→G) …Therefore, G≡¬I
  4. G ∨ I (2,1,CD)
  5. I ∨ G (4,COMM)
  6.  ¬I→G (5,IMP)
  7. ¬G ∨ ¬I (2,3,CD)
  8. G→¬I (7,IMP)
  9. (G→¬I) ∧ (¬I→G) (8,6,CONJ)
  10. G≡¬I (9,EQUIV)

25. In back of book.

26.

  1. Q ∨ (R ∧ S)
  2. (Q→T) ∧ (T→S) …Therefore, S
  3. Q→T (2,IMP)
  4. T→S (2,IMP)
  5. Q→S (3,4,HS)
  6. (Q ∨ R) ∧ (Q ∨ S) (1,DIST)
  7. Q ∨ S (6,SIMP)
  8. ¬S→¬Q (5,TRANS)
  9. ¬Q→S (7,IMP)
  10. ¬S→S (6,9,HS)
  11. S ∨ S (10,IMP)
  12. S (11,TAUT)

27.

  1. (U→V) ∧ (W→X) …Therefore, (U ∨ W)→(V ∨ X)
  2. U→V (CP)
  3. V ∨ X (2,1,CD)
  4. (U ∨ W)→(V ∨ X) (2-3,CP)

28.

  1. (Y→Z) ∧(A→B) …Therefore, (Y ∧ A)→(Z ∧ B)
  2. Y ∧ A (CP)
  3. Y→Z (1,SIMP)
  4. Y (2,SIMP)
  5. Z (4,3,MP)
  6. A→B (1,SIMP)
  7. A (2,SIMP)
  8. B (7,6,MP)
  9. Z ∧ B (5,8,CONJ)
  10. (Y ∧ A)→(Z ∧ B) (2-9,CP)

29.

  1. (C→D) ∧ (E→F)
  2. G→(C ∨ E) …Therefore, G→(D ∨ F)
  3. G (CP)
  4. C ∨ E (3,2,MP)
  5. D ∨ F (4,1,CD)
  6. G→(D ∨ F) (3-5,CP)

30.

  1. (H→I) ∧ (J→K)
  2. H ∨ J
  3. (H→¬K) ∧ (J→¬I)
  4. (I ∧ ¬K)→L
  5. K→(I ∨ M) …Therefore, L ∨ M
  6. I ∨ K (1,2,CD)
  7. ¬K ∨ ¬I (2,3,CD)
  8. ¬I→K (6,IMP)
  9. K→¬I (7,IMP)
  10. ¬I→¬I (8,9,HS)
  11. I ∨ ¬I (10,IMP)
  12. (¬K ∧ I)→L (4,COM)
  13. ¬K→(I→L) (12,EXP)
  14. I→¬K (9,TRANS)
  15. I→(I→L) (14,13,HS)
  16. (I ∧ I)→L (15,EXP)
  17. I→L (16,TAUT)
  18. ¬I→(I ∨ M) (8,5,HS)
  19. I ∨ (I ∨ M) (18,IMP)
  20. (I ∨ I) ∨ M (19,ASSOC)
  21. I ∨ M (20,TAUT)
  22. ¬I→M (21,IMP)
  23. (I→L) ∧ (¬I→M) (17,22,CONJ)
  24. L ∨ M (23,11,CD)
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