Use the complete total map Fin(1)={0,…,0}→Fin(1) with images=[0] in source-index order. Validate every image in the declared codomain first. Equal finite cardinalities do not make the given map bijective. Give (sorted swapped relation pairs [f(x),x], each codomain fiber size, inverse-function eligibility, inverse table or not-an-inverse-function).
G001 — Use the complete total map Fin(1)={0,…,0}→Fin(1) with images=[0] in source-index order. Validate every image in the declared codomain first. Equal finite cardinalities do not make the given map bijective. Give (sorted swapped relation pairs [f(x),x], each codomain fiber size, inverse-function eligibility, inverse table or not-an-inverse-function).
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.
Check THIS source interface:distance uses ONLY its complete coordinate frame;other families use a domain.Discrete operations use local hyphens,legacy banks underscores.Retain family-specific ordered output types and original bilingual data/status.
EVERY source image must lie in Fin(m),and the complete table has length n.A nonbijective total table is a legal input;do not assume an inverse function exists.
Swap EVERY ordered pair and sort by numeric(target,source)lexicographic order.ONLY singleton preimages at every target permit an inverse function.Empty-to-empty has an empty inverse;otherwise retain the tag without inventing0.
S004 · premise_and_result_audit — Reverse-check the ORIGINAL inputs,types and premises;the conclusion need not uniquely recover its premises.This source-bound candidate is not independent review or formal proof.
S003 · premise_and_result_audit — Reverse-check the ORIGINAL inputs,types and premises;the conclusion need not uniquely recover its premises.This source-bound candidate is not independent review or formal proof.
S002 · premise_and_result_audit — Reverse-check the ORIGINAL inputs,types and premises;the conclusion need not uniquely recover its premises.This source-bound candidate is not independent review or formal proof.
S001 · premise_and_result_audit — Reverse-check the ORIGINAL inputs,types and premises;the conclusion need not uniquely recover its premises.This source-bound candidate is not independent review or formal proof.
Result
([[0, 0]], [1], Yes, [0])
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.