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Question 35557: Finite Mapping as a Function

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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R0070-35557 · q-20260913-finite-function-mapping-0012

Original question and givens

Let X={x12-1, x12-2, x12-3, x12-4} and Y={y12-1, y12-2, y12-3, y12-4, y12-5}. The finite correspondence is x12-1↦y12-1, x12-2↦y12-1, x12-3↦z12, x12-4↦y12-1. Does it define a function f:X→Y? Keep the codomain Y distinct from the actual range.

  • G001 — Let X={x12-1, x12-2, x12-3, x12-4} and Y={y12-1, y12-2, y12-3, y12-4, y12-5}. The finite correspondence is x12-1↦y12-1, x12-2↦y12-1, x12-3↦z12, x12-4↦y12-1. Does it define a function f:X→Y? Keep the codomain Y distinct from the actual range.

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-finite-function-mapping, [codomain, domain, pairs, scenario]]

After

derived = [function, null, null]

Conditions: none

Use THIS bank’s actual fields,operation/domain and types.BASE retains permissible extra metadata.Do not import LDL SAFE/set limits or a different bank’s deduplication rules.

Step identifiers and dependencies

Rule: R70_SOURCE_bank_specific_interface · Uses: G001

S002 · implies

Before

source roles = [[x12-1, x12-2, x12-3, x12-4], [y12-1, y12-2, y12-3, y12-4, y12-5], [[x12-1, y12-1], [x12-2, y12-1], [x12-3, z12], [x12-4, y12-1]]]

After

derived = [one RAW assignment per domain input]

Conditions: none

THIS original finite-function bank requires exactly ONE RAW assignment row for every input.Do not borrow another bank’s duplicate-pair normalization.The codomain may exceed the range;check multiplicity,missing inputs and outside values directly.

Step identifiers and dependencies

Rule: Function_ORIGINAL_labels_RAW_rows_NO_legacy_dedup · Uses: G001, S001

S003 · implies

Before

source roles = [0, x12-1, y12-1]

After

derived = [Yes]

Conditions: none

Check EACH RAW row for x∈A and y∈B.Retain duplicate rows and row indices;never discard a bad row and then announce a function.

Step identifiers and dependencies

Rule: Function_EACH_RAW_assignment_domain_and_codomain · Uses: G001, S002

S004 · implies

Before

source roles = [1, x12-2, y12-1]

After

derived = [Yes]

Conditions: none

Check EACH RAW row for x∈A and y∈B.Retain duplicate rows and row indices;never discard a bad row and then announce a function.

Step identifiers and dependencies

Rule: Function_EACH_RAW_assignment_domain_and_codomain · Uses: G001, S003

S005 · implies

Before

source roles = [2, x12-3, z12]

After

derived = [No]

Conditions: none

Check EACH RAW row for x∈A and y∈B.Retain duplicate rows and row indices;never discard a bad row and then announce a function.

Step identifiers and dependencies

Rule: Function_EACH_RAW_assignment_domain_and_codomain · Uses: G001, S004

S006 · implies

Before

source roles = [3, x12-4, y12-1]

After

derived = [Yes]

Conditions: none

Check EACH RAW row for x∈A and y∈B.Retain duplicate rows and row indices;never discard a bad row and then announce a function.

Step identifiers and dependencies

Rule: Function_EACH_RAW_assignment_domain_and_codomain · Uses: G001, S005

S007 · implies

Before

source roles = [x12-1, [y12-1]]

After

derived = [1, Yes]

Conditions: none

For EVERY domain input,check EXACTLY ONE RAW assignment.Missing inputs fail;even two identical-output rows cannot silently deduplicate in this ORIGINAL contract.

Step identifiers and dependencies

Rule: Function_EACH_input_exactly_ONE_RAW_row · Uses: G001, S006

S008 · implies

Before

source roles = [x12-2, [y12-1]]

After

derived = [1, Yes]

Conditions: none

For EVERY domain input,check EXACTLY ONE RAW assignment.Missing inputs fail;even two identical-output rows cannot silently deduplicate in this ORIGINAL contract.

Step identifiers and dependencies

Rule: Function_EACH_input_exactly_ONE_RAW_row · Uses: G001, S007

S009 · implies

Before

source roles = [x12-3, [z12]]

After

derived = [1, Yes]

Conditions: none

For EVERY domain input,check EXACTLY ONE RAW assignment.Missing inputs fail;even two identical-output rows cannot silently deduplicate in this ORIGINAL contract.

Step identifiers and dependencies

Rule: Function_EACH_input_exactly_ONE_RAW_row · Uses: G001, S008

S010 · implies

Before

source roles = [x12-4, [y12-1]]

After

derived = [1, Yes]

Conditions: none

For EVERY domain input,check EXACTLY ONE RAW assignment.Missing inputs fail;even two identical-output rows cannot silently deduplicate in this ORIGINAL contract.

Step identifiers and dependencies

Rule: Function_EACH_input_exactly_ONE_RAW_row · Uses: G001, S009

S011 · implies

Before

source roles = [[], [], [[2, x12-3, z12]], [y12-1, z12], [y12-1, y12-2, y12-3, y12-4, y12-5]]

After

derived = [No]

Conditions: none

A function A→B requires absence of ALL three violations.Different inputs may share an output;injectivity/surjectivity is not required.The actual range does not replace the declared B.

Step identifiers and dependencies

Rule: Function_COMPLETE_total_single_valued_into_declared_codomain · Uses: G001, S010

Reverse checks

  • S011 · premise_and_result_audit — Reverse-audit THIS question’s original operands,roles and necessary premises;reconstruct the next step,without reversing a one-way implication.Numerical/finite-table checking is not formal proof of an arbitrary theorem.Original Q,bilingual sources,MKO,AST,status and history remain preserved.
  • S010 · premise_and_result_audit — Reverse-audit THIS question’s original operands,roles and necessary premises;reconstruct the next step,without reversing a one-way implication.Numerical/finite-table checking is not formal proof of an arbitrary theorem.Original Q,bilingual sources,MKO,AST,status and history remain preserved.
  • S009 · premise_and_result_audit — Reverse-audit THIS question’s original operands,roles and necessary premises;reconstruct the next step,without reversing a one-way implication.Numerical/finite-table checking is not formal proof of an arbitrary theorem.Original Q,bilingual sources,MKO,AST,status and history remain preserved.
  • S008 · premise_and_result_audit — Reverse-audit THIS question’s original operands,roles and necessary premises;reconstruct the next step,without reversing a one-way implication.Numerical/finite-table checking is not formal proof of an arbitrary theorem.Original Q,bilingual sources,MKO,AST,status and history remain preserved.
  • S007 · premise_and_result_audit — Reverse-audit THIS question’s original operands,roles and necessary premises;reconstruct the next step,without reversing a one-way implication.Numerical/finite-table checking is not formal proof of an arbitrary theorem.Original Q,bilingual sources,MKO,AST,status and history remain preserved.
  • S006 · premise_and_result_audit — Reverse-audit THIS question’s original operands,roles and necessary premises;reconstruct the next step,without reversing a one-way implication.Numerical/finite-table checking is not formal proof of an arbitrary theorem.Original Q,bilingual sources,MKO,AST,status and history remain preserved.
  • S005 · premise_and_result_audit — Reverse-audit THIS question’s original operands,roles and necessary premises;reconstruct the next step,without reversing a one-way implication.Numerical/finite-table checking is not formal proof of an arbitrary theorem.Original Q,bilingual sources,MKO,AST,status and history remain preserved.
  • S004 · premise_and_result_audit — Reverse-audit THIS question’s original operands,roles and necessary premises;reconstruct the next step,without reversing a one-way implication.Numerical/finite-table checking is not formal proof of an arbitrary theorem.Original Q,bilingual sources,MKO,AST,status and history remain preserved.
  • S003 · premise_and_result_audit — Reverse-audit THIS question’s original operands,roles and necessary premises;reconstruct the next step,without reversing a one-way implication.Numerical/finite-table checking is not formal proof of an arbitrary theorem.Original Q,bilingual sources,MKO,AST,status and history remain preserved.
  • S002 · premise_and_result_audit — Reverse-audit THIS question’s original operands,roles and necessary premises;reconstruct the next step,without reversing a one-way implication.Numerical/finite-table checking is not formal proof of an arbitrary theorem.Original Q,bilingual sources,MKO,AST,status and history remain preserved.
  • S001 · premise_and_result_audit — Reverse-audit THIS question’s original operands,roles and necessary premises;reconstruct the next step,without reversing a one-way implication.Numerical/finite-table checking is not formal proof of an arbitrary theorem.Original Q,bilingual sources,MKO,AST,status and history remain preserved.

Result

No

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: 328e996ee27e458e2127e7f782512baf788d48e1e2a46d04efdad5fc81f85d66
  • candidate record SHA-256: 622f3fd856fa2600237bcc39b84972609ff034a68cbd94fdabcc884a28406057
  • semantic review receipt SHA-256: 9820ac6529139bd53d516c8bfdfdd41184b0cd52dad69fa433344010db37af80
  • question-first witness SHA-256: c9c939762bbd4d3ea3403084c8e95541dc2c7e56eb82fc74eaba8c80a4517dab
  • Lean: NOT_RUN · human review: NOT_PERFORMED
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