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Question 10407: List Missing Closure Obligations without Guessing a Sigma-Algebra

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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R0019-10407 · q-20261013-advance-20261023-r05-sigma-algebra-measure-space-0032

Original question and givens

Use the explicit finite carrier Fin(7) and complete set family=[[], [0, 1, 2], [3, 4, 5, 6], [0, 1, 2, 3, 4, 5, 6]]. Each subset must have sorted distinct carrier elements, and the family must already be duplicate-free and ordered by increasing bit mask sum(2^element); do not reorder aligned measure values. Countable obligations may be reduced only for this specified finite carrier, not arbitrary infinite sigma-algebras or extended-real measures. Give (size,canonical family,missing [set,complement] records,missing [left,right,union] records,empty/full and both closure checks pass?).

  • G001 — Use the explicit finite carrier Fin(7) and complete set family=[[], [0, 1, 2], [3, 4, 5, 6], [0, 1, 2, 3, 4, 5, 6]]. Each subset must have sorted distinct carrier elements, and the family must already be duplicate-free and ordered by increasing bit mask sum(2^element); do not reorder aligned measure values. Countable obligations may be reduced only for this specified finite carrier, not arbitrary infinite sigma-algebras or extended-real measures. Give (size,canonical family,missing [set,complement] records,missing [left,right,union] records,empty/full and both closure checks pass?).

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

carrier Fin(7);raw family=[[], [0, 1, 2], [3, 4, 5, 6], [0, 1, 2, 3, 4, 5, 6]]

After

already canonical family=[[], [0, 1, 2], [3, 4, 5, 6], [0, 1, 2, 3, 4, 5, 6]]

Conditions: none

Each set must already be sorted/distinct,and the family mask-ordered and duplicate-free.This protocol rejects noncanonical data rather than reordering aligned measure values to repair it.

Step identifiers and dependencies

Rule: finite_sigma_canonical_alignment_guard · Uses: G001

S002 · implies

Before

source member=[];fixed carrier=[0, 1, 2, 3, 4, 5, 6]

After

complement=[0, 1, 2, 3, 4, 5, 6];present=Yes

Conditions: none

Take the complement in this same carrier for every member and retain every missing-complement record.

Step identifiers and dependencies

Rule: sigma_every_relative_complement · Uses: G001, S001

S003 · implies

Before

ordered left=[];ordered right=[]

After

union=[];present=Yes

Conditions: none

Check every ordered pair,retaining reverse-pair records as specified.Do not stop at a first failure or sample unions.

Step identifiers and dependencies

Rule: sigma_every_ordered_binary_union · Uses: G001, S002

S004 · implies

Before

ordered left=[];ordered right=[0, 1, 2]

After

union=[0, 1, 2];present=Yes

Conditions: none

Check every ordered pair,retaining reverse-pair records as specified.Do not stop at a first failure or sample unions.

Step identifiers and dependencies

Rule: sigma_every_ordered_binary_union · Uses: G001, S003

S005 · implies

Before

ordered left=[];ordered right=[3, 4, 5, 6]

After

union=[3, 4, 5, 6];present=Yes

Conditions: none

Check every ordered pair,retaining reverse-pair records as specified.Do not stop at a first failure or sample unions.

Step identifiers and dependencies

Rule: sigma_every_ordered_binary_union · Uses: G001, S004

S006 · implies

Before

ordered left=[];ordered right=[0, 1, 2, 3, 4, 5, 6]

After

union=[0, 1, 2, 3, 4, 5, 6];present=Yes

Conditions: none

Check every ordered pair,retaining reverse-pair records as specified.Do not stop at a first failure or sample unions.

Step identifiers and dependencies

Rule: sigma_every_ordered_binary_union · Uses: G001, S005

S007 · implies

Before

source member=[0, 1, 2];fixed carrier=[0, 1, 2, 3, 4, 5, 6]

After

complement=[3, 4, 5, 6];present=Yes

Conditions: none

Take the complement in this same carrier for every member and retain every missing-complement record.

Step identifiers and dependencies

Rule: sigma_every_relative_complement · Uses: G001, S006

S008 · implies

Before

ordered left=[0, 1, 2];ordered right=[]

After

union=[0, 1, 2];present=Yes

Conditions: none

Check every ordered pair,retaining reverse-pair records as specified.Do not stop at a first failure or sample unions.

Step identifiers and dependencies

Rule: sigma_every_ordered_binary_union · Uses: G001, S007

S009 · implies

Before

ordered left=[0, 1, 2];ordered right=[0, 1, 2]

After

union=[0, 1, 2];present=Yes

Conditions: none

Check every ordered pair,retaining reverse-pair records as specified.Do not stop at a first failure or sample unions.

Step identifiers and dependencies

Rule: sigma_every_ordered_binary_union · Uses: G001, S008

S010 · implies

Before

ordered left=[0, 1, 2];ordered right=[3, 4, 5, 6]

After

union=[0, 1, 2, 3, 4, 5, 6];present=Yes

Conditions: none

Check every ordered pair,retaining reverse-pair records as specified.Do not stop at a first failure or sample unions.

Step identifiers and dependencies

Rule: sigma_every_ordered_binary_union · Uses: G001, S009

S011 · implies

Before

ordered left=[0, 1, 2];ordered right=[0, 1, 2, 3, 4, 5, 6]

After

union=[0, 1, 2, 3, 4, 5, 6];present=Yes

Conditions: none

Check every ordered pair,retaining reverse-pair records as specified.Do not stop at a first failure or sample unions.

Step identifiers and dependencies

Rule: sigma_every_ordered_binary_union · Uses: G001, S010

S012 · implies

Before

source member=[3, 4, 5, 6];fixed carrier=[0, 1, 2, 3, 4, 5, 6]

After

complement=[0, 1, 2];present=Yes

Conditions: none

Take the complement in this same carrier for every member and retain every missing-complement record.

Step identifiers and dependencies

Rule: sigma_every_relative_complement · Uses: G001, S011

S013 · implies

Before

ordered left=[3, 4, 5, 6];ordered right=[]

After

union=[3, 4, 5, 6];present=Yes

Conditions: none

Check every ordered pair,retaining reverse-pair records as specified.Do not stop at a first failure or sample unions.

Step identifiers and dependencies

Rule: sigma_every_ordered_binary_union · Uses: G001, S012

S014 · implies

Before

ordered left=[3, 4, 5, 6];ordered right=[0, 1, 2]

After

union=[0, 1, 2, 3, 4, 5, 6];present=Yes

Conditions: none

Check every ordered pair,retaining reverse-pair records as specified.Do not stop at a first failure or sample unions.

Step identifiers and dependencies

Rule: sigma_every_ordered_binary_union · Uses: G001, S013

S015 · implies

Before

ordered left=[3, 4, 5, 6];ordered right=[3, 4, 5, 6]

After

union=[3, 4, 5, 6];present=Yes

Conditions: none

Check every ordered pair,retaining reverse-pair records as specified.Do not stop at a first failure or sample unions.

Step identifiers and dependencies

Rule: sigma_every_ordered_binary_union · Uses: G001, S014

S016 · implies

Before

ordered left=[3, 4, 5, 6];ordered right=[0, 1, 2, 3, 4, 5, 6]

After

union=[0, 1, 2, 3, 4, 5, 6];present=Yes

Conditions: none

Check every ordered pair,retaining reverse-pair records as specified.Do not stop at a first failure or sample unions.

Step identifiers and dependencies

Rule: sigma_every_ordered_binary_union · Uses: G001, S015

S017 · implies

Before

source member=[0, 1, 2, 3, 4, 5, 6];fixed carrier=[0, 1, 2, 3, 4, 5, 6]

After

complement=[];present=Yes

Conditions: none

Take the complement in this same carrier for every member and retain every missing-complement record.

Step identifiers and dependencies

Rule: sigma_every_relative_complement · Uses: G001, S016

S018 · implies

Before

ordered left=[0, 1, 2, 3, 4, 5, 6];ordered right=[]

After

union=[0, 1, 2, 3, 4, 5, 6];present=Yes

Conditions: none

Check every ordered pair,retaining reverse-pair records as specified.Do not stop at a first failure or sample unions.

Step identifiers and dependencies

Rule: sigma_every_ordered_binary_union · Uses: G001, S017

S019 · implies

Before

ordered left=[0, 1, 2, 3, 4, 5, 6];ordered right=[0, 1, 2]

After

union=[0, 1, 2, 3, 4, 5, 6];present=Yes

Conditions: none

Check every ordered pair,retaining reverse-pair records as specified.Do not stop at a first failure or sample unions.

Step identifiers and dependencies

Rule: sigma_every_ordered_binary_union · Uses: G001, S018

S020 · implies

Before

ordered left=[0, 1, 2, 3, 4, 5, 6];ordered right=[3, 4, 5, 6]

After

union=[0, 1, 2, 3, 4, 5, 6];present=Yes

Conditions: none

Check every ordered pair,retaining reverse-pair records as specified.Do not stop at a first failure or sample unions.

Step identifiers and dependencies

Rule: sigma_every_ordered_binary_union · Uses: G001, S019

S021 · implies

Before

ordered left=[0, 1, 2, 3, 4, 5, 6];ordered right=[0, 1, 2, 3, 4, 5, 6]

After

union=[0, 1, 2, 3, 4, 5, 6];present=Yes

Conditions: none

Check every ordered pair,retaining reverse-pair records as specified.Do not stop at a first failure or sample unions.

Step identifiers and dependencies

Rule: sigma_every_ordered_binary_union · Uses: G001, S020

S022 · implies

Before

empty present=Yes;full present=Yes;missing complements=[];missing unions=[]

After

finite countable reduction=Yes;sigma qualified=Yes

Conditions: none

A fixed finite carrier has finitely many distinct subsets,so a countable family's union equals a finite distinct-member union.Empty/whole membership,complements and binary unions therefore suffice here,not for arbitrary infinite carriers.

Step identifiers and dependencies

Rule: finite_sigma_countable_union_reduction · Uses: G001, S021

Reverse checks

  • S022 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S021 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S020 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S019 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S018 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S017 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S016 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S015 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S014 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S013 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S012 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S011 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • 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

(7, [[], [0, 1, 2], [3, 4, 5, 6], [0, 1, 2, 3, 4, 5, 6]], [], [], Yes)

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: 06abcb29dd57246e6274f22cec789fa7b18906954dea25367cd0b1f04e2786f8
  • candidate record SHA-256: b0d26265cee00fd1fa28980131d9b2f1ca083fed86540a5a2d26fed04df1e257
  • semantic review receipt SHA-256: eeb35dc5daab92234bf72827627392b47f38ead4cfb574d4373bebc43c09fbf5
  • question-first witness SHA-256: 3cf8e0d3a00b0df1994db2bfde81f4a31b382e170bdda83ea3d0b68da64e8419
  • Lean: NOT_RUN · human review: NOT_PERFORMED
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