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Logica en formele systemen 60 uitgewerkte afleidingen

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  • Course
  • Logica En Formele Systemen
  • Institution
  • Vrije Universiteit Brussel (VUB)
  • Book
  • Logica Voor Informatici 3Dr

Dit document bevat 60 volledig uitgewerkte afleidingen uit de propositielogica. Eindscore: 19/20, behaald op het examen van het vak "Logica en formele systemen" gegeven door professor Olga De Troyer. Vergeet zeker niet om een kijkje te nemen op het bestand met tips & tricks (en uitgewerkte voorb...

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  • September 18, 2021
  • 28
  • 2020/2021
  • Other
  • Unknown

Subjects

  • propositielogica
  • logica
  • afleidingen
  • deductie
  • natuurlijke deductie
  • gevolgtrekkingen
  • opgeloste oefeningen
  • uitgewerkte opgaven
  • logica voor informatici
  • logica en formele systemen

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  • Vrije Universiteit Brussel (VUB)
  • Computerwetenschappen
  • Logica En Formele Systemen
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Natuurlijke deductie (propositielogica): 60 volledig uitgewerkte afleidingen

1. 𝑝 → (𝑞 → 𝑟) ⊢ (𝑝 → 𝑞) → (𝑝 → 𝑟)
2. 𝑝 → (𝑞 → 𝑟) ⊢ 𝑞 → (𝑝 → 𝑟)
3. 𝑝 → (𝑞 ∧ 𝑟) ⊢ 𝑝 → 𝑞
4. 𝑝 → (𝑞 → 𝑟) ⊢ (𝑝 ∧ 𝑞) → 𝑟
5. (𝑝 ∧ 𝑞) → 𝑟 ⊢ 𝑝 → (𝑞 → 𝑟)
6. 𝑝 → (𝑞 ∧ 𝑟) ⊢ 𝑝 → 𝑟
7. 𝑝 ∨ (𝑝 ∧ 𝑞) ⊢ 𝑝 (Boolese axioma's voor absorptie 1.1)
8. 𝑝 ⊢ 𝑝 ∨ (𝑝 ∧ 𝑞) (Boolese axioma's voor absorptie 1.2)

9. (¬𝑝 ∧ ¬𝑞) ⊢ ¬(𝑝 ∨ 𝑞)
10. ¬(𝑝 ∨ 𝑞) ⊢ (¬𝑝 ∧ ¬𝑞)
11. ⊢ ¬¬𝑝 → 𝑝
12. ⊢ 𝑝 → ¬¬𝑝
13. ¬𝑝 ∨ 𝑞 ⊢ 𝑝 → 𝑞
14. 𝑝 → 𝑞 ⊢ ¬𝑝 ∨ 𝑞
15. 𝑝 → 𝑞 ⊢ ¬𝑞 → ¬𝑝
16. ¬𝑞 → ¬𝑝 ⊢ 𝑝 → 𝑞
17. 𝑝 ∧ (𝑝 ∨ 𝑞) ⊢ 𝑝 (Boolese axioma's voor absorptie 2.1)
18. 𝑝 ⊢ 𝑝 ∧ (𝑝 ∨ 𝑞) (Boolese axioma's voor absorptie 2.2)
19. {𝑝 → 𝑞, 𝑝 → 𝑟} ⊢ 𝑝 → (𝑞 ∧ 𝑟)
20. 𝑝 ∨ (𝑞 ∧ 𝑟) ⊢ (𝑝 ∨ 𝑞) ∧ (𝑝 ∨ 𝑟) (Boolese axioma's voor distributiviteit 1.1)
21. (𝑝 ∨ 𝑞) ∧ (𝑝 ∨ 𝑟) ⊢ 𝑝 ∨ (𝑞 ∧ 𝑟) (Boolese axioma's voor distributiviteit 1.2)

22. {𝑝 ∧ ¬𝑞, 𝑟 → 𝑞} ⊢ ¬𝑟
23. ¬𝑝 → ¬𝑞 ⊢ 𝑞 → 𝑝
24. (𝑝 ∨ 𝑞) → 𝑟 ⊢ 𝑝 → 𝑟
25. {𝑝 → (𝑞 ∧ 𝑟), 𝑞 → ¬𝑠} ⊢ 𝑠 → ¬𝑝
26. ((𝑝 ∨ 𝑞) ∨ 𝑟) ⊢ (𝑝 ∨ (𝑞 ∨ 𝑟))
27. ((𝑝 ∧ ¬𝑝) ∨ 𝑞) ⊢ 𝑞
28. {𝑝 ∧ 𝑞, 𝑝 → 𝑟, 𝑞 → 𝑠} ⊢ 𝑟 ∧ 𝑠
29. {𝑝, 𝑞 → (𝑝 → 𝑟)} ⊢ 𝑞 → 𝑟
30. {𝑝 → 𝑞, 𝑞 → 𝑟} ⊢ 𝑝 → 𝑟

,31. {𝑝 → 𝑞, ¬𝑞} ⊢ ¬𝑝
32. (𝑝 ∨ 𝑞) → 𝑟 ⊢ (𝑝 → 𝑟) ∧ (𝑞 → 𝑟)
33. (𝑝 → 𝑟) ∧ (𝑞 → 𝑟) ⊢ (𝑝 ∨ 𝑞) → 𝑟
34. 𝑝 ∧ (𝑞 ∨ 𝑟) ⊢ (𝑝 ∧ 𝑞) ∨ (𝑝 ∧ 𝑟) (Boolese axioma's voor distributiviteit 2.1)

35. (𝑝 ∧ 𝑞) ∨ (𝑝 ∧ 𝑟) ⊢ 𝑝 ∧ (𝑞 ∨ 𝑟) (Boolese axioma's voor distributiviteit 2.2)

36. {𝑝 ∨ 𝑟, 𝑞 → 𝑟} ⊢ (𝑝 → 𝑞) → 𝑟
37. ⊢ 𝑝 ∨ ¬𝑝
38. 𝑝 → ¬𝑞 ⊢ 𝑞 → ¬𝑝
39. {𝑝 ∨ 𝑞, ¬𝑞} ⊢ ¬(𝑝 → 𝑞)
40. ⊢ ¬𝑝 → (𝑝 → 𝑞)
41. {¬𝑝, ¬𝑞} ⊢ ¬((𝑝 → 𝑞) → 𝑞)
42. ¬𝑝 → 𝑞 ⊢ 𝑝 ∨ 𝑞
43. ⊢ (¬𝑝 ∧ ¬𝑞) → ¬(𝑝 ∨ 𝑞) (Wet van De Morgan 1.1)
44. ⊢ ¬(𝑝 ∨ 𝑞) → (¬𝑝 ∧ ¬𝑞) (Wet van De Morgan 1.2)

45. ⊢ ¬(𝑝 ∧ 𝑞) → (¬𝑝 ∨ ¬𝑞) (Wet van De Morgan 2.1)
46. ⊢ (¬𝑝 ∨ ¬𝑞) → ¬(𝑝 ∧ 𝑞) (Wet van De Morgan 2.2)

47. 𝑝 ∧ (𝑞 ∨ 𝑟) ⊢ (𝑝 ∧ 𝑞) ∨ (𝑝 ∧ 𝑟)(lineair)

48. (𝑝 ∧ 𝑞) ∨ (𝑝 ∧ 𝑟) ⊢ 𝑝 ∧ (𝑞 ∨ 𝑟)(lineair)
49. 𝑝 ∨ (𝑝 ∧ 𝑞) ⊢ 𝑝 (lineair)

50. 𝑝 ⊢ 𝑝 ∨ (𝑝 ∧ 𝑞) (lineair)
51. 𝑝 ∧ (𝑝 ∨ 𝑞) ⊢ 𝑝 (lineair)

52. 𝑝 ⊢ 𝑝 ∧ (𝑝 ∨ 𝑞) (lineair)
53. (𝑝 ∨ 𝑞) ∧ (𝑝 ∨ 𝑟) ⊢ 𝑝 ∨ (𝑞 ∧ 𝑟) (lineair)
54. 𝑝 ∨ (𝑞 ∧ 𝑟) ⊢ (𝑝 ∨ 𝑞) ∧ (𝑝 ∨ 𝑟) (lineair)
55. 𝑝 ∧ (𝑞 ∧ 𝑟) ⊢ 𝑟 ∧ 𝑝 (lineair)
56. {𝑝 → (𝑞 ∧ 𝑟), 𝑞 → 𝑠, 𝑝} ⊢ 𝑠 (lineair)

57. 𝑝 → (𝑝 → 𝑞) ⊢ 𝑝 → 𝑞 (lineair)

58. {(𝑝 ∧ ¬𝑞) → 𝑟, 𝑝} ⊢ ¬𝑞 → (𝑟 ∨ 𝑠) (lineair)
59. (𝑝 ∨ 𝑞) → 𝑟 ⊢ 𝑞 → 𝑟 (lineair)
60. {𝑝 ∨ 𝑞, 𝑝 → 𝑟, 𝑞 → 𝑠} ⊢ 𝑟 ∨ 𝑠 (lineair)

, 1. 𝑝 → (𝑞 → 𝑟) ⊢ (𝑝 → 𝑞) → (𝑝 → 𝑟)




2. 𝑝 → (𝑞 → 𝑟) ⊢ 𝑞 → (𝑝 → 𝑟)

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