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Question 33463: The Preimage of a Union Equals the Union of Preimages

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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← Question derivation index · Original question batch

R0065-33463 · q-20261013-advance-20261025-r05-measurable-function-0038

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=[0, 3, 0, 0], target events left=[0], right=[1, 2], 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=[0, 3, 0, 0], target events left=[0], right=[1, 2], 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.

Step-by-step derivation

S001 · implies

Before

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 dependencies

Rule: R65_SOURCE_bank_keys_operation_domain_ordered_output_roles · Uses: G001

S002 · implies

Before

source roles = [4, 4, [0, 3, 0, 0]]

After

derived = [[0, 3, 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 dependencies

Rule: Meas_RAW_complete_total_map_typed_NONEMPTY_carriers · Uses: G001, S001

S003 · implies

Before

source roles = [union LEFT event, 0, 0, [0]]

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 dependencies

Rule: Meas_PREIMAGE_EACH_source_label_target_membership · Uses: G001, S002

S004 · implies

Before

source roles = [union LEFT event, 1, 3, [0]]

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 dependencies

Rule: Meas_PREIMAGE_EACH_source_label_target_membership · Uses: G001, S003

S005 · implies

Before

source roles = [union LEFT event, 2, 0, [0]]

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 dependencies

Rule: Meas_PREIMAGE_EACH_source_label_target_membership · Uses: G001, S004

S006 · implies

Before

source roles = [union LEFT event, 3, 0, [0]]

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 dependencies

Rule: Meas_PREIMAGE_EACH_source_label_target_membership · Uses: G001, S005

S007 · implies

Before

source roles = [union RIGHT event, 0, 0, [1, 2]]

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 dependencies

Rule: Meas_PREIMAGE_EACH_source_label_target_membership · Uses: G001, S006

S008 · implies

Before

source roles = [union RIGHT event, 1, 3, [1, 2]]

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 dependencies

Rule: Meas_PREIMAGE_EACH_source_label_target_membership · Uses: G001, S007

S009 · implies

Before

source roles = [union RIGHT event, 2, 0, [1, 2]]

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 dependencies

Rule: Meas_PREIMAGE_EACH_source_label_target_membership · Uses: G001, S008

S010 · implies

Before

source roles = [union RIGHT event, 3, 0, [1, 2]]

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 dependencies

Rule: Meas_PREIMAGE_EACH_source_label_target_membership · Uses: G001, S009

S011 · implies

Before

source roles = [union TARGET union, 0, 0, [0, 1, 2]]

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 dependencies

Rule: Meas_PREIMAGE_EACH_source_label_target_membership · Uses: G001, S010

S012 · implies

Before

source roles = [union TARGET union, 1, 3, [0, 1, 2]]

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 dependencies

Rule: Meas_PREIMAGE_EACH_source_label_target_membership · Uses: G001, S011

S013 · implies

Before

source roles = [union TARGET union, 2, 0, [0, 1, 2]]

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 dependencies

Rule: Meas_PREIMAGE_EACH_source_label_target_membership · Uses: G001, S012

S014 · implies

Before

source roles = [union TARGET union, 3, 0, [0, 1, 2]]

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 dependencies

Rule: Meas_PREIMAGE_EACH_source_label_target_membership · Uses: G001, S013

S015 · implies

Before

source roles = [[0, 3, 0, 0], [0], [1, 2], [0, 1, 2]]

After

derived = [[0, 2, 3], [], [0, 2, 3], [0, 2, 3], 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 dependencies

Rule: 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

([0, 3, 0, 0], [0], [1, 2], [0, 1, 2], [0, 2, 3], [], [0, 2, 3], [0, 2, 3], 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: eb31d603319449dc0861dd397d322387491d4ce255f9a7cc78eba621e361fb1a
  • candidate record SHA-256: ed2499ff380d81a6c241cc44162d33cae06726f6336dc6f4069aac6fabd6d373
  • semantic review receipt SHA-256: 92bb5fa69818f30cd2806d74a26bd5ede6c48c80d2ecc3d31a5130d4ad7666b6
  • question-first witness SHA-256: 012b5aa905280d616be49aa9462daaa73bf7f6247d0a796e78bc5bcd24e6bd00
  • Lean: NOT_RUN · human review: NOT_PERFORMED
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