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R0011-06077 · q-20261009-advance-20261013-r06-event-0052
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.
G001 — 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.Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate.
Before
S=Fin(3)=[0, 1, 2];declared event model=partition_unions
After
finite S alone does not specify its event family or a probability law
Conditions: none
A finite sample space does not automatically choose F=P(S).Use the declared event family;set data alone give no probabilities,equiprobability or independence.
Rule: declared_finite_event_family · Uses: G001
Before
ordered blocks=[[0, 1], [2]];S=[0, 1, 2]
After
nonempty,pairwise disjoint blocks cover S exactly
Conditions: none
Verify nonempty disjoint blocks covering all S;an arbitrary family of subsets is not automatically a partition.
Rule: complete_event_partition_premise · Uses: G001, S001
Before
block=[0, 1];E selects=[0, 1]
After
hit=Yes;whole block selected=Yes
Conditions: none
A member of F must include or omit each entire block.A partial hit is still a subset of S but is not in this declared F.
Rule: partition_event_block_audit · Uses: G001, S002
Before
block=[2];E selects=[]
After
hit=No;whole block selected=No
Conditions: none
A member of F must include or omit each entire block.A partial hit is still a subset of S but is not in this declared F.
Rule: partition_event_block_audit · Uses: G001, S003
Before
partial blocks=[];all hit blocks=[[0, 1]]
After
qualified event=Yes;smallest containing F-set=[0, 1]
Conditions: none
Any block-union containing E must include every hit block.Their union itself belongs to F,so it is the smallest containing F-set,not a closure in an unspecified topology.
Rule: partition_union_minimal_event_hull · Uses: G001, S004
S005 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S004 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S003 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S002 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.(Yes, [], [0, 1])
The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.
Review date: 2026-10-26
Original questions, answers, step expressions, numeric ASTs and historical records are retained. Source candidate labels remain historical and are not overwritten as formal proof.
input commit: 0f7be61e2c22317a4075561dc822c87d881a50dboriginal question SHA-256: 666acfa66ab86b0df5561b2aaa93ff4d487cf54df55752f54e5daf2d79f920c2candidate record SHA-256: ef833f186b8208c1f7e9559536e8e2109d4726835c56c2d18586db753a87b12asemantic review receipt SHA-256: b9bbc1c947f477a9c4bb5a579368ac5265f2da4572ce612c30f3633ecaf4ae80question-first witness SHA-256: 93f720df3ab3af13a792a3e6b443728e3c2a56276df839911f36569333236c55Lean: NOT_RUN · human review: NOT_PERFORMED