# Question 35560: Finite Mapping as a Function [Return to catalogue page 178](https://ocme.evemisslab.com/en/questions/page/178/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20260913-foundational-language/#q-20260913-finite-function-mapping-0015) 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. R0070-35560 · q-20260913-finite-function-mapping-0015 ## Original question and givens Let X={x15-1, x15-2, x15-3, x15-4} and Y={y15-1, y15-2, y15-3, y15-4, y15-5}. The finite correspondence is x15-1↦y15-1, x15-2↦y15-1, x15-3↦y15-2. Does it define a function f:X→Y? Keep the codomain Y distinct from the actual range. - G001 — Let X={x15-1, x15-2, x15-3, x15-4} and Y={y15-1, y15-2, y15-3, y15-4, y15-5}. The finite correspondence is x15-1↦y15-1, x15-2↦y15-1, x15-3↦y15-2. 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. - [mko-function-mapping](https://ocme.evemisslab.com/en/mko/mko-function-mapping/) ## Step-by-step derivation S001 · impliesBefore 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 dependenciesRule: R70_SOURCE_bank_specific_interface · Uses: G001 S002 · impliesBefore source roles = [[x15-1, x15-2, x15-3, x15-4], [y15-1, y15-2, y15-3, y15-4, y15-5], [[x15-1, y15-1], [x15-2, y15-1], [x15-3, y15-2]]] 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 dependenciesRule: Function_ORIGINAL_labels_RAW_rows_NO_legacy_dedup · Uses: G001, S001 S003 · impliesBefore source roles = [0, x15-1, y15-1] 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 dependenciesRule: Function_EACH_RAW_assignment_domain_and_codomain · Uses: G001, S002 S004 · impliesBefore source roles = [1, x15-2, y15-1] 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 dependenciesRule: Function_EACH_RAW_assignment_domain_and_codomain · Uses: G001, S003 S005 · impliesBefore source roles = [2, x15-3, y15-2] 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 dependenciesRule: Function_EACH_RAW_assignment_domain_and_codomain · Uses: G001, S004 S006 · impliesBefore source roles = [x15-1, [y15-1]] 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 dependenciesRule: Function_EACH_input_exactly_ONE_RAW_row · Uses: G001, S005 S007 · impliesBefore source roles = [x15-2, [y15-1]] 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 dependenciesRule: Function_EACH_input_exactly_ONE_RAW_row · Uses: G001, S006 S008 · impliesBefore source roles = [x15-3, [y15-2]] 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 dependenciesRule: Function_EACH_input_exactly_ONE_RAW_row · Uses: G001, S007 S009 · impliesBefore source roles = [x15-4, []] After derived = [0, No] 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 dependenciesRule: Function_EACH_input_exactly_ONE_RAW_row · Uses: G001, S008 S010 · impliesBefore source roles = [[x15-4], [], [], [y15-1, y15-2], [y15-1, y15-2, y15-3, y15-4, y15-5]] 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 dependenciesRule: Function_COMPLETE_total_single_valued_into_declared_codomain · Uses: G001, S009 ## Reverse checks - 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: aa97661ed2ea7fd0a68363e5cc0c0b036980e03f2dedb1b4ac50da81989d8d8d - candidate record SHA-256: a2f693ab003573b9974a62aeac4cd5c64f252114e37a99bc8f801e10db7670fe - semantic review receipt SHA-256: 9820ac6529139bd53d516c8bfdfdd41184b0cd52dad69fa433344010db37af80 - question-first witness SHA-256: c9c939762bbd4d3ea3403084c8e95541dc2c7e56eb82fc74eaba8c80a4517dab - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20260913-finite-function-mapping-0015.json)