Suggested Problems for Proof Designer

  1. Hypotheses: A⊆B, A⊆C
    Conclusion: A⊆B∩C
  2. Hypotheses: A⊆B
    Conclusion: C∖B⊆C∖A
  3. Hypotheses: A∖B⊆C
    Conclusion: A∖C⊆B
  4. Hypotheses: none
    Conclusion: A∖(B∖C)⊆(A∖B)∪C
  5. Hypotheses: none
    Conclusion: A∖(B∩C)=(A∖B)∪(A∖C)
  6. Hypotheses: none
    Conclusion: A∩(B∪C)⊆(A∩B)∪C
  7. Hypotheses: none
    Conclusion: (A∪B)∖C⊆A∪(B∖C)
  8. Hypotheses: A∩(B∖C)=∅
    Conclusion: A∩B⊆C
  9. Hypotheses: A⊆B, A⊈C
    Conclusion: B⊈C
  10. Hypotheses: A⊆B, A∩C=∅
    Conclusion: A⊆B∖C
  11. Hypotheses: A⊆B∖C, A≠∅
    Conclusion: B⊈C
  12. Hypotheses: A∖B⊆C, A⊈C
    Conclusion: A∩B≠∅
  13. Hypotheses: A⊆B∖C
    Conclusion: A∩C=∅
  14. Hypotheses: none
    Conclusion: A∖C⊆(A∖B)∪(B∖C)
  15. Hypotheses: A∩C⊆B∩C, A∪C⊆B∪C
    Conclusion: A⊆B
  16. Hypotheses: none
    Conclusion: ∃!A∀B(A∪B=B)
  17. Hypotheses: none
    Conclusion: A⊆B↔𝒫(A)⊆𝒫(B)
  18. Hypotheses: none
    Conclusion: 𝒫(A∩B)=𝒫(A)∩(B)
  19. Hypotheses: none
    Conclusion: 𝒫(A)∪𝒫(B)⊆𝒫(A∪B)
  20. Hypotheses: 𝒫(A)∪𝒫(B)=𝒫(A∪B)
    Conclusion: A⊆B∨B⊆A
  21. Hypotheses: ∀x(x∈A→x⊆A)
    Conclusion: ∀x(x∈𝒫(A)→x⊆𝒫(A))
  22. Hypotheses: A∈F
    Conclusion: A⊆∪F
  23. Hypotheses: A∈F
    Conclusion: U∩∩F⊆A
  24. Hypotheses: F⊆G
    Conclusion: ∪F⊆∪G
  25. Hypotheses: F⊆G
    Conclusion: U∩∩G⊆U∩∩F
  26. Hypotheses: none
    Conclusion: ∪(F∪G)=(∪F)∪(∪G)
  27. Hypotheses: none
    Conclusion: ∪(F∩G)⊆(∪F)∩(∪G)
  28. Hypotheses: none
    Conclusion: U∩∩(F∪G)=(U∩∩F)∩(U∩∩G)
  29. Hypotheses: none
    Conclusion: A∩(∪F)=∪{A∩X|X∈F}
  30. Hypotheses: A⊆U
    Conclusion: A∪(U∩∩F)=U∩∩{A∪X|X∈F}
  31. Hypotheses: none
    Conclusion: U∖∪F=U∩∩{U∖X|X∈F}
  32. Hypotheses: A⊆U
    Conclusion: A∖(U∩∩F)=∪{A∖X|X∈F}
  33. Hypotheses: none
    Conclusion: ∪F∖∪G⊆∪(F∖G)
  34. Hypotheses: none
    Conclusion: ∪(F∖G)⊆∪F∖∪G→∪F∩∪G⊆∪(F∩G)
  35. Hypotheses: ∀A∈F∃B∈G(A∩B∈H)
    Conclusion: (∪F)∩∩G⊆∪H
  36. Hypotheses: ∀A∈F∃B∈G(A⊆B), ∃A∈F∀B∈G(B⊆A)
    Conclusion: F∩G≠∅
  37. Hypotheses: none
    Conclusion: F⊆𝒫(∪F)
  38. Hypotheses: none
    Conclusion: A=∪𝒫(A)
  39. Hypotheses: none
    Conclusion: U∩∩F∈𝒫(U)∩∩{𝒫(X)|X∈F}
  40. Hypotheses: none
    Conclusion: ∪{X∖A|X∈F}⊆∪{X∈F|X⊈A}
  41. Hypotheses: none
    Conclusion: (∪F)∩(∪G)=∅↔∀A∈F∀B∈G(A∩B=∅)
  42. Hypotheses: none
    Conclusion: ∪{𝒫(X)|X∈F}⊆𝒫(∪F)
  43. Hypotheses: none
    Conclusion: 𝒫(U)∩∩{𝒫(X)|X∈F}=𝒫(U∩∩F)
  44. Hypotheses: ∪{𝒫(X)|X∈F}=𝒫(∪F)
    Conclusion: ∃A∈F∀B∈F(B⊆A)
  45. Hypotheses: ∀F(∪F=A→A∈F)
    Conclusion: ∃x(A={x})
  46. Hypotheses: none
    Conclusion: 𝒫(A∖B)∖(𝒫(A)∖𝒫(B))={∅}
  47. Hypotheses: none
    Conclusion: A×(B∩C)=(A×B)∩(A×C)
  48. Hypotheses: none
    Conclusion: A×(B∪C)=(A×B)∪(A×C)
  49. Hypotheses: none
    Conclusion: (A△B)∩C=(A∩C)△(B∩C)
  50. Hypotheses: A△B⊆B
    Conclusion: A⊆B
  51. Hypotheses: none
    Conclusion: A△B⊆(A△C)∪(B△C)
  52. Hypotheses: none
    Conclusion: A△(A∩B)=A∖B
  53. Hypotheses: none
    Conclusion: A△(A∪B)=B∖A
  54. Hypotheses: none
    Conclusion: (A△B)△C=A△(B△C)
  55. Hypotheses: none
    Conclusion: A△A=∅
  56. Hypotheses: A△C=B△C
    Conclusion: A=B
  57. Hypotheses: none
    Conclusion: ∃!A∀B(A△B=B)
  58. Hypotheses: none
    Conclusion: ∀A∀B∃!C(A△C=B)
  59. Hypotheses: none
    Conclusion: ¬∃U∀A(A∈U)
  60. Hypotheses: none
    Conclusion: (R○S)−1=S−1○R−1
  61. Hypotheses: none
    Conclusion: (R○S)○T=R○(S○T)
  62. Hypotheses: S⊆T
    Conclusion: R○S⊆R○T
  63. Hypotheses: none
    Conclusion: (S∩T)○R⊆(S○R)∩(T○R)
  64. Hypotheses: none
    Conclusion: (S∪T)○R=(S○R)∪(T○R)
  65. Hypotheses: none
    Conclusion: (S○R)∖(T○R)⊆(S∖T)○R
  66. Hypotheses: none
    Conclusion: 𝒫(A∪B)=∪{{X∪Y|Y∈𝒫(B)}|X∈𝒫(A)}