# Question 8258: Three Closed-Interval Membership Tests and Signed Slacks [Return to catalogue page 42](https://ocme.evemisslab.com/en/questions/page/42/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261012-advance-20261017-r04-order-relation-foundations/#q-20261012-advance-20261017-r04-order-relation-0058) This teaching derivation has second-context AI mathematical and bilingual semantic review, limited to the model and assumptions stated in this question. It has no human review or new Lean proof and does not promote the original question or MKO to Canonical truth. R0015-08258 · q-20261012-advance-20261017-r04-order-relation-0058 ## 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=[-7/3, 7/2], queries=[0/1, -7/3, 8/2]. 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=[-7/3, 7/2], queries=[0/1, -7/3, 8/2]. 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](https://ocme.evemisslab.com/en/mko/mko-order-relation/) ## Step-by-step derivation S001 · impliesBefore usual total order on R;all original denominators positive After closed endpoints included;negative scaling reverses;zero collapses Conditions: none These exact rationals use the usual real total order,not complex numbers or an arbitrary partial order.Original denominators must be positive;normalizing a negative input denominator does not waive that contract. Step identifiers and dependenciesRule: real_rational_order_scope · Uses: G001 S002 · calculationBefore query0 x-lower := sub(0, -7/3) After query0 x-lower = 7/3 Conditions: none Substitute the source data exactly in the stated operand order;division requires a nonzero denominator. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S001 S003 · calculationBefore query0 upper-x := sub(7/2, 0) After query0 upper-x = 7/2 Conditions: none Substitute the source data exactly in the stated operand order;division requires a nonzero denominator. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S002 S004 · impliesBefore query0=0/1;signed slacks=[7/3, 7/2] After both nonnegative=Yes Conditions: none Both signed slacks must be nonnegative.Zero includes a closed endpoint;do not take absolute values of negative slacks. Step identifiers and dependenciesRule: closed_interval_both_signed_slacks · Uses: G001, S003 S005 · calculationBefore query1 x-lower := sub(-7/3, -7/3) After query1 x-lower = 0 Conditions: none Substitute the source data exactly in the stated operand order;division requires a nonzero denominator. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S004 S006 · calculationBefore query1 upper-x := sub(7/2, -7/3) After query1 upper-x = 35/6 Conditions: none Substitute the source data exactly in the stated operand order;division requires a nonzero denominator. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S005 S007 · impliesBefore query1=-7/3;signed slacks=[0/1, 35/6] After both nonnegative=Yes Conditions: none Both signed slacks must be nonnegative.Zero includes a closed endpoint;do not take absolute values of negative slacks. Step identifiers and dependenciesRule: closed_interval_both_signed_slacks · Uses: G001, S006 S008 · calculationBefore query2 x-lower := sub(4, -7/3) After query2 x-lower = 19/3 Conditions: none Substitute the source data exactly in the stated operand order;division requires a nonzero denominator. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S007 S009 · calculationBefore query2 upper-x := sub(7/2, 4) After query2 upper-x = -1/2 Conditions: none Substitute the source data exactly in the stated operand order;division requires a nonzero denominator. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S008 S010 · impliesBefore query2=4/1;signed slacks=[19/3, -1/2] After both nonnegative=No Conditions: none Both signed slacks must be nonnegative.Zero includes a closed endpoint;do not take absolute values of negative slacks. Step identifiers and dependenciesRule: closed_interval_both_signed_slacks · Uses: G001, S009 ## 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 ([-7/3, 7/2], [0/1, -7/3, 4/1], [Yes, Yes, No], [[7/3, 7/2], [0/1, 35/6], [19/3, -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. Source hashes and review receipts - input commit: 0f7be61e2c22317a4075561dc822c87d881a50db - original question SHA-256: 0a5987c9d932912a71006bff1415e1e158b52dcb62a79b035852312667ac8c51 - candidate record SHA-256: afee7f7d523174a29f7abae7a50c6a6df25cbe9af86c00bca2055705aeb196da - semantic review receipt SHA-256: 6ecad9d909199221a07e923f87425e80b3bdec3eed46fe5910d04fca694cae18 - question-first witness SHA-256: e8f00458977a09493b5f9f05ab9e6feec72debaf4c7d1c609d1fc95900a35a1b - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261012-advance-20261017-r04-order-relation-0058.json)