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Question 11484: The Inverse Image under a Complete Finite Total Map

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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R0021-11484 · q-20261013-advance-20261024-r02-measurable-function-0009

Original question and givens

Use carriers Fin(4) and Fin(4), complete total mapping=[0, 2, 0, 0] indexed by source labels, and target event=[0, 1] normalized to sorted distinct labels. Give (source size,target size,mapping,event,all source labels mapping into the event). This family computes a preimage without claiming a sigma-algebra qualification.

  • G001 — Use carriers Fin(4) and Fin(4), complete total mapping=[0, 2, 0, 0] indexed by source labels, and target event=[0, 1] normalized to sorted distinct labels. Give (source size,target size,mapping,event,all source labels mapping into the event). This family computes a preimage without claiming a sigma-algebra qualification.

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

raw target event=[0, 1];mapping=[0, 2, 0, 0];source Fin(4);target Fin(4)

After

sorted distinct target event=[0, 1]

Conditions: none

Normalize only the target event as a sorted set.Keep the total map indexed by source.This exercise has no sigma-algebra premise and computes a set preimage only,not measurability.

Step identifiers and dependencies

Rule: preimage_event_set_normalization · Uses: G001

S002 · implies

Before

source label=0;image=0;target event=[0, 1]

After

image in event=Yes;include source label=Yes

Conditions: none

Test image membership in the target event,but include the original source label,not the image value.

Step identifiers and dependencies

Rule: preimage_each_source_label · Uses: G001, S001

S003 · implies

Before

source label=1;image=2;target event=[0, 1]

After

image in event=No;include source label=No

Conditions: none

Test image membership in the target event,but include the original source label,not the image value.

Step identifiers and dependencies

Rule: preimage_each_source_label · Uses: G001, S002

S004 · implies

Before

source label=2;image=0;target event=[0, 1]

After

image in event=Yes;include source label=Yes

Conditions: none

Test image membership in the target event,but include the original source label,not the image value.

Step identifiers and dependencies

Rule: preimage_each_source_label · Uses: G001, S003

S005 · implies

Before

source label=3;image=0;target event=[0, 1]

After

image in event=Yes;include source label=Yes

Conditions: none

Test image membership in the target event,but include the original source label,not the image value.

Step identifiers and dependencies

Rule: preimage_each_source_label · Uses: G001, S004

S006 · implies

Before

complete source carrier=[0, 1, 2, 3];target event=[0, 1]

After

preimage=[0, 2, 3]

Conditions: none

The complete source carrier has been checked.Empty and full preimages are valid;source and target sizes need not match.

Step identifiers and dependencies

Rule: preimage_complete_source_inventory · Uses: G001, S005

Reverse checks

  • 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

(4, 4, [0, 2, 0, 0], [0, 1], [0, 2, 3])

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: 4fc7f4575964d06c69bc19afc637aa30ad78ff17a1b399f3a92a9ee019842307
  • candidate record SHA-256: e24a1e09f17b3a14f2ad7f45f11f5a1b5078e9ec5b9488d27016bce52b48dac7
  • semantic review receipt SHA-256: 47d23d3735987d68e27733b051bdfa80a3616d44fb223b82bf2e6c8e46d49258
  • question-first witness SHA-256: a857db558000724020d2d5e815cb6bc3e2c9e2933df295bf33eb478d3c695139
  • Lean: NOT_RUN · human review: NOT_PERFORMED
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