Question 8275: Three Closed-Interval Membership Tests and Signed Slacks

Original question and givens

Use the usual total order on R, restricted to exact rational inputs whose original denominators are positive. Output reduced fractions with positive denominators, retaining /1; do not extrapolate to arbitrary partial orders or complex numbers. Interval=[-24/3, 24/2], queries=[0/1, -24/3, 25/2]. Give (interval endpoints, normalized queries, three membership flags, each [query−lower, upper−query] signed slack pair).

• G001 — Use the usual total order on R, restricted to exact rational inputs whose original denominators are positive. Output reduced fractions with positive denominators, retaining /1; do not extrapolate to arbitrary partial orders or complex numbers. Interval=[-24/3, 24/2], queries=[0/1, -24/3, 25/2]. Give (interval endpoints, normalized queries, three membership flags, each [query−lower, upper−query] signed slack pair).

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-order-relation

Step-by-step derivation

usual total order on R;all original denominators positive

closed endpoints included;negative scaling reverses;zero collapses

query0 x-lower := sub(0, -8)

query0 x-lower = 8

query0 upper-x := sub(12, 0)

query0 upper-x = 12

query0=0/1;signed slacks=[8/1, 12/1]

both nonnegative=Yes

query1 x-lower := sub(-8, -8)

query1 x-lower = 0

query1 upper-x := sub(12, -8)

query1 upper-x = 20

query1=-8/1;signed slacks=[0/1, 20/1]

both nonnegative=Yes

query2 x-lower := sub(25/2, -8)

query2 x-lower = 41/2

query2 upper-x := sub(12, 25/2)

query2 upper-x = -1/2

query2=25/2;signed slacks=[41/2, -1/2]

both nonnegative=No

Reverse checks

• 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 — Recompute query2 upper-x with exact rational arithmetic, not rounded decimals.
• S008 · premise_and_result_audit — Recompute query2 x-lower with exact rational arithmetic, not rounded decimals.
• 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 — Recompute query1 upper-x with exact rational arithmetic, not rounded decimals.
• S005 · premise_and_result_audit — Recompute query1 x-lower with exact rational arithmetic, not rounded decimals.
• 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 — Recompute query0 upper-x with exact rational arithmetic, not rounded decimals.
• S002 · premise_and_result_audit — Recompute query0 x-lower with exact rational arithmetic, not rounded decimals.
• S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.

Result

([-8/1, 12/1], [0/1, -8/1, 25/2], [Yes, Yes, No], [[8/1, 12/1], [0/1, 20/1], [41/2, -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.