# Question 33460: The Preimage of a Union Equals the Union of Preimages [Return to catalogue page 168](https://ocme.evemisslab.com/en/questions/page/168/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261013-advance-20261025-r05-measurable-function-foundations/#q-20261013-advance-20261025-r05-measurable-function-0035) 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. R0065-33460 · q-20261013-advance-20261025-r05-measurable-function-0035 ## Original question and givens Use complete qualified finite sigma-algebras in their already-canonical increasing-bit-mask order and complete total mapping tables on labels starting at 0. All set laws concern inverse images. No continuity, integrability, measure preservation or arbitrary infinite-space conclusion is inferred. source size=4, target size=4, mapping=[1, 2, 0, 0], target events left=[1], right=[2, 3], normalized as sets. Give (mapping,left,right,target union,left preimage,right preimage,preimage of union,union of preimages,equal?). This is a complete finite set-law audit. - G001 — Use complete qualified finite sigma-algebras in their already-canonical increasing-bit-mask order and complete total mapping tables on labels starting at 0. All set laws concern inverse images. No continuity, integrability, measure preservation or arbitrary infinite-space conclusion is inferred. source size=4, target size=4, mapping=[1, 2, 0, 0], target events left=[1], right=[2, 3], normalized as sets. Give (mapping,left,right,target union,left preimage,right preimage,preimage of union,union of preimages,equal?). This is a complete finite set-law audit. ## 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-measurable-function](https://ocme.evemisslab.com/en/mko/mko-measurable-function/) ## Step-by-step derivation S001 · impliesBefore source roles = [qf-mod25-measdeep-preimage-union-law, measdeep-preimage-union-law, Finite_preimage_law_density, []] After derived = [[mapping, left, right, union, preimageL, preimageR, union_preimage, preimage_union, equal]] Conditions: none MOD25 uses LOCAL hyphens/named domains,legacy matrices underscores/reals;BASE logic acquires no domain/op.Output tuples,matrices,numeric fields and fraction texts follow THIS source,not a universal/1 rule.Retain originals/English/AST/status. Step identifiers and dependenciesRule: R65_SOURCE_bank_keys_operation_domain_ordered_output_roles · Uses: G001 S002 · impliesBefore source roles = [4, 4, [1, 2, 0, 0]] After derived = [[1, 2, 0, 0]] Conditions: none THIS source slice has carrier sizes1..7 and a typed target at EVERY original source index.Validate the WHOLE table before preimages;neither injectivity nor an inverse function is assumed. Step identifiers and dependenciesRule: Meas_RAW_complete_total_map_typed_NONEMPTY_carriers · Uses: G001, S001 S003 · impliesBefore source roles = [union LEFT event, 0, 1, [1]] After derived = [Yes] Conditions: none Source label i lies in the PREIMAGE IFF mapping[i]belongs to the TARGET event.Keep carrier roles distinct;do not substitute a forward image. Step identifiers and dependenciesRule: Meas_PREIMAGE_EACH_source_label_target_membership · Uses: G001, S002 S004 · impliesBefore source roles = [union LEFT event, 1, 2, [1]] After derived = [No] Conditions: none Source label i lies in the PREIMAGE IFF mapping[i]belongs to the TARGET event.Keep carrier roles distinct;do not substitute a forward image. Step identifiers and dependenciesRule: Meas_PREIMAGE_EACH_source_label_target_membership · Uses: G001, S003 S005 · impliesBefore source roles = [union LEFT event, 2, 0, [1]] After derived = [No] Conditions: none Source label i lies in the PREIMAGE IFF mapping[i]belongs to the TARGET event.Keep carrier roles distinct;do not substitute a forward image. Step identifiers and dependenciesRule: Meas_PREIMAGE_EACH_source_label_target_membership · Uses: G001, S004 S006 · impliesBefore source roles = [union LEFT event, 3, 0, [1]] After derived = [No] Conditions: none Source label i lies in the PREIMAGE IFF mapping[i]belongs to the TARGET event.Keep carrier roles distinct;do not substitute a forward image. Step identifiers and dependenciesRule: Meas_PREIMAGE_EACH_source_label_target_membership · Uses: G001, S005 S007 · impliesBefore source roles = [union RIGHT event, 0, 1, [2, 3]] After derived = [No] Conditions: none Source label i lies in the PREIMAGE IFF mapping[i]belongs to the TARGET event.Keep carrier roles distinct;do not substitute a forward image. Step identifiers and dependenciesRule: Meas_PREIMAGE_EACH_source_label_target_membership · Uses: G001, S006 S008 · impliesBefore source roles = [union RIGHT event, 1, 2, [2, 3]] After derived = [Yes] Conditions: none Source label i lies in the PREIMAGE IFF mapping[i]belongs to the TARGET event.Keep carrier roles distinct;do not substitute a forward image. Step identifiers and dependenciesRule: Meas_PREIMAGE_EACH_source_label_target_membership · Uses: G001, S007 S009 · impliesBefore source roles = [union RIGHT event, 2, 0, [2, 3]] After derived = [No] Conditions: none Source label i lies in the PREIMAGE IFF mapping[i]belongs to the TARGET event.Keep carrier roles distinct;do not substitute a forward image. Step identifiers and dependenciesRule: Meas_PREIMAGE_EACH_source_label_target_membership · Uses: G001, S008 S010 · impliesBefore source roles = [union RIGHT event, 3, 0, [2, 3]] After derived = [No] Conditions: none Source label i lies in the PREIMAGE IFF mapping[i]belongs to the TARGET event.Keep carrier roles distinct;do not substitute a forward image. Step identifiers and dependenciesRule: Meas_PREIMAGE_EACH_source_label_target_membership · Uses: G001, S009 S011 · impliesBefore source roles = [union TARGET union, 0, 1, [1, 2, 3]] After derived = [Yes] Conditions: none Source label i lies in the PREIMAGE IFF mapping[i]belongs to the TARGET event.Keep carrier roles distinct;do not substitute a forward image. Step identifiers and dependenciesRule: Meas_PREIMAGE_EACH_source_label_target_membership · Uses: G001, S010 S012 · impliesBefore source roles = [union TARGET union, 1, 2, [1, 2, 3]] After derived = [Yes] Conditions: none Source label i lies in the PREIMAGE IFF mapping[i]belongs to the TARGET event.Keep carrier roles distinct;do not substitute a forward image. Step identifiers and dependenciesRule: Meas_PREIMAGE_EACH_source_label_target_membership · Uses: G001, S011 S013 · impliesBefore source roles = [union TARGET union, 2, 0, [1, 2, 3]] After derived = [No] Conditions: none Source label i lies in the PREIMAGE IFF mapping[i]belongs to the TARGET event.Keep carrier roles distinct;do not substitute a forward image. Step identifiers and dependenciesRule: Meas_PREIMAGE_EACH_source_label_target_membership · Uses: G001, S012 S014 · impliesBefore source roles = [union TARGET union, 3, 0, [1, 2, 3]] After derived = [No] Conditions: none Source label i lies in the PREIMAGE IFF mapping[i]belongs to the TARGET event.Keep carrier roles distinct;do not substitute a forward image. Step identifiers and dependenciesRule: Meas_PREIMAGE_EACH_source_label_target_membership · Uses: G001, S013 S015 · impliesBefore source roles = [[1, 2, 0, 0], [1], [2, 3], [1, 2, 3]] After derived = [[0], [1], [0, 1], [0, 1], Yes] Conditions: none Compare BOTH routes:union THEN preimage versus separate preimages THEN union.Both results are SOURCE-label sets,not substituted forward-image statements. Step identifiers and dependenciesRule: Meas_UNION_BOTH_source_set_constructions_complete · Uses: G001, S014 ## Reverse checks - S015 · premise_and_result_audit — Reverse-check ORIGINAL data,types,ordered roles,domain and premises;the conclusion need not uniquely recover its premises.Keep finite source candidates,independent review,formal proof and publication separate. - S014 · premise_and_result_audit — Reverse-check ORIGINAL data,types,ordered roles,domain and premises;the conclusion need not uniquely recover its premises.Keep finite source candidates,independent review,formal proof and publication separate. - S013 · premise_and_result_audit — Reverse-check ORIGINAL data,types,ordered roles,domain and premises;the conclusion need not uniquely recover its premises.Keep finite source candidates,independent review,formal proof and publication separate. - S012 · premise_and_result_audit — Reverse-check ORIGINAL data,types,ordered roles,domain and premises;the conclusion need not uniquely recover its premises.Keep finite source candidates,independent review,formal proof and publication separate. - S011 · premise_and_result_audit — Reverse-check ORIGINAL data,types,ordered roles,domain and premises;the conclusion need not uniquely recover its premises.Keep finite source candidates,independent review,formal proof and publication separate. - S010 · premise_and_result_audit — Reverse-check ORIGINAL data,types,ordered roles,domain and premises;the conclusion need not uniquely recover its premises.Keep finite source candidates,independent review,formal proof and publication separate. - S009 · premise_and_result_audit — Reverse-check ORIGINAL data,types,ordered roles,domain and premises;the conclusion need not uniquely recover its premises.Keep finite source candidates,independent review,formal proof and publication separate. - S008 · premise_and_result_audit — Reverse-check ORIGINAL data,types,ordered roles,domain and premises;the conclusion need not uniquely recover its premises.Keep finite source candidates,independent review,formal proof and publication separate. - S007 · premise_and_result_audit — Reverse-check ORIGINAL data,types,ordered roles,domain and premises;the conclusion need not uniquely recover its premises.Keep finite source candidates,independent review,formal proof and publication separate. - S006 · premise_and_result_audit — Reverse-check ORIGINAL data,types,ordered roles,domain and premises;the conclusion need not uniquely recover its premises.Keep finite source candidates,independent review,formal proof and publication separate. - S005 · premise_and_result_audit — Reverse-check ORIGINAL data,types,ordered roles,domain and premises;the conclusion need not uniquely recover its premises.Keep finite source candidates,independent review,formal proof and publication separate. - S004 · premise_and_result_audit — Reverse-check ORIGINAL data,types,ordered roles,domain and premises;the conclusion need not uniquely recover its premises.Keep finite source candidates,independent review,formal proof and publication separate. - S003 · premise_and_result_audit — Reverse-check ORIGINAL data,types,ordered roles,domain and premises;the conclusion need not uniquely recover its premises.Keep finite source candidates,independent review,formal proof and publication separate. - S002 · premise_and_result_audit — Reverse-check ORIGINAL data,types,ordered roles,domain and premises;the conclusion need not uniquely recover its premises.Keep finite source candidates,independent review,formal proof and publication separate. - S001 · premise_and_result_audit — Reverse-check ORIGINAL data,types,ordered roles,domain and premises;the conclusion need not uniquely recover its premises.Keep finite source candidates,independent review,formal proof and publication separate. ## Result ([1, 2, 0, 0], [1], [2, 3], [1, 2, 3], [0], [1], [0, 1], [0, 1], 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: 4eb9b4b64503f503aac9927556d091725b1ec1aee3ec242bcb0a5f2d0979cfb6 - candidate record SHA-256: 945c1ad4963bd12c009b1e1b337eb7c09d7a2b4c5a06436bc2b058dc95e1ccb3 - semantic review receipt SHA-256: 92bb5fa69818f30cd2806d74a26bd5ede6c48c80d2ecc3d31a5130d4ad7666b6 - question-first witness SHA-256: 012b5aa905280d616be49aa9462daaa73bf7f6247d0a796e78bc5bcd24e6bd00 - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261013-advance-20261025-r05-measurable-function-0035.json)