Question 8214: Closed-Interval Intersection Including Touching Endpoints

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. Intervals are [1/2, 7/2] and [4/2, 10/2]. Give (first endpoints, second endpoints, maximum lower bound, minimum upper bound, nonempty?, intersection endpoints or empty-set).

• 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. Intervals are [1/2, 7/2] and [4/2, 10/2]. Give (first endpoints, second endpoints, maximum lower bound, minimum upper bound, nonempty?, intersection endpoints or empty-set).

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

lower bounds left cross := mul(1, 1)

lower bounds left cross = 1

lower bounds right cross := mul(2, 2)

lower bounds right cross = 4

lower bounds cross residual := sub(1, 4)

lower bounds cross residual = -3

lower bounds:fractions=[1/2, 2/1]

cross residual=-3;comparison=<

upper bounds left cross := mul(7, 1)

upper bounds left cross = 7

upper bounds right cross := mul(5, 2)

upper bounds right cross = 10

upper bounds cross residual := sub(7, 10)

upper bounds cross residual = -3

upper bounds:fractions=[7/2, 5/1]

cross residual=-3;comparison=<

intersection inner bounds left cross := mul(2, 2)

intersection inner bounds left cross = 4

intersection inner bounds right cross := mul(7, 1)

intersection inner bounds right cross = 7

intersection inner bounds cross residual := sub(4, 7)

intersection inner bounds cross residual = -3

intersection inner bounds:fractions=[2/1, 7/2]

cross residual=-3;comparison=<

intervals=[[1/2, 7/2], [2/1, 5/1]]

max lower=2/1;min upper=7/2;nonempty=Yes

Reverse checks

• S014 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S013 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S012 · premise_and_result_audit — Recompute intersection inner bounds cross residual with exact rational arithmetic, not rounded decimals.
• S011 · premise_and_result_audit — Recompute intersection inner bounds right cross with exact rational arithmetic, not rounded decimals.
• S010 · premise_and_result_audit — Recompute intersection inner bounds left cross with exact rational arithmetic, not rounded decimals.
• S009 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S008 · premise_and_result_audit — Recompute upper bounds cross residual with exact rational arithmetic, not rounded decimals.
• S007 · premise_and_result_audit — Recompute upper bounds right cross with exact rational arithmetic, not rounded decimals.
• S006 · premise_and_result_audit — Recompute upper bounds left cross with exact rational arithmetic, not rounded decimals.
• S005 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S004 · premise_and_result_audit — Recompute lower bounds cross residual with exact rational arithmetic, not rounded decimals.
• S003 · premise_and_result_audit — Recompute lower bounds right cross with exact rational arithmetic, not rounded decimals.
• S002 · premise_and_result_audit — Recompute lower bounds left cross 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

([1/2, 7/2], [2/1, 5/1], 2/1, 7/2, Yes, [2/1, 7/2])

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.