# Question 5585: Actual Range and Unused Codomain [Return to catalogue page 28](https://ocme.evemisslab.com/en/questions/page/28/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261008-advance-20261012-r07-function-mapping-foundations/#q-20261008-advance-20261012-r07-function-mapping-0035) 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. R0010-05585 · q-20261008-advance-20261012-r07-function-mapping-0035 ## Original question and givens The complete total map f:Fin(3)→Fin(8) has images=[0, 1, 2] indexed from 0; Fin(3)={0,…,2}. Every table entry must lie in the target type. Give (increasing actual range, increasing unused codomain elements, does the range equal the entire codomain?). - G001 — The complete total map f:Fin(3)→Fin(8) has images=[0, 1, 2] indexed from 0; Fin(3)={0,…,2}. Every table entry must lie in the target type. Give (increasing actual range, increasing unused codomain elements, does the range equal the entire codomain?). ## 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 f:Fin(3)→Fin(8);source indices start at0 After complete total map,not a partial program function Conditions: none Every legal source element has one codomain value.Fin(0) is empty;an outside query is ill-typed,not a hole of a partial function at a legal input. Step identifiers and dependenciesRule: total_finite_mapping_scope · Uses: G001 S002 · impliesBefore f input=0∈Fin(3) After f(0)=0∈Fin(8) Conditions: none Audit source and codomain eligibility for every entry;an unqueried entry may not be illegal. Step identifiers and dependenciesRule: complete_mapping_entry · Uses: G001, S001 S003 · impliesBefore f input=1∈Fin(3) After f(1)=1∈Fin(8) Conditions: none Audit source and codomain eligibility for every entry;an unqueried entry may not be illegal. Step identifiers and dependenciesRule: complete_mapping_entry · Uses: G001, S002 S004 · impliesBefore f input=2∈Fin(3) After f(2)=2∈Fin(8) Conditions: none Audit source and codomain eligibility for every entry;an unqueried entry may not be illegal. Step identifiers and dependenciesRule: complete_mapping_entry · Uses: G001, S003 S005 · impliesBefore codomain element=0 After preimages=[0];used=Yes Conditions: none Audit every codomain target;an unhit target remains in the codomain but not in the actual range. Step identifiers and dependenciesRule: codomain_fiber_inventory · Uses: G001, S004 S006 · impliesBefore codomain element=1 After preimages=[1];used=Yes Conditions: none Audit every codomain target;an unhit target remains in the codomain but not in the actual range. Step identifiers and dependenciesRule: codomain_fiber_inventory · Uses: G001, S005 S007 · impliesBefore codomain element=2 After preimages=[2];used=Yes Conditions: none Audit every codomain target;an unhit target remains in the codomain but not in the actual range. Step identifiers and dependenciesRule: codomain_fiber_inventory · Uses: G001, S006 S008 · impliesBefore codomain element=3 After preimages=[];used=No Conditions: none Audit every codomain target;an unhit target remains in the codomain but not in the actual range. Step identifiers and dependenciesRule: codomain_fiber_inventory · Uses: G001, S007 S009 · impliesBefore codomain element=4 After preimages=[];used=No Conditions: none Audit every codomain target;an unhit target remains in the codomain but not in the actual range. Step identifiers and dependenciesRule: codomain_fiber_inventory · Uses: G001, S008 S010 · impliesBefore codomain element=5 After preimages=[];used=No Conditions: none Audit every codomain target;an unhit target remains in the codomain but not in the actual range. Step identifiers and dependenciesRule: codomain_fiber_inventory · Uses: G001, S009 S011 · impliesBefore codomain element=6 After preimages=[];used=No Conditions: none Audit every codomain target;an unhit target remains in the codomain but not in the actual range. Step identifiers and dependenciesRule: codomain_fiber_inventory · Uses: G001, S010 S012 · impliesBefore codomain element=7 After preimages=[];used=No Conditions: none Audit every codomain target;an unhit target remains in the codomain but not in the actual range. Step identifiers and dependenciesRule: codomain_fiber_inventory · Uses: G001, S011 S013 · impliesBefore actual range=[0, 1, 2];unused codomain=[3, 4, 5, 6, 7] After range equals codomain=No Conditions: none Totality does not imply surjectivity;the range equals the codomain only if every target has a preimage.The empty map Fin(0)→Fin(0) satisfies this criterion. Step identifiers and dependenciesRule: range_vs_declared_codomain · Uses: G001, S012 ## Reverse checks - S013 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S012 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S011 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S010 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S009 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S008 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S007 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - 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 ([0, 1, 2], [3, 4, 5, 6, 7], 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: 1d8bd097d45136b8c7ea7c871993ed5b3cac2d8b6c6a04b27f2b43ffaf6359f2 - candidate record SHA-256: 61c9eae8ebd4cc6983dab1873cb16d1f3bae5e170fff9084ed5da8a3331e85ec - semantic review receipt SHA-256: 0328166a2e335aac7a5fb34556e68c80ba7d16612e822b3999c8ac348a9ca06f - question-first witness SHA-256: 0da18e4e6cf71ece5ec5f00dd0cd161217b6d251672bc0b46d79fe7e6253f5fc - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261008-advance-20261012-r07-function-mapping-0035.json)