Question 26168: A Many-One Reduction Preserves Both Yes and No Membership

Original question and givens

Use source carrier=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19], target carrier=[0, 1], aligned mapping=[0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1], source yes-set=[1, 3, 5, 7, 9, 11, 13, 15, 17, 19], target yes-set=[1]. Sort domain values with their associated images, not independently. A null or outside image is not a total map. Give (source_yes, target_yes, membership_rows, reduction, certificate).

• G001 — Use source carrier=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19], target carrier=[0, 1], aligned mapping=[0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1], source yes-set=[1, 3, 5, 7, 9, 11, 13, 15, 17, 19], target yes-set=[1]. Sort domain values with their associated images, not independently. A null or outside image is not a total map. Give (source_yes, target_yes, membership_rows, reduction, 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-iff-audit]

derived = [[source_yes, target_yes, membership_rows, reduction, certificate]]

source roles = [Reduction ORIGINAL map, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19], [0, 1], [0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1]]

derived = [[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19], [0, 1], [0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1], [[0, 0, Yes], [1, 1, Yes], [2, 0, Yes], [3, 1, Yes], [4, 0, Yes], [5, 1, Yes], [6, 0, Yes], [7, 1, Yes], [8, 0, Yes], [9, 1, Yes], [10, 0, Yes], [11, 1, Yes], [12, 0, Yes], [13, 1, Yes], [14, 0, Yes], [15, 1, Yes], [16, 0, Yes], [17, 1, Yes], [18, 0, Yes], [19, 1, Yes]], Yes, null]

source roles = [Reduction ORIGINAL map, [1, 3, 5, 7, 9, 11, 13, 15, 17, 19], [1], [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19], [0, 1], [0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1]]

derived = [[[0, No, 0, No, Yes], [1, Yes, 1, Yes, Yes], [2, No, 0, No, Yes], [3, Yes, 1, Yes, Yes], [4, No, 0, No, Yes], [5, Yes, 1, Yes, Yes], [6, No, 0, No, Yes], [7, Yes, 1, Yes, Yes], [8, No, 0, No, Yes], [9, Yes, 1, Yes, Yes], [10, No, 0, No, Yes], [11, Yes, 1, Yes, Yes], [12, No, 0, No, Yes], [13, Yes, 1, Yes, Yes], [14, No, 0, No, Yes], [15, Yes, 1, Yes, Yes], [16, No, 0, No, Yes], [17, Yes, 1, Yes, Yes], [18, No, 0, No, Yes], [19, Yes, 1, Yes, Yes]], Yes, Yes]

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

([1, 3, 5, 7, 9, 11, 13, 15, 17, 19], [1], [[0, No, 0, No, Yes], [1, Yes, 1, Yes, Yes], [2, No, 0, No, Yes], [3, Yes, 1, Yes, Yes], [4, No, 0, No, Yes], [5, Yes, 1, Yes, Yes], [6, No, 0, No, Yes], [7, Yes, 1, Yes, Yes], [8, No, 0, No, Yes], [9, Yes, 1, Yes, Yes], [10, No, 0, No, Yes], [11, Yes, 1, Yes, Yes], [12, No, 0, No, Yes], [13, Yes, 1, Yes, Yes], [14, No, 0, No, Yes], [15, Yes, 1, Yes, Yes], [16, No, 0, No, Yes], [17, Yes, 1, Yes, Yes], [18, No, 0, No, Yes], [19, Yes, 1, Yes, Yes]], Yes, x_in_A_iff_f_x_in_B_requires_both_yes_and_no_directions)

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.