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R0004-02885 · q-20261004-advance-20261005-r03-bijective-0010
The declared total function f:A→B has A={-3, -2, -1, 0}, B={0, 1, 2, 3}, and table {(-3, 1), (-2, 2), (-1, 3), (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, 3}, and table {(-3, 1), (-2, 2), (-1, 3), (0, 0)}. Is f both injective and surjective? Decide bijectivity from the actual table.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=[-3, -2, -1, 0]; B=[0, 1, 2, 3]; table=[[-3, 1], [-2, 2], [-1, 3], [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.
Rule: declared_total_function · Uses: G001
Before
input -3
After
f(-3)=1∈B; exactly one listed value
Conditions: none
Audit each domain row of the total function,not just set cardinalities.
Rule: total_function_row · Uses: G001, S001
Before
input -2
After
f(-2)=2∈B; exactly one listed value
Conditions: none
Audit each domain row of the total function,not just set cardinalities.
Rule: total_function_row · Uses: G001, S002
Before
input -1
After
f(-1)=3∈B; exactly one listed value
Conditions: none
Audit each domain row of the total function,not just set cardinalities.
Rule: total_function_row · Uses: G001, S003
Before
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.
Rule: total_function_row · Uses: G001, S004
Before
target 0; 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.
Rule: complete_codomain_fiber · Uses: G001, S005
Before
target 1; 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.
Rule: complete_codomain_fiber · Uses: G001, S006
Before
target 2; 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.
Rule: complete_codomain_fiber · Uses: G001, S007
Before
target 3; 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.
Rule: complete_codomain_fiber · Uses: G001, S008
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, S009
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, S010
Before
injective=Yes;surjective=Yes
After
requested result=Yes
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, S011
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.Yes
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: 1d5ef6738c6c90c34869cf104602d2d63485dcbbc1016b9e002fc7897ef24471candidate record SHA-256: 6a3d28f7716637dfa430c5a37aa2190e37ed47c74215880030fef06502f09359semantic review receipt SHA-256: 312ae86315346b8955842a547cb2d09246e1f826755f72c2a51f042e5c7d5c56question-first witness SHA-256: 831f449ed41ff9144a290e534aa32adf7252d15b81c16ed5886a7dfd65dee70cLean: NOT_RUN · human review: NOT_PERFORMED