# Question 2873: Complete Codomain Fiber Profile [Return to catalogue page 15](https://ocme.evemisslab.com/en/questions/page/15/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261004-advance-20261005-r03-bijective-foundations/#q-20261004-advance-20261005-r03-bijective-0073) 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. R0004-02873 · q-20261004-advance-20261005-r03-bijective-0073 ## Original question and givens The declared total function f:A→B has A={-6, -5, -4, -3, -2, -1, 0}, B={1, 3, 5, 7, 9, 11, 13, 15}, and table {(-6, 1), (-5, 3), (-4, 5), (-3, 7), (-2, 9), (-1, 11), (0, 13)}. In increasing target order, list every (b,|f⁻¹({b})|), including zero for targets not attained. - G001 — The declared total function f:A→B has A={-6, -5, -4, -3, -2, -1, 0}, B={1, 3, 5, 7, 9, 11, 13, 15}, and table {(-6, 1), (-5, 3), (-4, 5), (-3, 7), (-2, 9), (-1, 11), (0, 13)}. In increasing target order, list every (b,|f⁻¹({b})|), including zero for targets not attained. ## 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-bijective](https://ocme.evemisslab.com/en/mko/mko-bijective/) ## Step-by-step derivation S001 · impliesBefore A=[-6, -5, -4, -3, -2, -1, 0]; B=[1, 3, 5, 7, 9, 11, 13, 15]; table=[[-6, 1], [-5, 3], [-4, 5], [-3, 7], [-2, 9], [-1, 11], [0, 13]] After legal total function on the declared A→B Conditions: none First check exactly one table value for every A-element,and no input outside A or output outside B.Injectivity and surjectivity refer to this declared codomain. Step identifiers and dependenciesRule: declared_total_function · Uses: G001 S002 · impliesBefore input -6 After f(-6)=1∈B; exactly one listed value Conditions: none Audit each domain row of the total function,not just set cardinalities. Step identifiers and dependenciesRule: total_function_row · Uses: G001, S001 S003 · impliesBefore input -5 After f(-5)=3∈B; exactly one listed value Conditions: none Audit each domain row of the total function,not just set cardinalities. Step identifiers and dependenciesRule: total_function_row · Uses: G001, S002 S004 · impliesBefore input -4 After f(-4)=5∈B; exactly one listed value Conditions: none Audit each domain row of the total function,not just set cardinalities. Step identifiers and dependenciesRule: total_function_row · Uses: G001, S003 S005 · impliesBefore input -3 After f(-3)=7∈B; exactly one listed value Conditions: none Audit each domain row of the total function,not just set cardinalities. Step identifiers and dependenciesRule: total_function_row · Uses: G001, S004 S006 · impliesBefore input -2 After f(-2)=9∈B; exactly one listed value Conditions: none Audit each domain row of the total function,not just set cardinalities. Step identifiers and dependenciesRule: total_function_row · Uses: G001, S005 S007 · impliesBefore input -1 After f(-1)=11∈B; exactly one listed value Conditions: none Audit each domain row of the total function,not just set cardinalities. Step identifiers and dependenciesRule: total_function_row · Uses: G001, S006 S008 · impliesBefore input 0 After f(0)=13∈B; exactly one listed value Conditions: none Audit each domain row of the total function,not just set cardinalities. Step identifiers and dependenciesRule: total_function_row · Uses: G001, S007 S009 · impliesBefore target 1; all source rows examined After fiber=[-6]; size=1 Conditions: none List every preimage of this target;an unhit codomain element still has an empty fiber of size0. Step identifiers and dependenciesRule: complete_codomain_fiber · Uses: G001, S008 S010 · impliesBefore target 3; all source rows examined After fiber=[-5]; size=1 Conditions: none List every preimage of this target;an unhit codomain element still has an empty fiber of size0. Step identifiers and dependenciesRule: complete_codomain_fiber · Uses: G001, S009 S011 · impliesBefore target 5; all source rows examined After fiber=[-4]; size=1 Conditions: none List every preimage of this target;an unhit codomain element still has an empty fiber of size0. Step identifiers and dependenciesRule: complete_codomain_fiber · Uses: G001, S010 S012 · impliesBefore target 7; all source rows examined After fiber=[-3]; size=1 Conditions: none List every preimage of this target;an unhit codomain element still has an empty fiber of size0. Step identifiers and dependenciesRule: complete_codomain_fiber · Uses: G001, S011 S013 · impliesBefore target 9; all source rows examined After fiber=[-2]; size=1 Conditions: none List every preimage of this target;an unhit codomain element still has an empty fiber of size0. Step identifiers and dependenciesRule: complete_codomain_fiber · Uses: G001, S012 S014 · impliesBefore target 11; all source rows examined After fiber=[-1]; size=1 Conditions: none List every preimage of this target;an unhit codomain element still has an empty fiber of size0. Step identifiers and dependenciesRule: complete_codomain_fiber · Uses: G001, S013 S015 · impliesBefore target 13; all source rows examined After fiber=[0]; size=1 Conditions: none List every preimage of this target;an unhit codomain element still has an empty fiber of size0. Step identifiers and dependenciesRule: complete_codomain_fiber · Uses: G001, S014 S016 · impliesBefore target 15; all source rows examined After fiber=[]; size=0 Conditions: none List every preimage of this target;an unhit codomain element still has an empty fiber of size0. Step identifiers and dependenciesRule: complete_codomain_fiber · Uses: G001, S015 S017 · impliesBefore [] After all fibers have size≤1: Yes Conditions: none Injectivity forbids distinct inputs with the same output,exactly the condition that every fiber has size at most1.It holds vacuously for an empty domain. Step identifiers and dependenciesRule: injectivity_fiber_test · Uses: G001, S016 S018 · impliesBefore missing codomain targets=[15] After all codomain fibers nonempty: No Conditions: none Surjectivity requires a preimage for every codomain element.The universal condition is true for an empty codomain,but the total-function premise is still required. Step identifiers and dependenciesRule: surjectivity_fiber_test · Uses: G001, S017 S019 · impliesBefore injective=Yes;surjective=No After requested result={(1, 1), (3, 1), (5, 1), (7, 1), (9, 1), (11, 1), (13, 1), (15, 0)} Conditions: none A bijection is both injective and surjective;the fiber profile is numerically ordered over the entire codomain,including zero fibers. Step identifiers and dependenciesRule: bijection_requested_classification · Uses: G001, S018 ## Reverse checks - S019 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S018 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S017 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S016 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S015 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S014 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - 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 {(1, 1), (3, 1), (5, 1), (7, 1), (9, 1), (11, 1), (13, 1), (15, 0)} 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: f6132f0bee930932684f0680f5f07b32ea73c15be7c6e03bbb2e0858c4cde992 - candidate record SHA-256: f1914ad21befdfbee244719c6c9adf9dfbbb9b244ba5da0e3e15a7d09c18ef65 - semantic review receipt SHA-256: 312ae86315346b8955842a547cb2d09246e1f826755f72c2a51f042e5c7d5c56 - question-first witness SHA-256: 831f449ed41ff9144a290e534aa32adf7252d15b81c16ed5886a7dfd65dee70c - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261004-advance-20261005-r03-bijective-0073.json)