Question 10478: Every Complement and Binary Union on a Finite Carrier

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 (carrier size,canonical family,contains empty set?,contains whole carrier?,all missing [set,complement] records,all missing [left,right,union] records,finite countable-reduction qualified?,sigma-algebra?). Preserve nested family order in the witness lists.

• 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 (carrier size,canonical family,contains empty set?,contains whole carrier?,all missing [set,complement] records,all missing [left,right,union] records,finite countable-reduction qualified?,sigma-algebra?). Preserve nested family order in the witness lists.

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-sigma-algebra-measure-space

Step-by-step derivation

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

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

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

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

ordered left=[];ordered right=[]

union=[];present=Yes

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

union=[0, 1];present=Yes

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

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

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

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

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

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

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

union=[0, 1];present=Yes

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

union=[0, 1];present=Yes

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

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

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

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

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

complement=[0, 1];present=Yes

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

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

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

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

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

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

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

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

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

complement=[];present=Yes

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

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

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

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

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

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

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

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

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

finite countable reduction=Yes;sigma qualified=Yes

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, Yes, [], [], Yes, 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.