Question 2830: Four-Way Function Classification

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Original question and givens
Definition and scope
Step-by-step derivation
Reverse checks
Result
This question review and source binding

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R0004-02830 · q-20261004-advance-20261005-r03-bijective-0030

Original question and givens

The declared total function f:A→B has A={-1, 0}, B={-3, -2, -1}, and table {(-1, -1), (0, -1)}. Classify f as bijective, injective only, surjective only, or neither injective nor surjective.

G001 — The declared total function f:A→B has A={-1, 0}, B={-3, -2, -1}, and table {(-1, -1), (0, -1)}. Classify f as bijective, injective only, surjective only, or neither injective nor surjective.

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

Step-by-step derivation

S001 · implies

Before

A=[-1, 0]; B=[-3, -2, -1]; table=[[-1, -1], [0, -1]]

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 dependencies

Rule: declared_total_function · Uses: G001

S002 · implies

Before

input -1

After

f(-1)=-1∈B; exactly one listed value

Conditions: none

Audit each domain row of the total function,not just set cardinalities.

Step identifiers and dependencies

Rule: total_function_row · Uses: G001, S001

S003 · implies

Before

input 0

After

f(0)=-1∈B; exactly one listed value

Conditions: none

Audit each domain row of the total function,not just set cardinalities.

Step identifiers and dependencies

Rule: total_function_row · Uses: G001, S002

S004 · implies

Before

target -3; 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 dependencies

Rule: complete_codomain_fiber · Uses: G001, S003

S005 · implies

Before

target -2; 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 dependencies

Rule: complete_codomain_fiber · Uses: G001, S004

S006 · implies

Before

target -1; all source rows examined

After

fiber=[-1, 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 dependencies

Rule: complete_codomain_fiber · Uses: G001, S005

S007 · implies

Before

[[-1, [-1, 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 dependencies

Rule: injectivity_fiber_test · Uses: G001, S006

S008 · implies

Before

missing codomain targets=[-3, -2]

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 dependencies

Rule: surjectivity_fiber_test · Uses: G001, S007

S009 · implies

Before

injective=No;surjective=No

After

requested result=neither injective nor surjective

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 dependencies

Rule: bijection_requested_classification · Uses: G001, S008

Reverse checks

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

neither injective nor surjective

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: a332b1653c1cd41d7054c3afeb511cbe60136768dcb27d706215892101c8da7a
candidate record SHA-256: 57ea9edfdd7c24b2733a56868a03ca3a4ec9ce9cafcdb69bf424fd61b282c4f3
semantic review receipt SHA-256: 312ae86315346b8955842a547cb2d09246e1f826755f72c2a51f042e5c7d5c56
question-first witness SHA-256: 831f449ed41ff9144a290e534aa32adf7252d15b81c16ed5886a7dfd65dee70c
Lean: NOT_RUN · human review: NOT_PERFORMED

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