OCMEBilingual mathematics & derivations繁體中文

Explore related knowledge and questions →

Question 32726: Checking Both Inverse Identities

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

Return to catalogue page 164

← Question derivation index · Original question batch

R0064-32726 · q-20261004-advance-20261005-r04-inverse-function-0051

Original question and givens

The declared total function f:A→B has A=∅, B=∅, and table ∅. The function has been confirmed bijective. The candidate total g:B→A has table ∅. Check both identities to decide whether g is the inverse of f.

  • G001 — The declared total function f:A→B has A=∅, B=∅, and table ∅. The function has been confirmed bijective. The candidate total g:B→A has table ∅. Check both identities to decide whether g is the inverse of f.

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-candidate, inverse_candidate, no-required-domain, []]

After

derived = [[BOTH_restorations_YesNo_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, []]

After

derived = [[]]

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, []]

After

derived = [[]]

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, []]

After

derived = [[]]

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, [], [], []]

After

derived = [[], [], 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, S004

S006 · implies

Before

source roles = [ORIGINAL f, [], [], []]

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, S005

S007 · implies

Before

source roles = [CANDIDATE g domain, []]

After

derived = [[]]

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, S006

S008 · implies

Before

source roles = [CANDIDATE g codomain, []]

After

derived = [[]]

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, S007

S009 · implies

Before

source roles = [CANDIDATE g, []]

After

derived = [[]]

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, S008

S010 · implies

Before

source roles = [CANDIDATE g, [], [], []]

After

derived = [[], [], 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, S009

S011 · implies

Before

source roles = [CANDIDATE g, [], [], []]

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, S010

S012 · implies

Before

source roles = [[], [], [], []]

After

derived = [[], [], Yes]

Conditions: none

AFTER f is bijective and g is fully typed/total,BOTH complete restoration inventories must pass.A qualified wrong inverse is a legal No;missing/outside rows fail qualification instead.

Step identifiers and dependencies

Rule: Inverse_CANDIDATE_BOTH_typed_total_restoration_inventories · Uses: G001, S011

Reverse checks

  • 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

Yes

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: 0c52be45c5c324e301c825ec54172932aa2eda4c9f32858089008ca6a807a8ee
  • candidate record SHA-256: 58816cdaeb65a6a2c25240359bdc8cf30ea274eb084fa0a6309aaf3ca0a65985
  • semantic review receipt SHA-256: 8b9ea3ddfab468e9e3f0a0fc0f7682987d5e030be0d7c8860fd85df19cbe183b
  • question-first witness SHA-256: 066bfeab8fc279706861faade71b9a18a61a25235fc48700092f0488d2a6da6a
  • Lean: NOT_RUN · human review: NOT_PERFORMED
Same-source full JSON