Question 31017: Converse Relation before Inverse-Function Qualification

Original question and givens

Use the complete total map Fin(5)={0,…,4}→Fin(3) with images=[0, 0, 1, 0, 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(5)={0,…,4}→Fin(3) with images=[0, 0, 1, 0, 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.

• mko-bijective

Step-by-step derivation

source roles = [qf-density17-bij-swapped-relation-audit, bij-swapped-relation-audit, domain, Fin_total_table]

derived = [[swapped_pairs, fiber_sizes, bijective, inverse_or_tag]]

source roles = [5, 3, [0, 0, 1, 0, 0]]

derived = [[(0, 0), (1, 0), (2, 1), (3, 0), (4, 0)]]

source roles = [0, [0, 0, 1, 0, 0]]

derived = [[0, 1, 3, 4], 4]

source roles = [1, [0, 0, 1, 0, 0]]

derived = [[2], 1]

source roles = [2, [0, 0, 1, 0, 0]]

derived = [[], 0]

source roles = [5, 3, [0, 0, 1, 0, 0], [[0, 1, 3, 4], [2], []]]

derived = [[[0, 0], [0, 1], [0, 3], [0, 4], [1, 2]], [4, 1, 0], No, does not define an inverse function]

Reverse checks

• S006 · 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.
• S005 · 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.
• 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], [0, 1], [0, 3], [0, 4], [1, 2]], [4, 1, 0], No, not-an-inverse-function)

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.