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Question 7768: Every Declared Codomain Fiber and Missing Target

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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R0014-07768 · q-20261012-advance-20261016-r07-surjective-0018

Original question and givens

Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[2, 1, 1, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Give (original images, all fibers in codomain order 0,…,m−1, missing targets, surjectivity eligibility). An empty codomain has an empty fiber list.

  • G001 — Finite types start at 0: Fin(5)={0,…,4}→Fin(3), complete images=[2, 1, 1, 0, 1]. Check the whole original total table before any candidate evaluation; empty-function properties depend on the declared codomain. Give (original images, all fibers in codomain order 0,…,m−1, missing targets, surjectivity eligibility). An empty codomain has an empty fiber list.

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

n=5;m=3;images=[2, 1, 1, 0, 1]

After

all original inputs and outputs form a typed total table;labels start at0

Conditions: none

First validate the entire original table as a total function from Fin(n) to Fin(m),including cells not used by a query.Only then evaluate preimages for eligible targets and surjectivity.

Step identifiers and dependencies

Rule: finite_surjection_typed_scope · Uses: G001

S002 · implies

Before

phase=original;target=0;source labels=[0, 1, 2, 3, 4];images on labels=[2, 1, 1, 0, 1]

After

fiber=[3];missing=No

Conditions: none

For this eligible target scan every retained original input and list all preimages,not only the first.A target is missing exactly when its fiber is empty.

Step identifiers and dependencies

Rule: surjection_complete_target_fiber · Uses: G001, S001

S003 · implies

Before

phase=original;target=1;source labels=[0, 1, 2, 3, 4];images on labels=[2, 1, 1, 0, 1]

After

fiber=[1, 2, 4];missing=No

Conditions: none

For this eligible target scan every retained original input and list all preimages,not only the first.A target is missing exactly when its fiber is empty.

Step identifiers and dependencies

Rule: surjection_complete_target_fiber · Uses: G001, S002

S004 · implies

Before

phase=original;target=2;source labels=[0, 1, 2, 3, 4];images on labels=[2, 1, 1, 0, 1]

After

fiber=[0];missing=No

Conditions: none

For this eligible target scan every retained original input and list all preimages,not only the first.A target is missing exactly when its fiber is empty.

Step identifiers and dependencies

Rule: surjection_complete_target_fiber · Uses: G001, S003

S005 · implies

Before

phase=original;targets=[0, 1, 2];fibers=[[3], [1, 2, 4], [0]]

After

missing=[];surjective=Yes

Conditions: none

Surjectivity is relative to the declared codomain:every target needs a preimage.An empty codomain has no targets to test and satisfies the condition;an empty domain with a nonempty codomain misses every target.

Step identifiers and dependencies

Rule: surjection_all_declared_targets · Uses: G001, S004

Reverse checks

  • 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

([2, 1, 1, 0, 1], [[3], [1, 2, 4], [0]], [], 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: 2de0593a0481e9aa9da5ca4d6dfca941946b4a626d3e965c7f3e8c0c184e4df4
  • candidate record SHA-256: 08013aa7d23cf75de4089e6fb44df1f071725210eaf6622346cdc788b9cd361c
  • semantic review receipt SHA-256: 23ba8196e85a09c03add0a87f96bec5d67a00dc648d1d68eef89101525b41a77
  • question-first witness SHA-256: 17d59afc327b7cb8b6e588686d6dc1abbccd384d8ba32b8c046652f96821d705
  • Lean: NOT_RUN · human review: NOT_PERFORMED
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