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Question 2827: Four-Way Function Classification

On this page
  1. Original question and givens
  2. Definition and scope
  3. Step-by-step derivation
  4. Reverse checks
  5. Result
  6. This question review and source binding

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R0004-02827 · q-20261004-advance-20261005-r03-bijective-0027

Original question and givens

The declared total function f:A→B has A={-1, 0}, B={0, 1}, and table {(-1, 1), (0, 0)}. 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={0, 1}, and table {(-1, 1), (0, 0)}. 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.

Step-by-step derivation

S001 · implies

Before

A=[-1, 0]; B=[0, 1]; table=[[-1, 1], [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 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)=0∈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 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.

Step identifiers and dependencies

Rule: complete_codomain_fiber · Uses: G001, S003

S005 · implies

Before

target 1; 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 dependencies

Rule: complete_codomain_fiber · Uses: G001, S004

S006 · implies

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.

Step identifiers and dependencies

Rule: injectivity_fiber_test · Uses: G001, S005

S007 · implies

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.

Step identifiers and dependencies

Rule: surjectivity_fiber_test · Uses: G001, S006

S008 · implies

Before

injective=Yes;surjective=Yes

After

requested result=bijective

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, S007

Reverse checks

  • 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

bijective

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: 37c1e7571b75fb6fd9cc8d227995dc7e2e0da715957ebda018fc1f028f34d597
  • candidate record SHA-256: d88149ab3ac98e7df35eca7f3b1fb2b77604048c719deb451ff8ec5bc4fdae73
  • semantic review receipt SHA-256: 312ae86315346b8955842a547cb2d09246e1f826755f72c2a51f042e5c7d5c56
  • question-first witness SHA-256: 831f449ed41ff9144a290e534aa32adf7252d15b81c16ed5886a7dfd65dee70c
  • Lean: NOT_RUN · human review: NOT_PERFORMED
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