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Question 6012: Event Union, Intersection, and Outside Complement

On this page
  1. Original question and givens
  2. Definition and scope
  3. Step-by-step derivation
  4. Reverse checks
  5. Result
  6. This question review and source binding

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R0011-06012 · q-20261009-advance-20261013-r06-event-0037

Original question and givens

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.

  • G001 — 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.

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.

Step-by-step derivation

S001 · implies

Before

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 dependencies

Rule: declared_finite_event_family · Uses: G001

S002 · implies

Before

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 dependencies

Rule: event_boolean_membership_row · Uses: G001, S001

S003 · implies

Before

outcome=1;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 dependencies

Rule: event_boolean_membership_row · Uses: G001, S002

S004 · implies

Before

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 dependencies

Rule: event_boolean_membership_row · Uses: G001, S003

S005 · implies

Before

outcome=3;in A=Yes;in B=No

After

in union=Yes;in intersection=No;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 dependencies

Rule: event_boolean_membership_row · Uses: G001, S004

S006 · implies

Before

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 dependencies

Rule: 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

([1, 3], [1], [0, 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: 5a8ae43757c88677e2a1ef42d054a554188ea6a13534836905f996623c7609d3
  • candidate record SHA-256: 4e4cbcd3569981482d4e4ba65640e3394d9d331f5b3ccd8342385c354a378643
  • semantic review receipt SHA-256: b9bbc1c947f477a9c4bb5a579368ac5265f2da4572ce612c30f3633ecaf4ae80
  • question-first witness SHA-256: 93f720df3ab3af13a792a3e6b443728e3c2a56276df839911f36569333236c55
  • Lean: NOT_RUN · human review: NOT_PERFORMED
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