# Question 6010: Event Union, Intersection, and Outside Complement

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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.

R0011-06010 · q-20261009-advance-20261013-r06-event-0035



## Original question and givens

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.


- G001 — 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.





## 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

outcome=0;in A=No;in B=No

After

in union=No;in intersection=No;outside union=Yes

Conditions: none

Within the same S,evaluate inclusive union,intersection and the union's complement for each element;an elementary outcome is not an event set.

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





S003 · impliesBefore

outcome=1;in A=No;in B=No

After

in union=No;in intersection=No;outside union=Yes

Conditions: none

Within the same S,evaluate inclusive union,intersection and the union's complement for each element;an elementary outcome is not an event set.

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





S004 · impliesBefore

outcome=2;in A=No;in B=No

After

in union=No;in intersection=No;outside union=Yes

Conditions: none

Within the same S,evaluate inclusive union,intersection and the union's complement for each element;an elementary outcome is not an event set.

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





S005 · impliesBefore

outcome=3;in A=Yes;in B=Yes

After

in union=Yes;in intersection=Yes;outside union=No

Conditions: none

Within the same S,evaluate inclusive union,intersection and the union's complement for each element;an elementary outcome is not an event set.

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





S006 · impliesBefore

outcome=4;in A=No;in B=No

After

in union=No;in intersection=No;outside union=Yes

Conditions: none

Within the same S,evaluate inclusive union,intersection and the union's complement for each element;an elementary outcome is not an event set.

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









## Reverse checks


- 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

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

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: 61529c9d91ce16180b943e6cdac7bed56d3a34a5f48c6de073aaf25a5c500e0c
- candidate record SHA-256: 684a390acd6a2c6b441431b5643e8fb4d88dcbc2383208416458dd07bc2ce5ba
- 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-0035.json)