# Question 32507: Injectivity of a Finite Total Function [Return to catalogue page 163](https://ocme.evemisslab.com/en/questions/page/163/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261004-advance-20261005-r01-injective-foundations/#q-20261004-advance-20261005-r01-injective-0007) 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. R0064-32507 · q-20261004-advance-20261005-r01-injective-0007 ## Original question and givens The declared total function f:A→B has A={-2, -1, 0}, B={1, 3, 5, 7}, and table {(-2, 1), (-1, 3), (0, 5)}. Is f injective? Compare distinct permitted inputs rather than guessing from the codomain size. - G001 — The declared total function f:A→B has A={-2, -1, 0}, B={1, 3, 5, 7}, and table {(-2, 1), (-1, 3), (0, 5)}. Is f injective? Compare distinct permitted inputs rather than guessing from the codomain size. ## 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-injective](https://ocme.evemisslab.com/en/mko/mko-injective/) ## Step-by-step derivation S001 · impliesBefore source roles = [qf-injective-table-property, injective, no-required-domain, []] After derived = [[injective_YesNo_STRING]] Conditions: none Legacy banks enforce their exact source keys/operation/domain.BASE integers retain metadata without treating it as an oracle.Distinguish NUMBER,string,fraction pair and vector equality formats;retain original bilingual data/AST/status. Step identifiers and dependenciesRule: R64_SOURCE_exact_legacy_interface_BASE_metadata_AND_output_roles · Uses: G001 S002 · impliesBefore source roles = [ORIGINAL f domain, [-2, -1, 0]] After derived = [[-2, -1, 0]] Conditions: none RAW sets are distinct,SAFE and≤100.Sort only a derived work table,retaining original lists/hashes. Step identifiers and dependenciesRule: FiniteMap_RAW_distinct_SAFE_set_bound100 · Uses: G001, S001 S003 · impliesBefore source roles = [ORIGINAL f codomain, [1, 3, 5, 7]] After derived = [[1, 3, 5, 7]] Conditions: none RAW sets are distinct,SAFE and≤100.Sort only a derived work table,retaining original lists/hashes. Step identifiers and dependenciesRule: FiniteMap_RAW_distinct_SAFE_set_bound100 · Uses: G001, S002 S004 · impliesBefore source roles = [ORIGINAL f, [[-2, 1], [-1, 3], [0, 5]]] After derived = [[[-2, 1], [-1, 3], [0, 5]]] Conditions: none FIRST type EVERY raw pair and enforce raw length≤200,THEN normalize identical rows as one ordered pair.Do not merge different outputs of one input.Empty domains do not waive RAW checks. Step identifiers and dependenciesRule: FiniteMap_RAW200_bound_BEFORE_identical_row_ASSET_normalization · Uses: G001, S003 S005 · impliesBefore source roles = [ORIGINAL f, 0, -2, 1, [-2, -1, 0], [1, 3, 5, 7]] After derived = [Yes, Yes, Yes] Conditions: none EVERY row input belongs to the domain and output to the codomain,including orphan/unused rows. Step identifiers and dependenciesRule: FiniteMap_EACH_normalized_row_declared_domain_codomain_types · Uses: G001, S004 S006 · impliesBefore source roles = [ORIGINAL f, 1, -1, 3, [-2, -1, 0], [1, 3, 5, 7]] After derived = [Yes, Yes, Yes] Conditions: none EVERY row input belongs to the domain and output to the codomain,including orphan/unused rows. Step identifiers and dependenciesRule: FiniteMap_EACH_normalized_row_declared_domain_codomain_types · Uses: G001, S005 S007 · impliesBefore source roles = [ORIGINAL f, 2, 0, 5, [-2, -1, 0], [1, 3, 5, 7]] After derived = [Yes, Yes, Yes] Conditions: none EVERY row input belongs to the domain and output to the codomain,including orphan/unused rows. Step identifiers and dependenciesRule: FiniteMap_EACH_normalized_row_declared_domain_codomain_types · Uses: G001, S006 S008 · impliesBefore source roles = [ORIGINAL f, -2, [[-2, 1], [-1, 3], [0, 5]]] After derived = [1, Yes] Conditions: none EACH declared input needs EXACTLY one row for totality.A first lookup must not hide missing or multiple outputs. Step identifiers and dependenciesRule: FiniteMap_EACH_declared_input_exactly_one_output_row · Uses: G001, S007 S009 · impliesBefore source roles = [ORIGINAL f, -1, [[-2, 1], [-1, 3], [0, 5]]] After derived = [1, Yes] Conditions: none EACH declared input needs EXACTLY one row for totality.A first lookup must not hide missing or multiple outputs. Step identifiers and dependenciesRule: FiniteMap_EACH_declared_input_exactly_one_output_row · Uses: G001, S008 S010 · impliesBefore source roles = [ORIGINAL f, 0, [[-2, 1], [-1, 3], [0, 5]]] After derived = [1, Yes] Conditions: none EACH declared input needs EXACTLY one row for totality.A first lookup must not hide missing or multiple outputs. Step identifiers and dependenciesRule: FiniteMap_EACH_declared_input_exactly_one_output_row · Uses: G001, S009 S011 · impliesBefore source roles = [ORIGINAL f, [-2, -1, 0], [1, 3, 5, 7], [[-2, 1], [-1, 3], [0, 5]]] After derived = [[Yes, Yes, Yes], [1, 1, 1], Yes] Conditions: none Permit lookups ONLY after BOTH all-row typing and all-input totality.Empty-set vacuity cannot conceal an invalid row. Step identifiers and dependenciesRule: FiniteMap_TOTAL_qualification_BEFORE_lookups_even_empty_domain · Uses: G001, S010 S012 · impliesBefore source roles = [ORIGINAL f, 1, [-2, -1, 0], [[-2, 1], [-1, 3], [0, 5]]] After derived = [[-2], 1] Conditions: none List ALL preimages at EVERY codomain label,including empty fibers;do not delete unattained B labels. Step identifiers and dependenciesRule: FiniteMap_EVERY_codomain_fiber_including_unattained · Uses: G001, S011 S013 · impliesBefore source roles = [ORIGINAL f, 3, [-2, -1, 0], [[-2, 1], [-1, 3], [0, 5]]] After derived = [[-1], 1] Conditions: none List ALL preimages at EVERY codomain label,including empty fibers;do not delete unattained B labels. Step identifiers and dependenciesRule: FiniteMap_EVERY_codomain_fiber_including_unattained · Uses: G001, S012 S014 · impliesBefore source roles = [ORIGINAL f, 5, [-2, -1, 0], [[-2, 1], [-1, 3], [0, 5]]] After derived = [[0], 1] Conditions: none List ALL preimages at EVERY codomain label,including empty fibers;do not delete unattained B labels. Step identifiers and dependenciesRule: FiniteMap_EVERY_codomain_fiber_including_unattained · Uses: G001, S013 S015 · impliesBefore source roles = [ORIGINAL f, 7, [-2, -1, 0], [[-2, 1], [-1, 3], [0, 5]]] After derived = [[], 0] Conditions: none List ALL preimages at EVERY codomain label,including empty fibers;do not delete unattained B labels. Step identifiers and dependenciesRule: FiniteMap_EVERY_codomain_fiber_including_unattained · Uses: G001, S014 S016 · impliesBefore source roles = [ORIGINAL f, [-2, -1, 0], [1, 3, 5, 7], [[-2], [-1], [0], []]] After derived = [Yes, No, No] Conditions: none Injective means every fiber≤1,surjective every fiber≥1,and BOTH mean bijective.Unattained B does not defeat injectivity.Empty→empty is bijective;empty→nonempty is only injective. Step identifiers and dependenciesRule: FiniteMap_injective_surjective_BOTH_needed_for_inverse · Uses: G001, S015 ## Reverse checks - S016 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate. - S015 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate. - S014 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate. - S013 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate. - S012 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate. - S011 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate. - S010 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate. - S009 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate. - S008 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate. - S007 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate. - S006 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate. - S005 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate. - S004 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate. - S003 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate. - S002 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate. - S001 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate. ## Result 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: 5f3a187c005993b0e00a8fedfff235b6964aa242d694ddcbe1344192db15f568 - candidate record SHA-256: 2e2a51f4c6c859a5465c329f4cfa2c1dda6f9ffe027085bd65b9736cb8cf1581 - semantic review receipt SHA-256: 8b9ea3ddfab468e9e3f0a0fc0f7682987d5e030be0d7c8860fd85df19cbe183b - question-first witness SHA-256: 066bfeab8fc279706861faade71b9a18a61a25235fc48700092f0488d2a6da6a - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261004-advance-20261005-r01-injective-0007.json)