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Question 2843: 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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R0004-02843 · q-20261004-advance-20261005-r03-bijective-0043

Original question and givens

The declared total function f:A→B has A={-5, -4, -3, -2, -1, 0}, B={0, 1, 2, 3, 4, 5}, and table {(-5, 1), (-4, 2), (-3, 3), (-2, 4), (-1, 5), (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={-5, -4, -3, -2, -1, 0}, B={0, 1, 2, 3, 4, 5}, and table {(-5, 1), (-4, 2), (-3, 3), (-2, 4), (-1, 5), (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=[-5, -4, -3, -2, -1, 0]; B=[0, 1, 2, 3, 4, 5]; table=[[-5, 1], [-4, 2], [-3, 3], [-2, 4], [-1, 5], [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 -5

After

f(-5)=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 -4

After

f(-4)=2∈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

input -3

After

f(-3)=3∈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, S003

S005 · implies

Before

input -2

After

f(-2)=4∈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, S004

S006 · implies

Before

input -1

After

f(-1)=5∈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, S005

S007 · 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, S006

S008 · 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, S007

S009 · implies

Before

target 1; all source rows examined

After

fiber=[-5]; 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, S008

S010 · implies

Before

target 2; all source rows examined

After

fiber=[-4]; 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, S009

S011 · implies

Before

target 3; 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.

Step identifiers and dependencies

Rule: complete_codomain_fiber · Uses: G001, S010

S012 · implies

Before

target 4; 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 dependencies

Rule: complete_codomain_fiber · Uses: G001, S011

S013 · implies

Before

target 5; 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, S012

S014 · 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, S013

S015 · 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, S014

S016 · 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, S015

Reverse checks

  • S016 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S015 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S014 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S013 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • 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.

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: 87293d799011e4b52524fbd87ff7a1dda746ae4f181b0088f156a157db8c9440
  • candidate record SHA-256: dcd52d2306d768f7b8de57f68c372bb431ef582d22a2fe4d98867243b1438ebe
  • semantic review receipt SHA-256: 312ae86315346b8955842a547cb2d09246e1f826755f72c2a51f042e5c7d5c56
  • question-first witness SHA-256: 831f449ed41ff9144a290e534aa32adf7252d15b81c16ed5886a7dfd65dee70c
  • Lean: NOT_RUN · human review: NOT_PERFORMED
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