# Question 32098: Finite Set Union

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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.

R0063-32098 · q-20260915-finite-set-union-0023



## Original question and givens

Let A={221, 223, 225} and B={223, 224, 226}. Find A∪B, listing each element once in increasing order.


- G001 — Let A={221, 223, 225} and B={223, 224, 226}. Find A∪B, listing each element once in increasing order.





## 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-set-operations](https://ocme.evemisslab.com/en/mko/mko-set-operations/)





## Step-by-step derivation

S001 · impliesBefore

source roles = [qf-finite-set-union, no-required-operation, no-required-domain, []]

After

derived = [[numeric_sorted_set_STRING_or_empty]]

Conditions: none

BASE sources do not acquire a generic legacy operation/domain or unknown-metadata ban.Retain metadata WITHOUT using it as an answer oracle.PULSE/legacy enforce their exact source keys,operation/domain.Do not impose universal tuples,/1 or input bounds.

Step identifiers and dependenciesRule: R63_SOURCE_bank_keys_operation_domain_output_roles_metadata_not_oracle · Uses: G001





S002 · impliesBefore

source roles = [A ORIGINAL, [221, 223, 225]]

After

derived = [[221, 223, 225]]

Conditions: none

THIS BASE set bank requires distinct integer lists;do not silently repair duplicates.Sort ONLY derived display,retaining originals.Do not import another bank SAFE/100-element bound.

Step identifiers and dependenciesRule: Sets_BASE_distinct_integer_lists_no_SAFE_or100cap_no_silent_repair · Uses: G001, S001





S003 · impliesBefore

source roles = [B ORIGINAL, [223, 224, 226]]

After

derived = [[223, 224, 226]]

Conditions: none

THIS BASE set bank requires distinct integer lists;do not silently repair duplicates.Sort ONLY derived display,retaining originals.Do not import another bank SAFE/100-element bound.

Step identifiers and dependenciesRule: Sets_BASE_distinct_integer_lists_no_SAFE_or100cap_no_silent_repair · Uses: G001, S002





S004 · impliesBefore

source roles = [qf-finite-set-union, 221, [221, 223, 225], [223, 224, 226]]

After

derived = [Yes, No, Yes]

Conditions: none

For EVERY label of A∪B check membership in A and B SEPARATELY,then apply THIS union/intersection/A-and-not-B predicate.A\B is not B\A;do not infer from order or pictures.

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





S005 · impliesBefore

source roles = [qf-finite-set-union, 223, [221, 223, 225], [223, 224, 226]]

After

derived = [Yes, Yes, Yes]

Conditions: none

For EVERY label of A∪B check membership in A and B SEPARATELY,then apply THIS union/intersection/A-and-not-B predicate.A\B is not B\A;do not infer from order or pictures.

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





S006 · impliesBefore

source roles = [qf-finite-set-union, 224, [221, 223, 225], [223, 224, 226]]

After

derived = [No, Yes, Yes]

Conditions: none

For EVERY label of A∪B check membership in A and B SEPARATELY,then apply THIS union/intersection/A-and-not-B predicate.A\B is not B\A;do not infer from order or pictures.

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





S007 · impliesBefore

source roles = [qf-finite-set-union, 225, [221, 223, 225], [223, 224, 226]]

After

derived = [Yes, No, Yes]

Conditions: none

For EVERY label of A∪B check membership in A and B SEPARATELY,then apply THIS union/intersection/A-and-not-B predicate.A\B is not B\A;do not infer from order or pictures.

Step identifiers and dependenciesRule: Sets_EVERY_union_label_TWO_memberships_AND_directional_predicate · Uses: G001, S006





S008 · impliesBefore

source roles = [qf-finite-set-union, 226, [221, 223, 225], [223, 224, 226]]

After

derived = [No, Yes, Yes]

Conditions: none

For EVERY label of A∪B check membership in A and B SEPARATELY,then apply THIS union/intersection/A-and-not-B predicate.A\B is not B\A;do not infer from order or pictures.

Step identifiers and dependenciesRule: Sets_EVERY_union_label_TWO_memberships_AND_directional_predicate · Uses: G001, S007





S009 · impliesBefore

source roles = [qf-finite-set-union, [221, 223, 225], [223, 224, 226]]

After

derived = [[221, 223, 224, 225, 226], {221, 223, 224, 225, 226}, not-requested]

Conditions: none

AFTER every label,display the complete numeric increasing set once per element,or∅.For difference ALSO compute B\A to distinguish direction.Keep original A/B and source hashes.

Step identifiers and dependenciesRule: Sets_COMPLETE_extensional_result_numeric_order_reverse_difference_separate · Uses: G001, S008









## Reverse checks


- S009 · premise_and_result_audit — Reverse-check ORIGINAL inputs,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.
- S008 · premise_and_result_audit — Reverse-check ORIGINAL inputs,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.
- S007 · premise_and_result_audit — Reverse-check ORIGINAL inputs,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.
- S006 · premise_and_result_audit — Reverse-check ORIGINAL inputs,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.
- S005 · premise_and_result_audit — Reverse-check ORIGINAL inputs,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.
- S004 · premise_and_result_audit — Reverse-check ORIGINAL inputs,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.
- S003 · premise_and_result_audit — Reverse-check ORIGINAL inputs,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.
- S002 · premise_and_result_audit — Reverse-check ORIGINAL inputs,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.
- S001 · premise_and_result_audit — Reverse-check ORIGINAL inputs,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.





## Result

{221, 223, 224, 225, 226}

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: b86745e9fd533016645d8247f7c744faec9ece81d7f6dedb6ad495479350e280
- candidate record SHA-256: f2380e24a2bf5981a916e72bed459b6a7c4529b798b7fd2fe14c0978789dc14b
- semantic review receipt SHA-256: acf03b776f4855e3869810bcbecf1116e3a2b11dfb1e2a5dfbcf8135ad24a7d0
- question-first witness SHA-256: eadbbac3220988b52754f32450fb536d627217306d1ce28a49134f56c21bfa88
- Lean: NOT_RUN · human review: NOT_PERFORMED

[Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20260915-finite-set-union-0023.json)