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Question 32779: Inverse Table of a Finite Bijection

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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← Question derivation index · Original question batch

R0064-32779 · q-20261004-advance-20261005-r04-inverse-function-0004

Original question and givens

The declared total function f:A→B has A={-5, -4, -3, -2, -1}, B={1, 3, 5, 7, 9}, and table {(-5, 9), (-4, 1), (-3, 3), (-2, 5), (-1, 7)}. The function has been confirmed bijective. List the complete inverse table g:B→A, ordered by first and then second component with each pair once.

  • G001 — The declared total function f:A→B has A={-5, -4, -3, -2, -1}, B={1, 3, 5, 7, 9}, and table {(-5, 9), (-4, 1), (-3, 3), (-2, 5), (-1, 7)}. The function has been confirmed bijective. List the complete inverse table g:B→A, ordered by first and then second component with each pair once.

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

source roles = [qf-inverse-function-table, inverse_table, no-required-domain, []]

After

derived = [[complete_inverse_pairset_STRING]]

Conditions: none

Legacy banks enforce their exact source keys/operation/domain.BASE integers retain metadata without treating it as an oracle.Distinguish NUMBER,string,fraction pair and vector equality formats;retain original bilingual data/AST/status.

Step identifiers and dependencies

Rule: R64_SOURCE_exact_legacy_interface_BASE_metadata_AND_output_roles · Uses: G001

S002 · implies

Before

source roles = [ORIGINAL f domain, [-5, -4, -3, -2, -1]]

After

derived = [[-5, -4, -3, -2, -1]]

Conditions: none

RAW sets are distinct,SAFE and≤100.Sort only a derived work table,retaining original lists/hashes.

Step identifiers and dependencies

Rule: FiniteMap_RAW_distinct_SAFE_set_bound100 · Uses: G001, S001

S003 · implies

Before

source roles = [ORIGINAL f codomain, [1, 3, 5, 7, 9]]

After

derived = [[1, 3, 5, 7, 9]]

Conditions: none

RAW sets are distinct,SAFE and≤100.Sort only a derived work table,retaining original lists/hashes.

Step identifiers and dependencies

Rule: FiniteMap_RAW_distinct_SAFE_set_bound100 · Uses: G001, S002

S004 · implies

Before

source roles = [ORIGINAL f, [[-5, 9], [-4, 1], [-3, 3], [-2, 5], [-1, 7]]]

After

derived = [[[-5, 9], [-4, 1], [-3, 3], [-2, 5], [-1, 7]]]

Conditions: none

FIRST type EVERY raw pair and enforce raw length≤200,THEN normalize identical rows as one ordered pair.Do not merge different outputs of one input.Empty domains do not waive RAW checks.

Step identifiers and dependencies

Rule: FiniteMap_RAW200_bound_BEFORE_identical_row_ASSET_normalization · Uses: G001, S003

S005 · implies

Before

source roles = [ORIGINAL f, 0, -5, 9, [-5, -4, -3, -2, -1], [1, 3, 5, 7, 9]]

After

derived = [Yes, Yes, Yes]

Conditions: none

EVERY row input belongs to the domain and output to the codomain,including orphan/unused rows.

Step identifiers and dependencies

Rule: FiniteMap_EACH_normalized_row_declared_domain_codomain_types · Uses: G001, S004

S006 · implies

Before

source roles = [ORIGINAL f, 1, -4, 1, [-5, -4, -3, -2, -1], [1, 3, 5, 7, 9]]

After

derived = [Yes, Yes, Yes]

Conditions: none

EVERY row input belongs to the domain and output to the codomain,including orphan/unused rows.

Step identifiers and dependencies

Rule: FiniteMap_EACH_normalized_row_declared_domain_codomain_types · Uses: G001, S005

S007 · implies

Before

source roles = [ORIGINAL f, 2, -3, 3, [-5, -4, -3, -2, -1], [1, 3, 5, 7, 9]]

After

derived = [Yes, Yes, Yes]

Conditions: none

EVERY row input belongs to the domain and output to the codomain,including orphan/unused rows.

Step identifiers and dependencies

Rule: FiniteMap_EACH_normalized_row_declared_domain_codomain_types · Uses: G001, S006

S008 · implies

Before

source roles = [ORIGINAL f, 3, -2, 5, [-5, -4, -3, -2, -1], [1, 3, 5, 7, 9]]

After

derived = [Yes, Yes, Yes]

Conditions: none

EVERY row input belongs to the domain and output to the codomain,including orphan/unused rows.

Step identifiers and dependencies

Rule: FiniteMap_EACH_normalized_row_declared_domain_codomain_types · Uses: G001, S007

S009 · implies

Before

source roles = [ORIGINAL f, 4, -1, 7, [-5, -4, -3, -2, -1], [1, 3, 5, 7, 9]]

After

derived = [Yes, Yes, Yes]

Conditions: none

EVERY row input belongs to the domain and output to the codomain,including orphan/unused rows.

Step identifiers and dependencies

Rule: FiniteMap_EACH_normalized_row_declared_domain_codomain_types · Uses: G001, S008

S010 · implies

Before

source roles = [ORIGINAL f, -5, [[-5, 9], [-4, 1], [-3, 3], [-2, 5], [-1, 7]]]

After

derived = [1, Yes]

Conditions: none

EACH declared input needs EXACTLY one row for totality.A first lookup must not hide missing or multiple outputs.

Step identifiers and dependencies

Rule: FiniteMap_EACH_declared_input_exactly_one_output_row · Uses: G001, S009

S011 · implies

Before

source roles = [ORIGINAL f, -4, [[-5, 9], [-4, 1], [-3, 3], [-2, 5], [-1, 7]]]

After

derived = [1, Yes]

Conditions: none

EACH declared input needs EXACTLY one row for totality.A first lookup must not hide missing or multiple outputs.

Step identifiers and dependencies

Rule: FiniteMap_EACH_declared_input_exactly_one_output_row · Uses: G001, S010

S012 · implies

Before

source roles = [ORIGINAL f, -3, [[-5, 9], [-4, 1], [-3, 3], [-2, 5], [-1, 7]]]

After

derived = [1, Yes]

Conditions: none

EACH declared input needs EXACTLY one row for totality.A first lookup must not hide missing or multiple outputs.

Step identifiers and dependencies

Rule: FiniteMap_EACH_declared_input_exactly_one_output_row · Uses: G001, S011

S013 · implies

Before

source roles = [ORIGINAL f, -2, [[-5, 9], [-4, 1], [-3, 3], [-2, 5], [-1, 7]]]

After

derived = [1, Yes]

Conditions: none

EACH declared input needs EXACTLY one row for totality.A first lookup must not hide missing or multiple outputs.

Step identifiers and dependencies

Rule: FiniteMap_EACH_declared_input_exactly_one_output_row · Uses: G001, S012

S014 · implies

Before

source roles = [ORIGINAL f, -1, [[-5, 9], [-4, 1], [-3, 3], [-2, 5], [-1, 7]]]

After

derived = [1, Yes]

Conditions: none

EACH declared input needs EXACTLY one row for totality.A first lookup must not hide missing or multiple outputs.

Step identifiers and dependencies

Rule: FiniteMap_EACH_declared_input_exactly_one_output_row · Uses: G001, S013

S015 · implies

Before

source roles = [ORIGINAL f, [-5, -4, -3, -2, -1], [1, 3, 5, 7, 9], [[-5, 9], [-4, 1], [-3, 3], [-2, 5], [-1, 7]]]

After

derived = [[Yes, Yes, Yes, Yes, Yes], [1, 1, 1, 1, 1], Yes]

Conditions: none

Permit lookups ONLY after BOTH all-row typing and all-input totality.Empty-set vacuity cannot conceal an invalid row.

Step identifiers and dependencies

Rule: FiniteMap_TOTAL_qualification_BEFORE_lookups_even_empty_domain · Uses: G001, S014

S016 · implies

Before

source roles = [ORIGINAL f, 1, [-5, -4, -3, -2, -1], [[-5, 9], [-4, 1], [-3, 3], [-2, 5], [-1, 7]]]

After

derived = [[-4], 1]

Conditions: none

List ALL preimages at EVERY codomain label,including empty fibers;do not delete unattained B labels.

Step identifiers and dependencies

Rule: FiniteMap_EVERY_codomain_fiber_including_unattained · Uses: G001, S015

S017 · implies

Before

source roles = [ORIGINAL f, 3, [-5, -4, -3, -2, -1], [[-5, 9], [-4, 1], [-3, 3], [-2, 5], [-1, 7]]]

After

derived = [[-3], 1]

Conditions: none

List ALL preimages at EVERY codomain label,including empty fibers;do not delete unattained B labels.

Step identifiers and dependencies

Rule: FiniteMap_EVERY_codomain_fiber_including_unattained · Uses: G001, S016

S018 · implies

Before

source roles = [ORIGINAL f, 5, [-5, -4, -3, -2, -1], [[-5, 9], [-4, 1], [-3, 3], [-2, 5], [-1, 7]]]

After

derived = [[-2], 1]

Conditions: none

List ALL preimages at EVERY codomain label,including empty fibers;do not delete unattained B labels.

Step identifiers and dependencies

Rule: FiniteMap_EVERY_codomain_fiber_including_unattained · Uses: G001, S017

S019 · implies

Before

source roles = [ORIGINAL f, 7, [-5, -4, -3, -2, -1], [[-5, 9], [-4, 1], [-3, 3], [-2, 5], [-1, 7]]]

After

derived = [[-1], 1]

Conditions: none

List ALL preimages at EVERY codomain label,including empty fibers;do not delete unattained B labels.

Step identifiers and dependencies

Rule: FiniteMap_EVERY_codomain_fiber_including_unattained · Uses: G001, S018

S020 · implies

Before

source roles = [ORIGINAL f, 9, [-5, -4, -3, -2, -1], [[-5, 9], [-4, 1], [-3, 3], [-2, 5], [-1, 7]]]

After

derived = [[-5], 1]

Conditions: none

List ALL preimages at EVERY codomain label,including empty fibers;do not delete unattained B labels.

Step identifiers and dependencies

Rule: FiniteMap_EVERY_codomain_fiber_including_unattained · Uses: G001, S019

S021 · implies

Before

source roles = [ORIGINAL f, [-5, -4, -3, -2, -1], [1, 3, 5, 7, 9], [[-4], [-3], [-2], [-1], [-5]]]

After

derived = [Yes, Yes, Yes]

Conditions: none

Injective means every fiber≤1,surjective every fiber≥1,and BOTH mean bijective.Unattained B does not defeat injectivity.Empty→empty is bijective;empty→nonempty is only injective.

Step identifiers and dependencies

Rule: FiniteMap_injective_surjective_BOTH_needed_for_inverse · Uses: G001, S020

S022 · implies

Before

source roles = [[-5, -4, -3, -2, -1], [1, 3, 5, 7, 9], [[-5, 9], [-4, 1], [-3, 3], [-2, 5], [-1, 7]]]

After

derived = [[[1, -4], [3, -3], [5, -2], [7, -1], [9, -5]], {(1, -4), (3, -3), (5, -2), (7, -1), (9, -5)}]

Conditions: none

AFTER actual bijectivity,swap EVERY(x,y)to(y,x)and sort numericLEX for the COMPLETE B→A inverse table.The empty bijection has inverse∅;an arbitrary converse is not automatically an inverse function.

Step identifiers and dependencies

Rule: Inverse_TABLE_actual_bijection_complete_swapped_numeric_LEX_pairs · Uses: G001, S021

Reverse checks

  • S022 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.
  • S021 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.
  • S020 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.
  • S019 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.
  • S018 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.
  • S017 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.
  • S016 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.
  • S015 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.
  • S014 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.
  • S013 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.
  • S012 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.
  • S011 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.
  • S010 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.
  • S009 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.
  • S008 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.
  • S007 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.
  • S006 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.
  • S005 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.
  • S004 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.
  • S003 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.
  • S002 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.
  • S001 · premise_and_result_audit — Reverse-check ORIGINAL data,field roles,types,domain and premises;the conclusion need not uniquely recover its premises.Keep source-bound candidates,independent review,formal proof and publication separate.

Result

{(1, -4), (3, -3), (5, -2), (7, -1), (9, -5)}

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: ab22b137b8efa30df118072d7c36effdb3c21ac83876776c9d32737b084f52f0
  • candidate record SHA-256: c7ddecd8084e08a6b9e97571d5c7ac8a9ff05198cb6c565cb03980abafc9817b
  • semantic review receipt SHA-256: 8b9ea3ddfab468e9e3f0a0fc0f7682987d5e030be0d7c8860fd85df19cbe183b
  • question-first witness SHA-256: 066bfeab8fc279706861faade71b9a18a61a25235fc48700092f0488d2a6da6a
  • Lean: NOT_RUN · human review: NOT_PERFORMED
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