Declared Event Families and Typed Occurrences
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This question bank has corresponding bilingual step-by-step derivations. Source candidate and English ai_candidate states are retained. AI review, finite computation and Lean formal proof are labelled separately.
batch-20261009-advance-20261013-r06-event-foundations · 2026-10-09 · 100 questions
This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.
1. Declared Event and Relative Complement Partition
S=Fin(1)={0,…,0}; the event family is explicitly F=P(S). E=[]⊆S. Give (increasing E, increasing S\E, n).
View answer and explanation
([], [0], 1)
Take the complement inside the explicitly declared full event family. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is ([], [0], 1).
Reviewed step-by-step derivation
Event · qf-density13-event-complement-partition · candidate
2. Declared Event and Relative Complement Partition
S=Fin(1)={0,…,0}; the event family is explicitly F=P(S). E=[0]⊆S. Give (increasing E, increasing S\E, n).
View answer and explanation
([0], [], 1)
Take the complement inside the explicitly declared full event family. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is ([0], [], 1).
Reviewed step-by-step derivation
Event · qf-density13-event-complement-partition · candidate
3. Declared Event and Relative Complement Partition
S=Fin(5)={0,…,4}; the event family is explicitly F=P(S). E=[0]⊆S. Give (increasing E, increasing S\E, n).
View answer and explanation
([0], [1, 2, 3, 4], 5)
Take the complement inside the explicitly declared full event family. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is ([0], [1, 2, 3, 4], 5).
Reviewed step-by-step derivation
Event · qf-density13-event-complement-partition · candidate
4. Declared Event and Relative Complement Partition
S=Fin(5)={0,…,4}; the event family is explicitly F=P(S). E=[1]⊆S. Give (increasing E, increasing S\E, n).
View answer and explanation
([1], [0, 2, 3, 4], 5)
Take the complement inside the explicitly declared full event family. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is ([1], [0, 2, 3, 4], 5).
Reviewed step-by-step derivation
Event · qf-density13-event-complement-partition · candidate
5. Declared Event and Relative Complement Partition
S=Fin(5)={0,…,4}; the event family is explicitly F=P(S). E=[0, 1]⊆S. Give (increasing E, increasing S\E, n).
View answer and explanation
([0, 1], [2, 3, 4], 5)
Take the complement inside the explicitly declared full event family. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is ([0, 1], [2, 3, 4], 5).
Reviewed step-by-step derivation
Event · qf-density13-event-complement-partition · candidate
6. Declared Event and Relative Complement Partition
S=Fin(5)={0,…,4}; the event family is explicitly F=P(S). E=[2]⊆S. Give (increasing E, increasing S\E, n).
View answer and explanation
([2], [0, 1, 3, 4], 5)
Take the complement inside the explicitly declared full event family. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is ([2], [0, 1, 3, 4], 5).
Reviewed step-by-step derivation
Event · qf-density13-event-complement-partition · candidate
7. Declared Event and Relative Complement Partition
S=Fin(5)={0,…,4}; the event family is explicitly F=P(S). E=[0, 2]⊆S. Give (increasing E, increasing S\E, n).
View answer and explanation
([0, 2], [1, 3, 4], 5)
Take the complement inside the explicitly declared full event family. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is ([0, 2], [1, 3, 4], 5).
Reviewed step-by-step derivation
Event · qf-density13-event-complement-partition · candidate
8. Declared Event and Relative Complement Partition
S=Fin(5)={0,…,4}; the event family is explicitly F=P(S). E=[1, 2]⊆S. Give (increasing E, increasing S\E, n).
View answer and explanation
([1, 2], [0, 3, 4], 5)
Take the complement inside the explicitly declared full event family. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is ([1, 2], [0, 3, 4], 5).
Reviewed step-by-step derivation
Event · qf-density13-event-complement-partition · candidate
9. Declared Event and Relative Complement Partition
S=Fin(5)={0,…,4}; the event family is explicitly F=P(S). E=[0, 1, 2]⊆S. Give (increasing E, increasing S\E, n).
View answer and explanation
([0, 1, 2], [3, 4], 5)
Take the complement inside the explicitly declared full event family. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is ([0, 1, 2], [3, 4], 5).
Reviewed step-by-step derivation
Event · qf-density13-event-complement-partition · candidate
10. Declared Event and Relative Complement Partition
S=Fin(5)={0,…,4}; the event family is explicitly F=P(S). E=[3]⊆S. Give (increasing E, increasing S\E, n).
View answer and explanation
([3], [0, 1, 2, 4], 5)
Take the complement inside the explicitly declared full event family. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is ([3], [0, 1, 2, 4], 5).
Reviewed step-by-step derivation
Event · qf-density13-event-complement-partition · candidate
11. Declared Event and Relative Complement Partition
S=Fin(5)={0,…,4}; the event family is explicitly F=P(S). E=[0, 3]⊆S. Give (increasing E, increasing S\E, n).
View answer and explanation
([0, 3], [1, 2, 4], 5)
Take the complement inside the explicitly declared full event family. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is ([0, 3], [1, 2, 4], 5).
Reviewed step-by-step derivation
Event · qf-density13-event-complement-partition · candidate
12. Declared Event and Relative Complement Partition
S=Fin(5)={0,…,4}; the event family is explicitly F=P(S). E=[1, 3]⊆S. Give (increasing E, increasing S\E, n).
View answer and explanation
([1, 3], [0, 2, 4], 5)
Take the complement inside the explicitly declared full event family. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is ([1, 3], [0, 2, 4], 5).
Reviewed step-by-step derivation
Event · qf-density13-event-complement-partition · candidate
13. Declared Event and Relative Complement Partition
S=Fin(5)={0,…,4}; the event family is explicitly F=P(S). E=[0, 1, 3]⊆S. Give (increasing E, increasing S\E, n).
View answer and explanation
([0, 1, 3], [2, 4], 5)
Take the complement inside the explicitly declared full event family. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is ([0, 1, 3], [2, 4], 5).
Reviewed step-by-step derivation
Event · qf-density13-event-complement-partition · candidate
14. Declared Event and Relative Complement Partition
S=Fin(5)={0,…,4}; the event family is explicitly F=P(S). E=[2, 3]⊆S. Give (increasing E, increasing S\E, n).
View answer and explanation
([2, 3], [0, 1, 4], 5)
Take the complement inside the explicitly declared full event family. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is ([2, 3], [0, 1, 4], 5).
Reviewed step-by-step derivation
Event · qf-density13-event-complement-partition · candidate
15. Declared Event and Relative Complement Partition
S=Fin(5)={0,…,4}; the event family is explicitly F=P(S). E=[0, 2, 3]⊆S. Give (increasing E, increasing S\E, n).
View answer and explanation
([0, 2, 3], [1, 4], 5)
Take the complement inside the explicitly declared full event family. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is ([0, 2, 3], [1, 4], 5).
Reviewed step-by-step derivation
Event · qf-density13-event-complement-partition · candidate
16. Declared Event and Relative Complement Partition
S=Fin(5)={0,…,4}; the event family is explicitly F=P(S). E=[1, 2, 3]⊆S. Give (increasing E, increasing S\E, n).
View answer and explanation
([1, 2, 3], [0, 4], 5)
Take the complement inside the explicitly declared full event family. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is ([1, 2, 3], [0, 4], 5).
Reviewed step-by-step derivation
Event · qf-density13-event-complement-partition · candidate
17. Declared Event and Relative Complement Partition
S=Fin(5)={0,…,4}; the event family is explicitly F=P(S). E=[0, 1, 2, 3]⊆S. Give (increasing E, increasing S\E, n).
View answer and explanation
([0, 1, 2, 3], [4], 5)
Take the complement inside the explicitly declared full event family. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is ([0, 1, 2, 3], [4], 5).
Reviewed step-by-step derivation
Event · qf-density13-event-complement-partition · candidate
18. Declared Event and Relative Complement Partition
S=Fin(5)={0,…,4}; the event family is explicitly F=P(S). E=[4]⊆S. Give (increasing E, increasing S\E, n).
View answer and explanation
([4], [0, 1, 2, 3], 5)
Take the complement inside the explicitly declared full event family. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is ([4], [0, 1, 2, 3], 5).
Reviewed step-by-step derivation
Event · qf-density13-event-complement-partition · candidate
19. Declared Event and Relative Complement Partition
S=Fin(5)={0,…,4}; the event family is explicitly F=P(S). E=[0, 4]⊆S. Give (increasing E, increasing S\E, n).
View answer and explanation
([0, 4], [1, 2, 3], 5)
Take the complement inside the explicitly declared full event family. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is ([0, 4], [1, 2, 3], 5).
Reviewed step-by-step derivation
Event · qf-density13-event-complement-partition · candidate
20. Declared Event and Relative Complement Partition
S=Fin(5)={0,…,4}; the event family is explicitly F=P(S). E=[1, 4]⊆S. Give (increasing E, increasing S\E, n).
View answer and explanation
([1, 4], [0, 2, 3], 5)
Take the complement inside the explicitly declared full event family. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is ([1, 4], [0, 2, 3], 5).
Reviewed step-by-step derivation
Event · qf-density13-event-complement-partition · candidate
21. Declared Event and Relative Complement Partition
S=Fin(5)={0,…,4}; the event family is explicitly F=P(S). E=[0, 1, 4]⊆S. Give (increasing E, increasing S\E, n).
View answer and explanation
([0, 1, 4], [2, 3], 5)
Take the complement inside the explicitly declared full event family. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is ([0, 1, 4], [2, 3], 5).
Reviewed step-by-step derivation
Event · qf-density13-event-complement-partition · candidate
22. Declared Event and Relative Complement Partition
S=Fin(5)={0,…,4}; the event family is explicitly F=P(S). E=[2, 4]⊆S. Give (increasing E, increasing S\E, n).
View answer and explanation
([2, 4], [0, 1, 3], 5)
Take the complement inside the explicitly declared full event family. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is ([2, 4], [0, 1, 3], 5).
Reviewed step-by-step derivation
Event · qf-density13-event-complement-partition · candidate
23. Declared Event and Relative Complement Partition
S=Fin(5)={0,…,4}; the event family is explicitly F=P(S). E=[0, 2, 4]⊆S. Give (increasing E, increasing S\E, n).
View answer and explanation
([0, 2, 4], [1, 3], 5)
Take the complement inside the explicitly declared full event family. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is ([0, 2, 4], [1, 3], 5).
Reviewed step-by-step derivation
Event · qf-density13-event-complement-partition · candidate
24. Declared Event and Relative Complement Partition
S=Fin(5)={0,…,4}; the event family is explicitly F=P(S). E=[1, 2, 4]⊆S. Give (increasing E, increasing S\E, n).
View answer and explanation
([1, 2, 4], [0, 3], 5)
Take the complement inside the explicitly declared full event family. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is ([1, 2, 4], [0, 3], 5).
Reviewed step-by-step derivation
Event · qf-density13-event-complement-partition · candidate
25. Declared Event and Relative Complement Partition
S=Fin(5)={0,…,4}; the event family is explicitly F=P(S). E=[0, 1, 2, 4]⊆S. Give (increasing E, increasing S\E, n).
View answer and explanation
([0, 1, 2, 4], [3], 5)
Take the complement inside the explicitly declared full event family. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is ([0, 1, 2, 4], [3], 5).
Reviewed step-by-step derivation
Event · qf-density13-event-complement-partition · candidate
26. Event Union, Intersection, and Outside Complement
S=Fin(1)={0,…,0}; the event family is explicitly F=P(S). A=[], B=[] are subsets of S. Give (A∪B, A∩B, S\(A∪B)), each deduplicated and increasing.
View answer and explanation
([], [], [0])
Enumerate all three sets in the same sample space without confusing an outcome with an event. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is ([], [], [0]).
Reviewed step-by-step derivation
Event · qf-density13-event-boolean-algebra-pair · candidate
27. Event Union, Intersection, and Outside Complement
S=Fin(1)={0,…,0}; the event family is explicitly F=P(S). A=[0], B=[0] are subsets of S. Give (A∪B, A∩B, S\(A∪B)), each deduplicated and increasing.
View answer and explanation
([0], [0], [])
Enumerate all three sets in the same sample space without confusing an outcome with an event. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is ([0], [0], []).
Reviewed step-by-step derivation
Event · qf-density13-event-boolean-algebra-pair · candidate
28. Event Union, Intersection, and Outside Complement
S=Fin(5)={0,…,4}; the event family is explicitly F=P(S). A=[0], B=[] are subsets of S. Give (A∪B, A∩B, S\(A∪B)), each deduplicated and increasing.
View answer and explanation
([0], [], [1, 2, 3, 4])
Enumerate all three sets in the same sample space without confusing an outcome with an event. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is ([0], [], [1, 2, 3, 4]).
Reviewed step-by-step derivation
Event · qf-density13-event-boolean-algebra-pair · candidate
29. Event Union, Intersection, and Outside Complement
S=Fin(5)={0,…,4}; the event family is explicitly F=P(S). A=[1], B=[1] are subsets of S. Give (A∪B, A∩B, S\(A∪B)), each deduplicated and increasing.
View answer and explanation
([1], [1], [0, 2, 3, 4])
Enumerate all three sets in the same sample space without confusing an outcome with an event. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is ([1], [1], [0, 2, 3, 4]).
Reviewed step-by-step derivation
Event · qf-density13-event-boolean-algebra-pair · candidate
30. Event Union, Intersection, and Outside Complement
S=Fin(5)={0,…,4}; the event family is explicitly F=P(S). A=[0, 1], B=[] are subsets of S. Give (A∪B, A∩B, S\(A∪B)), each deduplicated and increasing.
View answer and explanation
([0, 1], [], [2, 3, 4])
Enumerate all three sets in the same sample space without confusing an outcome with an event. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is ([0, 1], [], [2, 3, 4]).
Reviewed step-by-step derivation
Event · qf-density13-event-boolean-algebra-pair · candidate
31. Event Union, Intersection, and Outside Complement
S=Fin(5)={0,…,4}; the event family is explicitly F=P(S). A=[2], B=[2] are subsets of S. Give (A∪B, A∩B, S\(A∪B)), each deduplicated and increasing.
View answer and explanation
([2], [2], [0, 1, 3, 4])
Enumerate all three sets in the same sample space without confusing an outcome with an event. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is ([2], [2], [0, 1, 3, 4]).
Reviewed step-by-step derivation
Event · qf-density13-event-boolean-algebra-pair · candidate
32. Event Union, Intersection, and Outside Complement
S=Fin(5)={0,…,4}; the event family is explicitly F=P(S). A=[0, 2], B=[] are subsets of S. Give (A∪B, A∩B, S\(A∪B)), each deduplicated and increasing.
View answer and explanation
([0, 2], [], [1, 3, 4])
Enumerate all three sets in the same sample space without confusing an outcome with an event. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is ([0, 2], [], [1, 3, 4]).
Reviewed step-by-step derivation
Event · qf-density13-event-boolean-algebra-pair · candidate
33. Event Union, Intersection, and Outside Complement
S=Fin(5)={0,…,4}; the event family is explicitly F=P(S). A=[1, 2], B=[1] are subsets of S. Give (A∪B, A∩B, S\(A∪B)), each deduplicated and increasing.
View answer and explanation
([1, 2], [1], [0, 3, 4])
Enumerate all three sets in the same sample space without confusing an outcome with an event. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is ([1, 2], [1], [0, 3, 4]).
Reviewed step-by-step derivation
Event · qf-density13-event-boolean-algebra-pair · candidate
34. Event Union, Intersection, and Outside Complement
S=Fin(5)={0,…,4}; the event family is explicitly F=P(S). A=[0, 1, 2], B=[] are subsets of S. Give (A∪B, A∩B, S\(A∪B)), each deduplicated and increasing.
View answer and explanation
([0, 1, 2], [], [3, 4])
Enumerate all three sets in the same sample space without confusing an outcome with an event. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is ([0, 1, 2], [], [3, 4]).
Reviewed step-by-step derivation
Event · qf-density13-event-boolean-algebra-pair · candidate
35. Event Union, Intersection, and Outside Complement
S=Fin(5)={0,…,4}; the event family is explicitly F=P(S). A=[3], B=[3] are subsets of S. Give (A∪B, A∩B, S\(A∪B)), each deduplicated and increasing.
View answer and explanation
([3], [3], [0, 1, 2, 4])
Enumerate all three sets in the same sample space without confusing an outcome with an event. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is ([3], [3], [0, 1, 2, 4]).
Reviewed step-by-step derivation
Event · qf-density13-event-boolean-algebra-pair · candidate
36. Event Union, Intersection, and Outside Complement
S=Fin(5)={0,…,4}; the event family is explicitly F=P(S). A=[0, 3], B=[] are subsets of S. Give (A∪B, A∩B, S\(A∪B)), each deduplicated and increasing.
View answer and explanation
([0, 3], [], [1, 2, 4])
Enumerate all three sets in the same sample space without confusing an outcome with an event. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is ([0, 3], [], [1, 2, 4]).
Reviewed step-by-step derivation
Event · qf-density13-event-boolean-algebra-pair · candidate
37. Event Union, Intersection, and Outside Complement
S=Fin(5)={0,…,4}; the event family is explicitly F=P(S). A=[1, 3], B=[1] are subsets of S. Give (A∪B, A∩B, S\(A∪B)), each deduplicated and increasing.
View answer and explanation
([1, 3], [1], [0, 2, 4])
Enumerate all three sets in the same sample space without confusing an outcome with an event. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is ([1, 3], [1], [0, 2, 4]).
Reviewed step-by-step derivation
Event · qf-density13-event-boolean-algebra-pair · candidate
38. Event Union, Intersection, and Outside Complement
S=Fin(5)={0,…,4}; the event family is explicitly F=P(S). A=[0, 1, 3], B=[] are subsets of S. Give (A∪B, A∩B, S\(A∪B)), each deduplicated and increasing.
View answer and explanation
([0, 1, 3], [], [2, 4])
Enumerate all three sets in the same sample space without confusing an outcome with an event. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is ([0, 1, 3], [], [2, 4]).
Reviewed step-by-step derivation
Event · qf-density13-event-boolean-algebra-pair · candidate
39. Event Union, Intersection, and Outside Complement
S=Fin(5)={0,…,4}; the event family is explicitly F=P(S). A=[2, 3], B=[2] are subsets of S. Give (A∪B, A∩B, S\(A∪B)), each deduplicated and increasing.
View answer and explanation
([2, 3], [2], [0, 1, 4])
Enumerate all three sets in the same sample space without confusing an outcome with an event. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is ([2, 3], [2], [0, 1, 4]).
Reviewed step-by-step derivation
Event · qf-density13-event-boolean-algebra-pair · candidate
40. Event Union, Intersection, and Outside Complement
S=Fin(5)={0,…,4}; the event family is explicitly F=P(S). A=[0, 2, 3], B=[] are subsets of S. Give (A∪B, A∩B, S\(A∪B)), each deduplicated and increasing.
View answer and explanation
([0, 2, 3], [], [1, 4])
Enumerate all three sets in the same sample space without confusing an outcome with an event. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is ([0, 2, 3], [], [1, 4]).
Reviewed step-by-step derivation
Event · qf-density13-event-boolean-algebra-pair · candidate
41. Event Union, Intersection, and Outside Complement
S=Fin(5)={0,…,4}; the event family is explicitly F=P(S). A=[1, 2, 3], B=[1] are subsets of S. Give (A∪B, A∩B, S\(A∪B)), each deduplicated and increasing.
View answer and explanation
([1, 2, 3], [1], [0, 4])
Enumerate all three sets in the same sample space without confusing an outcome with an event. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is ([1, 2, 3], [1], [0, 4]).
Reviewed step-by-step derivation
Event · qf-density13-event-boolean-algebra-pair · candidate
42. Event Union, Intersection, and Outside Complement
S=Fin(5)={0,…,4}; the event family is explicitly F=P(S). A=[0, 1, 2, 3], B=[] are subsets of S. Give (A∪B, A∩B, S\(A∪B)), each deduplicated and increasing.
View answer and explanation
([0, 1, 2, 3], [], [4])
Enumerate all three sets in the same sample space without confusing an outcome with an event. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is ([0, 1, 2, 3], [], [4]).
Reviewed step-by-step derivation
Event · qf-density13-event-boolean-algebra-pair · candidate
43. Event Union, Intersection, and Outside Complement
S=Fin(5)={0,…,4}; the event family is explicitly F=P(S). A=[4], B=[4] are subsets of S. Give (A∪B, A∩B, S\(A∪B)), each deduplicated and increasing.
View answer and explanation
([4], [4], [0, 1, 2, 3])
Enumerate all three sets in the same sample space without confusing an outcome with an event. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is ([4], [4], [0, 1, 2, 3]).
Reviewed step-by-step derivation
Event · qf-density13-event-boolean-algebra-pair · candidate
44. Event Union, Intersection, and Outside Complement
S=Fin(5)={0,…,4}; the event family is explicitly F=P(S). A=[0, 4], B=[] are subsets of S. Give (A∪B, A∩B, S\(A∪B)), each deduplicated and increasing.
View answer and explanation
([0, 4], [], [1, 2, 3])
Enumerate all three sets in the same sample space without confusing an outcome with an event. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is ([0, 4], [], [1, 2, 3]).
Reviewed step-by-step derivation
Event · qf-density13-event-boolean-algebra-pair · candidate
45. Event Union, Intersection, and Outside Complement
S=Fin(5)={0,…,4}; the event family is explicitly F=P(S). A=[1, 4], B=[1] are subsets of S. Give (A∪B, A∩B, S\(A∪B)), each deduplicated and increasing.
View answer and explanation
([1, 4], [1], [0, 2, 3])
Enumerate all three sets in the same sample space without confusing an outcome with an event. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is ([1, 4], [1], [0, 2, 3]).
Reviewed step-by-step derivation
Event · qf-density13-event-boolean-algebra-pair · candidate
46. Event Union, Intersection, and Outside Complement
S=Fin(5)={0,…,4}; the event family is explicitly F=P(S). A=[0, 1, 4], B=[] are subsets of S. Give (A∪B, A∩B, S\(A∪B)), each deduplicated and increasing.
View answer and explanation
([0, 1, 4], [], [2, 3])
Enumerate all three sets in the same sample space without confusing an outcome with an event. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is ([0, 1, 4], [], [2, 3]).
Reviewed step-by-step derivation
Event · qf-density13-event-boolean-algebra-pair · candidate
47. Event Union, Intersection, and Outside Complement
S=Fin(5)={0,…,4}; the event family is explicitly F=P(S). A=[2, 4], B=[2] are subsets of S. Give (A∪B, A∩B, S\(A∪B)), each deduplicated and increasing.
View answer and explanation
([2, 4], [2], [0, 1, 3])
Enumerate all three sets in the same sample space without confusing an outcome with an event. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is ([2, 4], [2], [0, 1, 3]).
Reviewed step-by-step derivation
Event · qf-density13-event-boolean-algebra-pair · candidate
48. Event Union, Intersection, and Outside Complement
S=Fin(5)={0,…,4}; the event family is explicitly F=P(S). A=[0, 2, 4], B=[] are subsets of S. Give (A∪B, A∩B, S\(A∪B)), each deduplicated and increasing.
View answer and explanation
([0, 2, 4], [], [1, 3])
Enumerate all three sets in the same sample space without confusing an outcome with an event. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is ([0, 2, 4], [], [1, 3]).
Reviewed step-by-step derivation
Event · qf-density13-event-boolean-algebra-pair · candidate
49. Event Union, Intersection, and Outside Complement
S=Fin(5)={0,…,4}; the event family is explicitly F=P(S). A=[1, 2, 4], B=[1] are subsets of S. Give (A∪B, A∩B, S\(A∪B)), each deduplicated and increasing.
View answer and explanation
([1, 2, 4], [1], [0, 3])
Enumerate all three sets in the same sample space without confusing an outcome with an event. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is ([1, 2, 4], [1], [0, 3]).
Reviewed step-by-step derivation
Event · qf-density13-event-boolean-algebra-pair · candidate
50. Event Union, Intersection, and Outside Complement
S=Fin(5)={0,…,4}; the event family is explicitly F=P(S). A=[0, 1, 2, 4], B=[] are subsets of S. Give (A∪B, A∩B, S\(A∪B)), each deduplicated and increasing.
View answer and explanation
([0, 1, 2, 4], [], [3])
Enumerate all three sets in the same sample space without confusing an outcome with an event. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is ([0, 1, 2, 4], [], [3]).
Reviewed step-by-step derivation
Event · qf-density13-event-boolean-algebra-pair · candidate
51. Partition-Union Event Eligibility and Least Containing Event
S=Fin(1)={0,…,0}; F consists of unions of complete disjoint nonempty blocks=[[0]]. Candidate E=[]⊆S. Give (is E a union of whole blocks?, partially hit block-member lists, smallest F set containing E). Sort inside each block and order blocks numerically lexicographically.
View answer and explanation
(Yes, [], [])
Each block is taken wholly or not at all; the least containing event takes every block hit by E, while fully hit blocks are not partial hits. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is (Yes, [], []).
Reviewed step-by-step derivation
Event · qf-density13-event-partition-family-qualification · candidate
52. Partition-Union Event Eligibility and Least Containing Event
S=Fin(3)={0,…,2}; F consists of unions of complete disjoint nonempty blocks=[[0, 1], [2]]. Candidate E=[0, 1]⊆S. Give (is E a union of whole blocks?, partially hit block-member lists, smallest F set containing E). Sort inside each block and order blocks numerically lexicographically.
View answer and explanation
(Yes, [], [0, 1])
Each block is taken wholly or not at all; the least containing event takes every block hit by E, while fully hit blocks are not partial hits. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is (Yes, [], [0, 1]).
Reviewed step-by-step derivation
Event · qf-density13-event-partition-family-qualification · candidate
53. Partition-Union Event Eligibility and Least Containing Event
S=Fin(3)={0,…,2}; F consists of unions of complete disjoint nonempty blocks=[[0, 1], [2]]. Candidate E=[0]⊆S. Give (is E a union of whole blocks?, partially hit block-member lists, smallest F set containing E). Sort inside each block and order blocks numerically lexicographically.
View answer and explanation
(No, [[0, 1]], [0, 1])
Each block is taken wholly or not at all; the least containing event takes every block hit by E, while fully hit blocks are not partial hits. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is (No, [[0, 1]], [0, 1]).
Reviewed step-by-step derivation
Event · qf-density13-event-partition-family-qualification · candidate
54. Partition-Union Event Eligibility and Least Containing Event
S=Fin(8)={0,…,7}; F consists of unions of complete disjoint nonempty blocks=[[0, 2], [1], [3], [4], [5], [6], [7]]. Candidate E=[0]⊆S. Give (is E a union of whole blocks?, partially hit block-member lists, smallest F set containing E). Sort inside each block and order blocks numerically lexicographically.
View answer and explanation
(No, [[0, 2]], [0, 2])
Each block is taken wholly or not at all; the least containing event takes every block hit by E, while fully hit blocks are not partial hits. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is (No, [[0, 2]], [0, 2]).
Reviewed step-by-step derivation
Event · qf-density13-event-partition-family-qualification · candidate
55. Partition-Union Event Eligibility and Least Containing Event
S=Fin(8)={0,…,7}; F consists of unions of complete disjoint nonempty blocks=[[1, 2], [0], [3], [4], [5], [6], [7]]. Candidate E=[1]⊆S. Give (is E a union of whole blocks?, partially hit block-member lists, smallest F set containing E). Sort inside each block and order blocks numerically lexicographically.
View answer and explanation
(No, [[1, 2]], [1, 2])
Each block is taken wholly or not at all; the least containing event takes every block hit by E, while fully hit blocks are not partial hits. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is (No, [[1, 2]], [1, 2]).
Reviewed step-by-step derivation
Event · qf-density13-event-partition-family-qualification · candidate
56. Partition-Union Event Eligibility and Least Containing Event
S=Fin(8)={0,…,7}; F consists of unions of complete disjoint nonempty blocks=[[0, 1, 2], [3], [4], [5], [6], [7]]. Candidate E=[0]⊆S. Give (is E a union of whole blocks?, partially hit block-member lists, smallest F set containing E). Sort inside each block and order blocks numerically lexicographically.
View answer and explanation
(No, [[0, 1, 2]], [0, 1, 2])
Each block is taken wholly or not at all; the least containing event takes every block hit by E, while fully hit blocks are not partial hits. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is (No, [[0, 1, 2]], [0, 1, 2]).
Reviewed step-by-step derivation
Event · qf-density13-event-partition-family-qualification · candidate
57. Partition-Union Event Eligibility and Least Containing Event
S=Fin(8)={0,…,7}; F consists of unions of complete disjoint nonempty blocks=[[0, 3], [1], [2], [4], [5], [6], [7]]. Candidate E=[0]⊆S. Give (is E a union of whole blocks?, partially hit block-member lists, smallest F set containing E). Sort inside each block and order blocks numerically lexicographically.
View answer and explanation
(No, [[0, 3]], [0, 3])
Each block is taken wholly or not at all; the least containing event takes every block hit by E, while fully hit blocks are not partial hits. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is (No, [[0, 3]], [0, 3]).
Reviewed step-by-step derivation
Event · qf-density13-event-partition-family-qualification · candidate
58. Partition-Union Event Eligibility and Least Containing Event
S=Fin(8)={0,…,7}; F consists of unions of complete disjoint nonempty blocks=[[1, 3], [0], [2], [4], [5], [6], [7]]. Candidate E=[1]⊆S. Give (is E a union of whole blocks?, partially hit block-member lists, smallest F set containing E). Sort inside each block and order blocks numerically lexicographically.
View answer and explanation
(No, [[1, 3]], [1, 3])
Each block is taken wholly or not at all; the least containing event takes every block hit by E, while fully hit blocks are not partial hits. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is (No, [[1, 3]], [1, 3]).
Reviewed step-by-step derivation
Event · qf-density13-event-partition-family-qualification · candidate
59. Partition-Union Event Eligibility and Least Containing Event
S=Fin(8)={0,…,7}; F consists of unions of complete disjoint nonempty blocks=[[0, 1, 3], [2], [4], [5], [6], [7]]. Candidate E=[0]⊆S. Give (is E a union of whole blocks?, partially hit block-member lists, smallest F set containing E). Sort inside each block and order blocks numerically lexicographically.
View answer and explanation
(No, [[0, 1, 3]], [0, 1, 3])
Each block is taken wholly or not at all; the least containing event takes every block hit by E, while fully hit blocks are not partial hits. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is (No, [[0, 1, 3]], [0, 1, 3]).
Reviewed step-by-step derivation
Event · qf-density13-event-partition-family-qualification · candidate
60. Partition-Union Event Eligibility and Least Containing Event
S=Fin(8)={0,…,7}; F consists of unions of complete disjoint nonempty blocks=[[2, 3], [0], [1], [4], [5], [6], [7]]. Candidate E=[2]⊆S. Give (is E a union of whole blocks?, partially hit block-member lists, smallest F set containing E). Sort inside each block and order blocks numerically lexicographically.
View answer and explanation
(No, [[2, 3]], [2, 3])
Each block is taken wholly or not at all; the least containing event takes every block hit by E, while fully hit blocks are not partial hits. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is (No, [[2, 3]], [2, 3]).
Reviewed step-by-step derivation
Event · qf-density13-event-partition-family-qualification · candidate
61. Partition-Union Event Eligibility and Least Containing Event
S=Fin(8)={0,…,7}; F consists of unions of complete disjoint nonempty blocks=[[0, 2, 3], [1], [4], [5], [6], [7]]. Candidate E=[0]⊆S. Give (is E a union of whole blocks?, partially hit block-member lists, smallest F set containing E). Sort inside each block and order blocks numerically lexicographically.
View answer and explanation
(No, [[0, 2, 3]], [0, 2, 3])
Each block is taken wholly or not at all; the least containing event takes every block hit by E, while fully hit blocks are not partial hits. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is (No, [[0, 2, 3]], [0, 2, 3]).
Reviewed step-by-step derivation
Event · qf-density13-event-partition-family-qualification · candidate
62. Partition-Union Event Eligibility and Least Containing Event
S=Fin(8)={0,…,7}; F consists of unions of complete disjoint nonempty blocks=[[1, 2, 3], [0], [4], [5], [6], [7]]. Candidate E=[1]⊆S. Give (is E a union of whole blocks?, partially hit block-member lists, smallest F set containing E). Sort inside each block and order blocks numerically lexicographically.
View answer and explanation
(No, [[1, 2, 3]], [1, 2, 3])
Each block is taken wholly or not at all; the least containing event takes every block hit by E, while fully hit blocks are not partial hits. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is (No, [[1, 2, 3]], [1, 2, 3]).
Reviewed step-by-step derivation
Event · qf-density13-event-partition-family-qualification · candidate
63. Partition-Union Event Eligibility and Least Containing Event
S=Fin(8)={0,…,7}; F consists of unions of complete disjoint nonempty blocks=[[0, 1, 2, 3], [4], [5], [6], [7]]. Candidate E=[0]⊆S. Give (is E a union of whole blocks?, partially hit block-member lists, smallest F set containing E). Sort inside each block and order blocks numerically lexicographically.
View answer and explanation
(No, [[0, 1, 2, 3]], [0, 1, 2, 3])
Each block is taken wholly or not at all; the least containing event takes every block hit by E, while fully hit blocks are not partial hits. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is (No, [[0, 1, 2, 3]], [0, 1, 2, 3]).
Reviewed step-by-step derivation
Event · qf-density13-event-partition-family-qualification · candidate
64. Partition-Union Event Eligibility and Least Containing Event
S=Fin(8)={0,…,7}; F consists of unions of complete disjoint nonempty blocks=[[0, 4], [1], [2], [3], [5], [6], [7]]. Candidate E=[0]⊆S. Give (is E a union of whole blocks?, partially hit block-member lists, smallest F set containing E). Sort inside each block and order blocks numerically lexicographically.
View answer and explanation
(No, [[0, 4]], [0, 4])
Each block is taken wholly or not at all; the least containing event takes every block hit by E, while fully hit blocks are not partial hits. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is (No, [[0, 4]], [0, 4]).
Reviewed step-by-step derivation
Event · qf-density13-event-partition-family-qualification · candidate
65. Partition-Union Event Eligibility and Least Containing Event
S=Fin(8)={0,…,7}; F consists of unions of complete disjoint nonempty blocks=[[1, 4], [0], [2], [3], [5], [6], [7]]. Candidate E=[1]⊆S. Give (is E a union of whole blocks?, partially hit block-member lists, smallest F set containing E). Sort inside each block and order blocks numerically lexicographically.
View answer and explanation
(No, [[1, 4]], [1, 4])
Each block is taken wholly or not at all; the least containing event takes every block hit by E, while fully hit blocks are not partial hits. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is (No, [[1, 4]], [1, 4]).
Reviewed step-by-step derivation
Event · qf-density13-event-partition-family-qualification · candidate
66. Partition-Union Event Eligibility and Least Containing Event
S=Fin(8)={0,…,7}; F consists of unions of complete disjoint nonempty blocks=[[0, 1, 4], [2], [3], [5], [6], [7]]. Candidate E=[0]⊆S. Give (is E a union of whole blocks?, partially hit block-member lists, smallest F set containing E). Sort inside each block and order blocks numerically lexicographically.
View answer and explanation
(No, [[0, 1, 4]], [0, 1, 4])
Each block is taken wholly or not at all; the least containing event takes every block hit by E, while fully hit blocks are not partial hits. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is (No, [[0, 1, 4]], [0, 1, 4]).
Reviewed step-by-step derivation
Event · qf-density13-event-partition-family-qualification · candidate
67. Partition-Union Event Eligibility and Least Containing Event
S=Fin(8)={0,…,7}; F consists of unions of complete disjoint nonempty blocks=[[2, 4], [0], [1], [3], [5], [6], [7]]. Candidate E=[2]⊆S. Give (is E a union of whole blocks?, partially hit block-member lists, smallest F set containing E). Sort inside each block and order blocks numerically lexicographically.
View answer and explanation
(No, [[2, 4]], [2, 4])
Each block is taken wholly or not at all; the least containing event takes every block hit by E, while fully hit blocks are not partial hits. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is (No, [[2, 4]], [2, 4]).
Reviewed step-by-step derivation
Event · qf-density13-event-partition-family-qualification · candidate
68. Partition-Union Event Eligibility and Least Containing Event
S=Fin(8)={0,…,7}; F consists of unions of complete disjoint nonempty blocks=[[0, 2, 4], [1], [3], [5], [6], [7]]. Candidate E=[0]⊆S. Give (is E a union of whole blocks?, partially hit block-member lists, smallest F set containing E). Sort inside each block and order blocks numerically lexicographically.
View answer and explanation
(No, [[0, 2, 4]], [0, 2, 4])
Each block is taken wholly or not at all; the least containing event takes every block hit by E, while fully hit blocks are not partial hits. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is (No, [[0, 2, 4]], [0, 2, 4]).
Reviewed step-by-step derivation
Event · qf-density13-event-partition-family-qualification · candidate
69. Partition-Union Event Eligibility and Least Containing Event
S=Fin(8)={0,…,7}; F consists of unions of complete disjoint nonempty blocks=[[1, 2, 4], [0], [3], [5], [6], [7]]. Candidate E=[1]⊆S. Give (is E a union of whole blocks?, partially hit block-member lists, smallest F set containing E). Sort inside each block and order blocks numerically lexicographically.
View answer and explanation
(No, [[1, 2, 4]], [1, 2, 4])
Each block is taken wholly or not at all; the least containing event takes every block hit by E, while fully hit blocks are not partial hits. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is (No, [[1, 2, 4]], [1, 2, 4]).
Reviewed step-by-step derivation
Event · qf-density13-event-partition-family-qualification · candidate
70. Partition-Union Event Eligibility and Least Containing Event
S=Fin(8)={0,…,7}; F consists of unions of complete disjoint nonempty blocks=[[0, 1, 2, 4], [3], [5], [6], [7]]. Candidate E=[0]⊆S. Give (is E a union of whole blocks?, partially hit block-member lists, smallest F set containing E). Sort inside each block and order blocks numerically lexicographically.
View answer and explanation
(No, [[0, 1, 2, 4]], [0, 1, 2, 4])
Each block is taken wholly or not at all; the least containing event takes every block hit by E, while fully hit blocks are not partial hits. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is (No, [[0, 1, 2, 4]], [0, 1, 2, 4]).
Reviewed step-by-step derivation
Event · qf-density13-event-partition-family-qualification · candidate
71. Partition-Union Event Eligibility and Least Containing Event
S=Fin(8)={0,…,7}; F consists of unions of complete disjoint nonempty blocks=[[3, 4], [0], [1], [2], [5], [6], [7]]. Candidate E=[3]⊆S. Give (is E a union of whole blocks?, partially hit block-member lists, smallest F set containing E). Sort inside each block and order blocks numerically lexicographically.
View answer and explanation
(No, [[3, 4]], [3, 4])
Each block is taken wholly or not at all; the least containing event takes every block hit by E, while fully hit blocks are not partial hits. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is (No, [[3, 4]], [3, 4]).
Reviewed step-by-step derivation
Event · qf-density13-event-partition-family-qualification · candidate
72. Partition-Union Event Eligibility and Least Containing Event
S=Fin(8)={0,…,7}; F consists of unions of complete disjoint nonempty blocks=[[0, 3, 4], [1], [2], [5], [6], [7]]. Candidate E=[0]⊆S. Give (is E a union of whole blocks?, partially hit block-member lists, smallest F set containing E). Sort inside each block and order blocks numerically lexicographically.
View answer and explanation
(No, [[0, 3, 4]], [0, 3, 4])
Each block is taken wholly or not at all; the least containing event takes every block hit by E, while fully hit blocks are not partial hits. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is (No, [[0, 3, 4]], [0, 3, 4]).
Reviewed step-by-step derivation
Event · qf-density13-event-partition-family-qualification · candidate
73. Partition-Union Event Eligibility and Least Containing Event
S=Fin(8)={0,…,7}; F consists of unions of complete disjoint nonempty blocks=[[1, 3, 4], [0], [2], [5], [6], [7]]. Candidate E=[1]⊆S. Give (is E a union of whole blocks?, partially hit block-member lists, smallest F set containing E). Sort inside each block and order blocks numerically lexicographically.
View answer and explanation
(No, [[1, 3, 4]], [1, 3, 4])
Each block is taken wholly or not at all; the least containing event takes every block hit by E, while fully hit blocks are not partial hits. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is (No, [[1, 3, 4]], [1, 3, 4]).
Reviewed step-by-step derivation
Event · qf-density13-event-partition-family-qualification · candidate
74. Partition-Union Event Eligibility and Least Containing Event
S=Fin(8)={0,…,7}; F consists of unions of complete disjoint nonempty blocks=[[0, 1, 3, 4], [2], [5], [6], [7]]. Candidate E=[0]⊆S. Give (is E a union of whole blocks?, partially hit block-member lists, smallest F set containing E). Sort inside each block and order blocks numerically lexicographically.
View answer and explanation
(No, [[0, 1, 3, 4]], [0, 1, 3, 4])
Each block is taken wholly or not at all; the least containing event takes every block hit by E, while fully hit blocks are not partial hits. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is (No, [[0, 1, 3, 4]], [0, 1, 3, 4]).
Reviewed step-by-step derivation
Event · qf-density13-event-partition-family-qualification · candidate
75. Partition-Union Event Eligibility and Least Containing Event
S=Fin(8)={0,…,7}; F consists of unions of complete disjoint nonempty blocks=[[2, 3, 4], [0], [1], [5], [6], [7]]. Candidate E=[2]⊆S. Give (is E a union of whole blocks?, partially hit block-member lists, smallest F set containing E). Sort inside each block and order blocks numerically lexicographically.
View answer and explanation
(No, [[2, 3, 4]], [2, 3, 4])
Each block is taken wholly or not at all; the least containing event takes every block hit by E, while fully hit blocks are not partial hits. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is (No, [[2, 3, 4]], [2, 3, 4]).
Reviewed step-by-step derivation
Event · qf-density13-event-partition-family-qualification · candidate
76. Typed Observations, Event Occurrence, and First Hit
S=Fin(1)={0,…,0}; the event family is explicitly F=P(S). E=[], ordered outcomes=[]. Give (sample-eligibility mask, True/False/invalid-outcome occurrence list, first legal E hit’s zero-based index or none).
View answer and explanation
([], [], none)
Only a legal sample-space value belonging to E is an occurrence; an outside candidate is invalid-outcome, not False. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is ([], [], none).
Reviewed step-by-step derivation
Event · qf-density13-event-occurrence-first-hit · candidate
77. Typed Observations, Event Occurrence, and First Hit
S=Fin(3)={0,…,2}; the event family is explicitly F=P(S). E=[1], ordered outcomes=[100, 101, 102]. Give (sample-eligibility mask, True/False/invalid-outcome occurrence list, first legal E hit’s zero-based index or none).
View answer and explanation
([No, No, No], [invalid-outcome, invalid-outcome, invalid-outcome], none)
Only a legal sample-space value belonging to E is an occurrence; an outside candidate is invalid-outcome, not False. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is ([No, No, No], [invalid-outcome, invalid-outcome, invalid-outcome], none).
Reviewed step-by-step derivation
Event · qf-density13-event-occurrence-first-hit · candidate
78. Typed Observations, Event Occurrence, and First Hit
S=Fin(3)={0,…,2}; the event family is explicitly F=P(S). E=[1], ordered outcomes=[0, 2, 0]. Give (sample-eligibility mask, True/False/invalid-outcome occurrence list, first legal E hit’s zero-based index or none).
View answer and explanation
([Yes, Yes, Yes], [False, False, False], none)
Only a legal sample-space value belonging to E is an occurrence; an outside candidate is invalid-outcome, not False. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is ([Yes, Yes, Yes], [False, False, False], none).
Reviewed step-by-step derivation
Event · qf-density13-event-occurrence-first-hit · candidate
79. Typed Observations, Event Occurrence, and First Hit
S=Fin(3)={0,…,2}; the event family is explicitly F=P(S). E=[1], ordered outcomes=[1, 1, 1]. Give (sample-eligibility mask, True/False/invalid-outcome occurrence list, first legal E hit’s zero-based index or none).
View answer and explanation
([Yes, Yes, Yes], [True, True, True], 0)
Only a legal sample-space value belonging to E is an occurrence; an outside candidate is invalid-outcome, not False. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is ([Yes, Yes, Yes], [True, True, True], 0).
Reviewed step-by-step derivation
Event · qf-density13-event-occurrence-first-hit · candidate
80. Typed Observations, Event Occurrence, and First Hit
S=Fin(3)={0,…,2}; the event family is explicitly F=P(S). E=[1], ordered outcomes=[0, 101, 102]. Give (sample-eligibility mask, True/False/invalid-outcome occurrence list, first legal E hit’s zero-based index or none).
View answer and explanation
([Yes, No, No], [False, invalid-outcome, invalid-outcome], none)
Only a legal sample-space value belonging to E is an occurrence; an outside candidate is invalid-outcome, not False. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is ([Yes, No, No], [False, invalid-outcome, invalid-outcome], none).
Reviewed step-by-step derivation
Event · qf-density13-event-occurrence-first-hit · candidate
81. Typed Observations, Event Occurrence, and First Hit
S=Fin(3)={0,…,2}; the event family is explicitly F=P(S). E=[1], ordered outcomes=[1, 101, 102]. Give (sample-eligibility mask, True/False/invalid-outcome occurrence list, first legal E hit’s zero-based index or none).
View answer and explanation
([Yes, No, No], [True, invalid-outcome, invalid-outcome], 0)
Only a legal sample-space value belonging to E is an occurrence; an outside candidate is invalid-outcome, not False. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is ([Yes, No, No], [True, invalid-outcome, invalid-outcome], 0).
Reviewed step-by-step derivation
Event · qf-density13-event-occurrence-first-hit · candidate
82. Typed Observations, Event Occurrence, and First Hit
S=Fin(3)={0,…,2}; the event family is explicitly F=P(S). E=[1], ordered outcomes=[100, 2, 102]. Give (sample-eligibility mask, True/False/invalid-outcome occurrence list, first legal E hit’s zero-based index or none).
View answer and explanation
([No, Yes, No], [invalid-outcome, False, invalid-outcome], none)
Only a legal sample-space value belonging to E is an occurrence; an outside candidate is invalid-outcome, not False. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is ([No, Yes, No], [invalid-outcome, False, invalid-outcome], none).
Reviewed step-by-step derivation
Event · qf-density13-event-occurrence-first-hit · candidate
83. Typed Observations, Event Occurrence, and First Hit
S=Fin(3)={0,…,2}; the event family is explicitly F=P(S). E=[1], ordered outcomes=[0, 2, 102]. Give (sample-eligibility mask, True/False/invalid-outcome occurrence list, first legal E hit’s zero-based index or none).
View answer and explanation
([Yes, Yes, No], [False, False, invalid-outcome], none)
Only a legal sample-space value belonging to E is an occurrence; an outside candidate is invalid-outcome, not False. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is ([Yes, Yes, No], [False, False, invalid-outcome], none).
Reviewed step-by-step derivation
Event · qf-density13-event-occurrence-first-hit · candidate
84. Typed Observations, Event Occurrence, and First Hit
S=Fin(3)={0,…,2}; the event family is explicitly F=P(S). E=[1], ordered outcomes=[1, 2, 102]. Give (sample-eligibility mask, True/False/invalid-outcome occurrence list, first legal E hit’s zero-based index or none).
View answer and explanation
([Yes, Yes, No], [True, False, invalid-outcome], 0)
Only a legal sample-space value belonging to E is an occurrence; an outside candidate is invalid-outcome, not False. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is ([Yes, Yes, No], [True, False, invalid-outcome], 0).
Reviewed step-by-step derivation
Event · qf-density13-event-occurrence-first-hit · candidate
85. Typed Observations, Event Occurrence, and First Hit
S=Fin(3)={0,…,2}; the event family is explicitly F=P(S). E=[1], ordered outcomes=[100, 1, 102]. Give (sample-eligibility mask, True/False/invalid-outcome occurrence list, first legal E hit’s zero-based index or none).
View answer and explanation
([No, Yes, No], [invalid-outcome, True, invalid-outcome], 1)
Only a legal sample-space value belonging to E is an occurrence; an outside candidate is invalid-outcome, not False. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is ([No, Yes, No], [invalid-outcome, True, invalid-outcome], 1).
Reviewed step-by-step derivation
Event · qf-density13-event-occurrence-first-hit · candidate
86. Typed Observations, Event Occurrence, and First Hit
S=Fin(3)={0,…,2}; the event family is explicitly F=P(S). E=[1], ordered outcomes=[0, 1, 102]. Give (sample-eligibility mask, True/False/invalid-outcome occurrence list, first legal E hit’s zero-based index or none).
View answer and explanation
([Yes, Yes, No], [False, True, invalid-outcome], 1)
Only a legal sample-space value belonging to E is an occurrence; an outside candidate is invalid-outcome, not False. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is ([Yes, Yes, No], [False, True, invalid-outcome], 1).
Reviewed step-by-step derivation
Event · qf-density13-event-occurrence-first-hit · candidate
87. Typed Observations, Event Occurrence, and First Hit
S=Fin(3)={0,…,2}; the event family is explicitly F=P(S). E=[1], ordered outcomes=[1, 1, 102]. Give (sample-eligibility mask, True/False/invalid-outcome occurrence list, first legal E hit’s zero-based index or none).
View answer and explanation
([Yes, Yes, No], [True, True, invalid-outcome], 0)
Only a legal sample-space value belonging to E is an occurrence; an outside candidate is invalid-outcome, not False. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is ([Yes, Yes, No], [True, True, invalid-outcome], 0).
Reviewed step-by-step derivation
Event · qf-density13-event-occurrence-first-hit · candidate
88. Typed Observations, Event Occurrence, and First Hit
S=Fin(3)={0,…,2}; the event family is explicitly F=P(S). E=[1], ordered outcomes=[100, 101, 0]. Give (sample-eligibility mask, True/False/invalid-outcome occurrence list, first legal E hit’s zero-based index or none).
View answer and explanation
([No, No, Yes], [invalid-outcome, invalid-outcome, False], none)
Only a legal sample-space value belonging to E is an occurrence; an outside candidate is invalid-outcome, not False. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is ([No, No, Yes], [invalid-outcome, invalid-outcome, False], none).
Reviewed step-by-step derivation
Event · qf-density13-event-occurrence-first-hit · candidate
89. Typed Observations, Event Occurrence, and First Hit
S=Fin(3)={0,…,2}; the event family is explicitly F=P(S). E=[1], ordered outcomes=[0, 101, 0]. Give (sample-eligibility mask, True/False/invalid-outcome occurrence list, first legal E hit’s zero-based index or none).
View answer and explanation
([Yes, No, Yes], [False, invalid-outcome, False], none)
Only a legal sample-space value belonging to E is an occurrence; an outside candidate is invalid-outcome, not False. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is ([Yes, No, Yes], [False, invalid-outcome, False], none).
Reviewed step-by-step derivation
Event · qf-density13-event-occurrence-first-hit · candidate
90. Typed Observations, Event Occurrence, and First Hit
S=Fin(3)={0,…,2}; the event family is explicitly F=P(S). E=[1], ordered outcomes=[1, 101, 0]. Give (sample-eligibility mask, True/False/invalid-outcome occurrence list, first legal E hit’s zero-based index or none).
View answer and explanation
([Yes, No, Yes], [True, invalid-outcome, False], 0)
Only a legal sample-space value belonging to E is an occurrence; an outside candidate is invalid-outcome, not False. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is ([Yes, No, Yes], [True, invalid-outcome, False], 0).
Reviewed step-by-step derivation
Event · qf-density13-event-occurrence-first-hit · candidate
91. Typed Observations, Event Occurrence, and First Hit
S=Fin(3)={0,…,2}; the event family is explicitly F=P(S). E=[1], ordered outcomes=[100, 2, 0]. Give (sample-eligibility mask, True/False/invalid-outcome occurrence list, first legal E hit’s zero-based index or none).
View answer and explanation
([No, Yes, Yes], [invalid-outcome, False, False], none)
Only a legal sample-space value belonging to E is an occurrence; an outside candidate is invalid-outcome, not False. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is ([No, Yes, Yes], [invalid-outcome, False, False], none).
Reviewed step-by-step derivation
Event · qf-density13-event-occurrence-first-hit · candidate
92. Typed Observations, Event Occurrence, and First Hit
S=Fin(3)={0,…,2}; the event family is explicitly F=P(S). E=[1], ordered outcomes=[1, 2, 0]. Give (sample-eligibility mask, True/False/invalid-outcome occurrence list, first legal E hit’s zero-based index or none).
View answer and explanation
([Yes, Yes, Yes], [True, False, False], 0)
Only a legal sample-space value belonging to E is an occurrence; an outside candidate is invalid-outcome, not False. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is ([Yes, Yes, Yes], [True, False, False], 0).
Reviewed step-by-step derivation
Event · qf-density13-event-occurrence-first-hit · candidate
93. Typed Observations, Event Occurrence, and First Hit
S=Fin(3)={0,…,2}; the event family is explicitly F=P(S). E=[1], ordered outcomes=[100, 1, 0]. Give (sample-eligibility mask, True/False/invalid-outcome occurrence list, first legal E hit’s zero-based index or none).
View answer and explanation
([No, Yes, Yes], [invalid-outcome, True, False], 1)
Only a legal sample-space value belonging to E is an occurrence; an outside candidate is invalid-outcome, not False. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is ([No, Yes, Yes], [invalid-outcome, True, False], 1).
Reviewed step-by-step derivation
Event · qf-density13-event-occurrence-first-hit · candidate
94. Typed Observations, Event Occurrence, and First Hit
S=Fin(3)={0,…,2}; the event family is explicitly F=P(S). E=[1], ordered outcomes=[0, 1, 0]. Give (sample-eligibility mask, True/False/invalid-outcome occurrence list, first legal E hit’s zero-based index or none).
View answer and explanation
([Yes, Yes, Yes], [False, True, False], 1)
Only a legal sample-space value belonging to E is an occurrence; an outside candidate is invalid-outcome, not False. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is ([Yes, Yes, Yes], [False, True, False], 1).
Reviewed step-by-step derivation
Event · qf-density13-event-occurrence-first-hit · candidate
95. Typed Observations, Event Occurrence, and First Hit
S=Fin(3)={0,…,2}; the event family is explicitly F=P(S). E=[1], ordered outcomes=[1, 1, 0]. Give (sample-eligibility mask, True/False/invalid-outcome occurrence list, first legal E hit’s zero-based index or none).
View answer and explanation
([Yes, Yes, Yes], [True, True, False], 0)
Only a legal sample-space value belonging to E is an occurrence; an outside candidate is invalid-outcome, not False. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is ([Yes, Yes, Yes], [True, True, False], 0).
Reviewed step-by-step derivation
Event · qf-density13-event-occurrence-first-hit · candidate
96. Typed Observations, Event Occurrence, and First Hit
S=Fin(3)={0,…,2}; the event family is explicitly F=P(S). E=[1], ordered outcomes=[100, 101, 1]. Give (sample-eligibility mask, True/False/invalid-outcome occurrence list, first legal E hit’s zero-based index or none).
View answer and explanation
([No, No, Yes], [invalid-outcome, invalid-outcome, True], 2)
Only a legal sample-space value belonging to E is an occurrence; an outside candidate is invalid-outcome, not False. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is ([No, No, Yes], [invalid-outcome, invalid-outcome, True], 2).
Reviewed step-by-step derivation
Event · qf-density13-event-occurrence-first-hit · candidate
97. Typed Observations, Event Occurrence, and First Hit
S=Fin(3)={0,…,2}; the event family is explicitly F=P(S). E=[1], ordered outcomes=[0, 101, 1]. Give (sample-eligibility mask, True/False/invalid-outcome occurrence list, first legal E hit’s zero-based index or none).
View answer and explanation
([Yes, No, Yes], [False, invalid-outcome, True], 2)
Only a legal sample-space value belonging to E is an occurrence; an outside candidate is invalid-outcome, not False. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is ([Yes, No, Yes], [False, invalid-outcome, True], 2).
Reviewed step-by-step derivation
Event · qf-density13-event-occurrence-first-hit · candidate
98. Typed Observations, Event Occurrence, and First Hit
S=Fin(3)={0,…,2}; the event family is explicitly F=P(S). E=[1], ordered outcomes=[1, 101, 1]. Give (sample-eligibility mask, True/False/invalid-outcome occurrence list, first legal E hit’s zero-based index or none).
View answer and explanation
([Yes, No, Yes], [True, invalid-outcome, True], 0)
Only a legal sample-space value belonging to E is an occurrence; an outside candidate is invalid-outcome, not False. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is ([Yes, No, Yes], [True, invalid-outcome, True], 0).
Reviewed step-by-step derivation
Event · qf-density13-event-occurrence-first-hit · candidate
99. Typed Observations, Event Occurrence, and First Hit
S=Fin(3)={0,…,2}; the event family is explicitly F=P(S). E=[1], ordered outcomes=[100, 2, 1]. Give (sample-eligibility mask, True/False/invalid-outcome occurrence list, first legal E hit’s zero-based index or none).
View answer and explanation
([No, Yes, Yes], [invalid-outcome, False, True], 2)
Only a legal sample-space value belonging to E is an occurrence; an outside candidate is invalid-outcome, not False. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is ([No, Yes, Yes], [invalid-outcome, False, True], 2).
Reviewed step-by-step derivation
Event · qf-density13-event-occurrence-first-hit · candidate
100. Typed Observations, Event Occurrence, and First Hit
S=Fin(3)={0,…,2}; the event family is explicitly F=P(S). E=[1], ordered outcomes=[0, 2, 1]. Give (sample-eligibility mask, True/False/invalid-outcome occurrence list, first legal E hit’s zero-based index or none).
View answer and explanation
([Yes, Yes, Yes], [False, False, True], 2)
Only a legal sample-space value belonging to E is an occurrence; an outside candidate is invalid-outcome, not False. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is ([Yes, Yes, Yes], [False, False, True], 2).
Reviewed step-by-step derivation
Event · qf-density13-event-occurrence-first-hit · candidate
Canonical question JSONL · English JSONL · Validation report