Question 26183: Complete Total Mapping Qualification Precedes Membership Equivalence

Original question and givens

Use source carrier=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9], target carrier=[0, 1, 2], aligned mapping=[1, 2, 0, 1, 2, 0, 1, 2, 0, 1]. Sort domain values with their associated images, not independently. A null or outside image is not a total map. Give (source_carrier, target_carrier, mapping_rows, total, first_invalid_image_or_null, certificate).

• G001 — Use source carrier=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9], target carrier=[0, 1, 2], aligned mapping=[1, 2, 0, 1, 2, 0, 1, 2, 0, 1]. Sort domain values with their associated images, not independently. A null or outside image is not a total map. Give (source_carrier, target_carrier, mapping_rows, total, first_invalid_image_or_null, certificate).

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-many-one-reduction

Step-by-step derivation

source roles = [mreduction-total-map-qualification]

derived = [[source_carrier, target_carrier, mapping_rows, total, first_invalid_image_or_null, certificate]]

source roles = [Reduction ORIGINAL map, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [0, 1, 2], [1, 2, 0, 1, 2, 0, 1, 2, 0, 1]]

derived = [[0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [0, 1, 2], [1, 2, 0, 1, 2, 0, 1, 2, 0, 1], [[0, 1, Yes], [1, 2, Yes], [2, 0, Yes], [3, 1, Yes], [4, 2, Yes], [5, 0, Yes], [6, 1, Yes], [7, 2, Yes], [8, 0, Yes], [9, 1, Yes]], Yes, null]

Reverse checks

• 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

([0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [0, 1, 2], [[0, 1, Yes], [1, 2, Yes], [2, 0, Yes], [3, 1, Yes], [4, 2, Yes], [5, 0, Yes], [6, 1, Yes], [7, 2, Yes], [8, 0, Yes], [9, 1, Yes]], Yes, null, many_one_reduction_requires_a_complete_total_map_before_membership_audit)

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.