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R0004-02851 · q-20261004-advance-20261005-r03-bijective-0051
The declared total function f:A→B has A=∅, B=∅, and table ∅. 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=∅, B=∅, and table ∅. In increasing target order, list every (b,|f⁻¹({b})|), including zero for targets not attained.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.
Before
A=[]; B=[]; table=[]
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.
Rule: declared_total_function · Uses: G001
Before
[]
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.
Rule: injectivity_fiber_test · Uses: G001, S001
Before
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.
Rule: surjectivity_fiber_test · Uses: G001, S002
Before
injective=Yes;surjective=Yes
After
requested result=∅
Conditions: none
A bijection is both injective and surjective;the fiber profile is numerically ordered over the entire codomain,including zero fibers.
Rule: bijection_requested_classification · Uses: G001, S003
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.∅
The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.
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.
input commit: 0f7be61e2c22317a4075561dc822c87d881a50dboriginal question SHA-256: 46fe8918568e17b32c0b60cea76bc3cb9b22b99d1d5e25a6e754406124e7ca78candidate record SHA-256: aa219975eeebf627379e5662cafa453c5a423c6de2468eabe6b171b6aa8b5ac9semantic review receipt SHA-256: 312ae86315346b8955842a547cb2d09246e1f826755f72c2a51f042e5c7d5c56question-first witness SHA-256: 831f449ed41ff9144a290e534aa32adf7252d15b81c16ed5886a7dfd65dee70cLean: NOT_RUN · human review: NOT_PERFORMED