Question 30996: Equal Cardinality versus the Given Total Table

Original question and givens

Use the complete total map Fin(5)={0,…,4}→Fin(3) with images=[1, 1, 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 ([n,m], images, n=m?, every fiber, is the given map bijective?).

• G001 — Use the complete total map Fin(5)={0,…,4}→Fin(3) with images=[1, 1, 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 ([n,m], images, n=m?, every fiber, is the given map bijective?).

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-size-obstruction-with-table, bij-size-obstruction-with-table, Fin_total_table]

derived = [[dimensions, images, equal_sizes, ALL_fibers, actual_bijective]]

source roles = [Bijection ORIGINAL complete table, 5, 3, [1, 1, 1, 0, 0]]

derived = [[1, 1, 1, 0, 0]]

Bijection ORIGINAL reverse ALL fiber sizes through target0 := sum(0, 2)

Bijection ORIGINAL reverse ALL fiber sizes through target0 = 2

source roles = [Bijection ORIGINAL, [1, 1, 1, 0, 0], 3, 0]

derived = [[3, 4], 2]

Bijection ORIGINAL reverse ALL fiber sizes through target1 := sum(2, 3)

Bijection ORIGINAL reverse ALL fiber sizes through target1 = 5

source roles = [Bijection ORIGINAL, [1, 1, 1, 0, 0], 3, 1]

derived = [[0, 1, 2], 3]

Bijection ORIGINAL reverse ALL fiber sizes through target2 := sum(5, 0)

Bijection ORIGINAL reverse ALL fiber sizes through target2 = 5

source roles = [Bijection ORIGINAL, [1, 1, 1, 0, 0], 3, 2]

derived = [[], 0]

source roles = [Bijection ORIGINAL, [1, 1, 1, 0, 0], 3]

derived = [[[3, 4], [0, 1, 2], []], [[0, 1], [0, 2], [1, 2], [3, 4]], [2], No, No, No, 5]

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

derived = [No, [[3, 4], [0, 1, 2], []], No]

Reverse checks

• S010 · premise_and_result_audit — Reverse-check using the ORIGINAL data,types,indices and premises.Consistency auditing does not claim the conclusion uniquely determines its inputs.Keep finite candidates,independent review,formal proof and publication separate.
• S009 · premise_and_result_audit — Reverse-check using the ORIGINAL data,types,indices and premises.Consistency auditing does not claim the conclusion uniquely determines its inputs.Keep finite candidates,independent review,formal proof and publication separate.
• S008 · premise_and_result_audit — Reverse-check using the ORIGINAL data,types,indices and premises.Consistency auditing does not claim the conclusion uniquely determines its inputs.Keep finite candidates,independent review,formal proof and publication separate.
• S007 · premise_and_result_audit — Recompute Bijection ORIGINAL reverse ALL fiber sizes through target2 with exact rational arithmetic, not rounded decimals.
• S006 · premise_and_result_audit — Reverse-check using the ORIGINAL data,types,indices and premises.Consistency auditing does not claim the conclusion uniquely determines its inputs.Keep finite candidates,independent review,formal proof and publication separate.
• S005 · premise_and_result_audit — Recompute Bijection ORIGINAL reverse ALL fiber sizes through target1 with exact rational arithmetic, not rounded decimals.
• S004 · premise_and_result_audit — Reverse-check using the ORIGINAL data,types,indices and premises.Consistency auditing does not claim the conclusion uniquely determines its inputs.Keep finite candidates,independent review,formal proof and publication separate.
• S003 · premise_and_result_audit — Recompute Bijection ORIGINAL reverse ALL fiber sizes through target0 with exact rational arithmetic, not rounded decimals.
• S002 · premise_and_result_audit — Reverse-check using the ORIGINAL data,types,indices and premises.Consistency auditing does not claim the conclusion uniquely determines its inputs.Keep finite candidates,independent review,formal proof and publication separate.
• S001 · premise_and_result_audit — Reverse-check using the ORIGINAL data,types,indices and premises.Consistency auditing does not claim the conclusion uniquely determines its inputs.Keep finite candidates,independent review,formal proof and publication separate.

Result

([5, 3], [1, 1, 1, 0, 0], No, [[3, 4], [0, 1, 2], []], No)

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.