Question 7251: Two Typed Codomain Candidates and Their Preimages

Original question and givens

Fin(0)=∅ is the source and Fin(0)=∅ the codomain; complete typed images=[] are indexed from 0. All entries must be typed, even unused ones. Ordered codomain candidates=[0, -1]. Give (candidate-type eligibility, preimage index lists or outside-codomain, sizes or outside-codomain). An outside value is not a legal empty preimage.

• G001 — Fin(0)=∅ is the source and Fin(0)=∅ the codomain; complete typed images=[] are indexed from 0. All entries must be typed, even unused ones. Ordered codomain candidates=[0, -1]. Give (candidate-type eligibility, preimage index lists or outside-codomain, sizes or outside-codomain). An outside value is not a legal empty preimage.

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-domain-codomain-range

Step-by-step derivation

declared types Fin(0)→Fin(0);0-based source labels

domain,codomain and actual range remain distinct

target candidate=0;inside Fin(0)=No

preimage=outside the codomain;size/status=outside the codomain

target candidate=-1;inside Fin(0)=No

preimage=outside the codomain;size/status=outside the codomain

Reverse checks

• 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

([No, No], [outside-codomain, outside-codomain], [outside-codomain, outside-codomain])

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.