# Question 6038: Declared Event and Relative Complement Partition

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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.

R0011-06038 · q-20261009-advance-20261013-r06-event-0013



## Original question and givens

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).


- G001 — 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).





## 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(5)=[0, 1, 2, 3, 4];declared event model=full_powerset

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

E=[0, 1, 3];S=[0, 1, 2, 3, 4]

After

S\E=[2, 4]

Conditions: none

Take the complement only inside S,without outside values;the explicitly declared F=P(S) contains this complement.

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





S003 · impliesBefore

([0, 1, 3], [2, 4])

After

disjoint union is S;|S|=5

Conditions: none

The event and complement are disjoint and cover S.The final field is n,not a probability;no probability law was supplied.

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









## Reverse checks


- 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

([0, 1, 3], [2, 4], 5)

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: 8e58e89eb240ad020648ab0fe43ad10a9d3c8eb436523ea4a228110c7ff95a0a
- candidate record SHA-256: fe50bca51bf2318dafbc4e122f7bac3a28e3a0e200f070de96a05aba52e6a65d
- 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-0013.json)