Question 7270: Two Typed Codomain Candidates and Their Preimages

Original question and givens

Fin(5)={0,…,4} is the source and Fin(3)={0,…,2} the codomain; complete typed images=[0, 0, 0, 0, 1] 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(5)={0,…,4} is the source and Fin(3)={0,…,2} the codomain; complete typed images=[0, 0, 0, 0, 1] 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(5)→Fin(3);0-based source labels

domain,codomain and actual range remain distinct

source index=0∈Fin(5)

image[0]=0∈Fin(3)

source index=1∈Fin(5)

image[1]=0∈Fin(3)

source index=2∈Fin(5)

image[2]=0∈Fin(3)

source index=3∈Fin(5)

image[3]=0∈Fin(3)

source index=4∈Fin(5)

image[4]=1∈Fin(3)

target candidate=0;inside Fin(3)=Yes

preimage=[0, 1, 2, 3];size/status=4

target candidate=1;inside Fin(3)=Yes

preimage=[4];size/status=1

Reverse checks

• S008 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S007 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S006 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S005 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S004 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• 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

([Yes, Yes], [[0, 1, 2, 3], [4]], [4, 1])

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.