# Question 6082: Partition-Union Event Eligibility and Least Containing Event

[Return to catalogue page 31](https://ocme.evemisslab.com/en/questions/page/31/)

[← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261009-advance-20261013-r06-event-foundations/#q-20261009-advance-20261013-r06-event-0057)

This teaching derivation has second-context AI mathematical and bilingual semantic review, limited to the model and assumptions stated in this question. It has no human review or new Lean proof and does not promote the original question or MKO to Canonical truth.

R0011-06082 · q-20261009-advance-20261013-r06-event-0057



## Original question and givens

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.


- G001 — 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.





## Definition and scope

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.


- [mko-event](https://ocme.evemisslab.com/en/mko/mko-event/)





## Step-by-step derivation

S001 · impliesBefore

S=Fin(8)=[0, 1, 2, 3, 4, 5, 6, 7];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.

Step identifiers and dependenciesRule: declared_finite_event_family · Uses: G001





S002 · impliesBefore

ordered blocks=[[0, 3], [1], [2], [4], [5], [6], [7]];S=[0, 1, 2, 3, 4, 5, 6, 7]

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.

Step identifiers and dependenciesRule: complete_event_partition_premise · Uses: G001, S001





S003 · impliesBefore

block=[0, 3];E selects=[0]

After

hit=Yes;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.

Step identifiers and dependenciesRule: partition_event_block_audit · Uses: G001, S002





S004 · impliesBefore

block=[1];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.

Step identifiers and dependenciesRule: partition_event_block_audit · Uses: G001, S003





S005 · impliesBefore

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.

Step identifiers and dependenciesRule: partition_event_block_audit · Uses: G001, S004





S006 · impliesBefore

block=[4];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.

Step identifiers and dependenciesRule: partition_event_block_audit · Uses: G001, S005





S007 · impliesBefore

block=[5];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.

Step identifiers and dependenciesRule: partition_event_block_audit · Uses: G001, S006





S008 · impliesBefore

block=[6];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.

Step identifiers and dependenciesRule: partition_event_block_audit · Uses: G001, S007





S009 · impliesBefore

block=[7];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.

Step identifiers and dependenciesRule: partition_event_block_audit · Uses: G001, S008





S010 · impliesBefore

partial blocks=[[0, 3]];all hit blocks=[[0, 3]]

After

qualified event=No;smallest containing F-set=[0, 3]

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.

Step identifiers and dependenciesRule: partition_union_minimal_event_hull · Uses: G001, S009









## Reverse checks


- S010 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S009 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S008 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S007 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S006 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- 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.





## Result

(No, [[0, 3]], [0, 3])

The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.





## This question review and source binding

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.

Source hashes and review receipts
- input commit: 0f7be61e2c22317a4075561dc822c87d881a50db
- original question SHA-256: efdfae3fb8c7229f1f2a542875027bdc351da5c583a4b4d780d9984548744b4b
- candidate record SHA-256: f100fae05eb6e98eb7262cc48144d7dfb7046617c54df2253e0fb8a90de91b68
- semantic review receipt SHA-256: b9bbc1c947f477a9c4bb5a579368ac5265f2da4572ce612c30f3633ecaf4ae80
- question-first witness SHA-256: 93f720df3ab3af13a792a3e6b443728e3c2a56276df839911f36569333236c55
- Lean: NOT_RUN · human review: NOT_PERFORMED

[Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261009-advance-20261013-r06-event-0057.json)