Question 5257: Finite Principal-Filter Image Generator

Original question and givens

The complete total map f:Fin(8)→Fin(8) has images=[0, 1, 2, 3, 4, 5, 6, 7] indexed from 0; Fin(8)={0,…,7}. Every table entry must lie in the target type. S=[1, 2]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. List f(S) as an increasing set without repetition.

• G001 — The complete total map f:Fin(8)→Fin(8) has images=[0, 1, 2, 3, 4, 5, 6, 7] indexed from 0; Fin(8)={0,…,7}. Every table entry must lie in the target type. S=[1, 2]. principal(S) contains all source subsets containing S; the empty generator is allowed, and no NeBot premise is added. List f(S) as an increasing set without repetition.

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-tends-to-relation

Step-by-step derivation

source index=0∈Fin(8)

image=0∈Fin(8)

source index=1∈Fin(8)

image=1∈Fin(8)

source index=2∈Fin(8)

image=2∈Fin(8)

source index=3∈Fin(8)

image=3∈Fin(8)

source index=4∈Fin(8)

image=4∈Fin(8)

source index=5∈Fin(8)

image=5∈Fin(8)

source index=6∈Fin(8)

image=6∈Fin(8)

source index=7∈Fin(8)

image=7∈Fin(8)

Fin(8)→Fin(8);S=[1, 2];full map=[0, 1, 2, 3, 4, 5, 6, 7]

principal(S)={A:S⊆A};empty generators allowed,no NeBot

source generator x=1∈S

f(x)=1

source generator x=2∈S

f(x)=2

[1, 2]

A⊇f(S) iff f⁻¹(A)⊇S

Reverse checks

• S012 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S011 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S010 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S009 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• 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

[1, 2]

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.