# Question 2887: Bijectivity of a Finite Total Function [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-0012) 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. R0004-02887 · q-20261004-advance-20261005-r03-bijective-0012 ## Original question and givens The declared total function f:A→B has A={-3, -2, -1, 0}, B={0, 1, 2}, and table {(-3, 0), (-2, 1), (-1, 2), (0, 0)}. Is f both injective and surjective? Decide bijectivity from the actual table. - G001 — The declared total function f:A→B has A={-3, -2, -1, 0}, B={0, 1, 2}, and table {(-3, 0), (-2, 1), (-1, 2), (0, 0)}. Is f both injective and surjective? Decide bijectivity from the actual table. ## 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=[-3, -2, -1, 0]; B=[0, 1, 2]; table=[[-3, 0], [-2, 1], [-1, 2], [0, 0]] 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 -3 After f(-3)=0∈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 -2 After f(-2)=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, S002 S004 · impliesBefore input -1 After f(-1)=2∈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 0 After f(0)=0∈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 target 0; all source rows examined After fiber=[-3, 0]; size=2 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, S005 S007 · impliesBefore target 1; 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, S006 S008 · impliesBefore target 2; 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, S007 S009 · impliesBefore [[0, [-3, 0]]] After all fibers have size≤1: No 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, S008 S010 · impliesBefore missing codomain targets=[] After all codomain fibers nonempty: Yes 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, S009 S011 · impliesBefore injective=No;surjective=Yes After requested result=No 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, S010 ## Reverse checks - 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 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: de6df9218a99f915e8da43f9c89eaa592458ea7fbb35edea99edfcbc01df10b0 - candidate record SHA-256: 685e32d6ddc822766e92b75b4822ae97f29304b2149f32967be0b992de36463a - 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-0012.json)