# Question 11481: The Inverse Image under a Complete Finite Total Map

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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.

R0021-11481 · q-20261013-advance-20261024-r02-measurable-function-0006



## Original question and givens

Use carriers Fin(4) and Fin(4), complete total mapping=[1, 1, 0, 0] indexed by source labels, and target event=[1, 2] 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=[1, 1, 0, 0] indexed by source labels, and target event=[1, 2] 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.


- [mko-measurable-function](https://ocme.evemisslab.com/en/mko/mko-measurable-function/)





## Step-by-step derivation

S001 · impliesBefore

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

After

sorted distinct target event=[1, 2]

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 dependenciesRule: preimage_event_set_normalization · Uses: G001





S002 · impliesBefore

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

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 dependenciesRule: preimage_each_source_label · Uses: G001, S001





S003 · impliesBefore

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

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 dependenciesRule: preimage_each_source_label · Uses: G001, S002





S004 · impliesBefore

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

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 dependenciesRule: preimage_each_source_label · Uses: G001, S003





S005 · impliesBefore

source label=3;image=0;target event=[1, 2]

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 dependenciesRule: preimage_each_source_label · Uses: G001, S004





S006 · impliesBefore

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

After

preimage=[0, 1]

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 dependenciesRule: 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, [1, 1, 0, 0], [1, 2], [0, 1])

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: 0ba13a869693f7145bf4b0f03bf8ed7ab1850fa2284104ccb81ea888c9cd215d
- candidate record SHA-256: 3492054a656c9d80c90da70ba90a4ff00d3fb4271bbfca7f286c676028cd5a1d
- semantic review receipt SHA-256: 47d23d3735987d68e27733b051bdfa80a3616d44fb223b82bf2e6c8e46d49258
- question-first witness SHA-256: a857db558000724020d2d5e815cb6bc3e2c9e2933df295bf33eb478d3c695139
- Lean: NOT_RUN · human review: NOT_PERFORMED

[Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261013-advance-20261024-r02-measurable-function-0006.json)