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Exact Rational Order and Closed-Interval Boundaries

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This question bank has corresponding bilingual step-by-step derivations. Source candidate and English ai_candidate states are retained. AI review, finite computation and Lean formal proof are labelled separately.

batch-20261012-advance-20261017-r04-order-relation-foundations · 2026-10-12 · 100 questions

This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.

1. Stable Rational Sort with Original Index Witnesses

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. Ordered inputs=[0/1, 0/2, 0/3]. Give (normalized inputs, stable sorted original indices, sorted values, adjacent comparison symbols). Equal values retain their original index order.

View answer and explanation

([0/1, 0/1, 0/1], [0, 1, 2], [0/1, 0/1, 0/1], [=, =])

Compare exact rational cross-products and use original indices to break equal-value ties. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0/1, 0/1, 0/1], [0, 1, 2], [0/1, 0/1, 0/1], [=, =]).

Reviewed step-by-step derivation

Usual order on the real numbers · qf-density17-order-three-rational-sort · candidate

2. Stable Rational Sort with Original Index Witnesses

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. Ordered inputs=[-1/2, 1/3, -2/4]. Give (normalized inputs, stable sorted original indices, sorted values, adjacent comparison symbols). Equal values retain their original index order.

View answer and explanation

([-1/2, 1/3, -1/2], [0, 2, 1], [-1/2, -1/2, 1/3], [=, <])

Compare exact rational cross-products and use original indices to break equal-value ties. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([-1/2, 1/3, -1/2], [0, 2, 1], [-1/2, -1/2, 1/3], [=, <]).

Reviewed step-by-step derivation

Usual order on the real numbers · qf-density17-order-three-rational-sort · candidate

3. Stable Rational Sort with Original Index Witnesses

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. Ordered inputs=[1/2, -1/2, 0/1]. Give (normalized inputs, stable sorted original indices, sorted values, adjacent comparison symbols). Equal values retain their original index order.

View answer and explanation

([1/2, -1/2, 0/1], [1, 2, 0], [-1/2, 0/1, 1/2], [<, <])

Compare exact rational cross-products and use original indices to break equal-value ties. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1/2, -1/2, 0/1], [1, 2, 0], [-1/2, 0/1, 1/2], [<, <]).

Reviewed step-by-step derivation

Usual order on the real numbers · qf-density17-order-three-rational-sort · candidate

4. Stable Rational Sort with Original Index Witnesses

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. Ordered inputs=[-9/3, 1/2, -1/4]. Give (normalized inputs, stable sorted original indices, sorted values, adjacent comparison symbols). Equal values retain their original index order.

View answer and explanation

([-3/1, 1/2, -1/4], [0, 2, 1], [-3/1, -1/4, 1/2], [<, <])

Compare exact rational cross-products and use original indices to break equal-value ties. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([-3/1, 1/2, -1/4], [0, 2, 1], [-3/1, -1/4, 1/2], [<, <]).

Reviewed step-by-step derivation

Usual order on the real numbers · qf-density17-order-three-rational-sort · candidate

5. Stable Rational Sort with Original Index Witnesses

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. Ordered inputs=[-8/3, 1/2, -1/4]. Give (normalized inputs, stable sorted original indices, sorted values, adjacent comparison symbols). Equal values retain their original index order.

View answer and explanation

([-8/3, 1/2, -1/4], [0, 2, 1], [-8/3, -1/4, 1/2], [<, <])

Compare exact rational cross-products and use original indices to break equal-value ties. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([-8/3, 1/2, -1/4], [0, 2, 1], [-8/3, -1/4, 1/2], [<, <]).

Reviewed step-by-step derivation

Usual order on the real numbers · qf-density17-order-three-rational-sort · candidate

6. Stable Rational Sort with Original Index Witnesses

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. Ordered inputs=[-7/3, 1/2, -1/4]. Give (normalized inputs, stable sorted original indices, sorted values, adjacent comparison symbols). Equal values retain their original index order.

View answer and explanation

([-7/3, 1/2, -1/4], [0, 2, 1], [-7/3, -1/4, 1/2], [<, <])

Compare exact rational cross-products and use original indices to break equal-value ties. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([-7/3, 1/2, -1/4], [0, 2, 1], [-7/3, -1/4, 1/2], [<, <]).

Reviewed step-by-step derivation

Usual order on the real numbers · qf-density17-order-three-rational-sort · candidate

7. Stable Rational Sort with Original Index Witnesses

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. Ordered inputs=[-6/3, 1/2, -1/4]. Give (normalized inputs, stable sorted original indices, sorted values, adjacent comparison symbols). Equal values retain their original index order.

View answer and explanation

([-2/1, 1/2, -1/4], [0, 2, 1], [-2/1, -1/4, 1/2], [<, <])

Compare exact rational cross-products and use original indices to break equal-value ties. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([-2/1, 1/2, -1/4], [0, 2, 1], [-2/1, -1/4, 1/2], [<, <]).

Reviewed step-by-step derivation

Usual order on the real numbers · qf-density17-order-three-rational-sort · candidate

8. Stable Rational Sort with Original Index Witnesses

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. Ordered inputs=[-5/3, 1/2, -1/4]. Give (normalized inputs, stable sorted original indices, sorted values, adjacent comparison symbols). Equal values retain their original index order.

View answer and explanation

([-5/3, 1/2, -1/4], [0, 2, 1], [-5/3, -1/4, 1/2], [<, <])

Compare exact rational cross-products and use original indices to break equal-value ties. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([-5/3, 1/2, -1/4], [0, 2, 1], [-5/3, -1/4, 1/2], [<, <]).

Reviewed step-by-step derivation

Usual order on the real numbers · qf-density17-order-three-rational-sort · candidate

9. Stable Rational Sort with Original Index Witnesses

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. Ordered inputs=[-4/3, 1/2, -1/4]. Give (normalized inputs, stable sorted original indices, sorted values, adjacent comparison symbols). Equal values retain their original index order.

View answer and explanation

([-4/3, 1/2, -1/4], [0, 2, 1], [-4/3, -1/4, 1/2], [<, <])

Compare exact rational cross-products and use original indices to break equal-value ties. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([-4/3, 1/2, -1/4], [0, 2, 1], [-4/3, -1/4, 1/2], [<, <]).

Reviewed step-by-step derivation

Usual order on the real numbers · qf-density17-order-three-rational-sort · candidate

10. Stable Rational Sort with Original Index Witnesses

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. Ordered inputs=[-3/3, 1/2, -1/4]. Give (normalized inputs, stable sorted original indices, sorted values, adjacent comparison symbols). Equal values retain their original index order.

View answer and explanation

([-1/1, 1/2, -1/4], [0, 2, 1], [-1/1, -1/4, 1/2], [<, <])

Compare exact rational cross-products and use original indices to break equal-value ties. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([-1/1, 1/2, -1/4], [0, 2, 1], [-1/1, -1/4, 1/2], [<, <]).

Reviewed step-by-step derivation

Usual order on the real numbers · qf-density17-order-three-rational-sort · candidate

11. Stable Rational Sort with Original Index Witnesses

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. Ordered inputs=[-2/3, 1/2, -1/4]. Give (normalized inputs, stable sorted original indices, sorted values, adjacent comparison symbols). Equal values retain their original index order.

View answer and explanation

([-2/3, 1/2, -1/4], [0, 2, 1], [-2/3, -1/4, 1/2], [<, <])

Compare exact rational cross-products and use original indices to break equal-value ties. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([-2/3, 1/2, -1/4], [0, 2, 1], [-2/3, -1/4, 1/2], [<, <]).

Reviewed step-by-step derivation

Usual order on the real numbers · qf-density17-order-three-rational-sort · candidate

12. Stable Rational Sort with Original Index Witnesses

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. Ordered inputs=[-1/3, 1/2, -1/4]. Give (normalized inputs, stable sorted original indices, sorted values, adjacent comparison symbols). Equal values retain their original index order.

View answer and explanation

([-1/3, 1/2, -1/4], [0, 2, 1], [-1/3, -1/4, 1/2], [<, <])

Compare exact rational cross-products and use original indices to break equal-value ties. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([-1/3, 1/2, -1/4], [0, 2, 1], [-1/3, -1/4, 1/2], [<, <]).

Reviewed step-by-step derivation

Usual order on the real numbers · qf-density17-order-three-rational-sort · candidate

13. Stable Rational Sort with Original Index Witnesses

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. Ordered inputs=[0/3, 1/2, -1/4]. Give (normalized inputs, stable sorted original indices, sorted values, adjacent comparison symbols). Equal values retain their original index order.

View answer and explanation

([0/1, 1/2, -1/4], [2, 0, 1], [-1/4, 0/1, 1/2], [<, <])

Compare exact rational cross-products and use original indices to break equal-value ties. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0/1, 1/2, -1/4], [2, 0, 1], [-1/4, 0/1, 1/2], [<, <]).

Reviewed step-by-step derivation

Usual order on the real numbers · qf-density17-order-three-rational-sort · candidate

14. Stable Rational Sort with Original Index Witnesses

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. Ordered inputs=[1/3, 1/2, -1/4]. Give (normalized inputs, stable sorted original indices, sorted values, adjacent comparison symbols). Equal values retain their original index order.

View answer and explanation

([1/3, 1/2, -1/4], [2, 0, 1], [-1/4, 1/3, 1/2], [<, <])

Compare exact rational cross-products and use original indices to break equal-value ties. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1/3, 1/2, -1/4], [2, 0, 1], [-1/4, 1/3, 1/2], [<, <]).

Reviewed step-by-step derivation

Usual order on the real numbers · qf-density17-order-three-rational-sort · candidate

15. Stable Rational Sort with Original Index Witnesses

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. Ordered inputs=[2/3, 1/2, -1/4]. Give (normalized inputs, stable sorted original indices, sorted values, adjacent comparison symbols). Equal values retain their original index order.

View answer and explanation

([2/3, 1/2, -1/4], [2, 1, 0], [-1/4, 1/2, 2/3], [<, <])

Compare exact rational cross-products and use original indices to break equal-value ties. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([2/3, 1/2, -1/4], [2, 1, 0], [-1/4, 1/2, 2/3], [<, <]).

Reviewed step-by-step derivation

Usual order on the real numbers · qf-density17-order-three-rational-sort · candidate

16. Stable Rational Sort with Original Index Witnesses

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. Ordered inputs=[3/3, 1/2, -1/4]. Give (normalized inputs, stable sorted original indices, sorted values, adjacent comparison symbols). Equal values retain their original index order.

View answer and explanation

([1/1, 1/2, -1/4], [2, 1, 0], [-1/4, 1/2, 1/1], [<, <])

Compare exact rational cross-products and use original indices to break equal-value ties. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1/1, 1/2, -1/4], [2, 1, 0], [-1/4, 1/2, 1/1], [<, <]).

Reviewed step-by-step derivation

Usual order on the real numbers · qf-density17-order-three-rational-sort · candidate

17. Stable Rational Sort with Original Index Witnesses

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. Ordered inputs=[4/3, 1/2, -1/4]. Give (normalized inputs, stable sorted original indices, sorted values, adjacent comparison symbols). Equal values retain their original index order.

View answer and explanation

([4/3, 1/2, -1/4], [2, 1, 0], [-1/4, 1/2, 4/3], [<, <])

Compare exact rational cross-products and use original indices to break equal-value ties. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([4/3, 1/2, -1/4], [2, 1, 0], [-1/4, 1/2, 4/3], [<, <]).

Reviewed step-by-step derivation

Usual order on the real numbers · qf-density17-order-three-rational-sort · candidate

18. Stable Rational Sort with Original Index Witnesses

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. Ordered inputs=[5/3, 1/2, -1/4]. Give (normalized inputs, stable sorted original indices, sorted values, adjacent comparison symbols). Equal values retain their original index order.

View answer and explanation

([5/3, 1/2, -1/4], [2, 1, 0], [-1/4, 1/2, 5/3], [<, <])

Compare exact rational cross-products and use original indices to break equal-value ties. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([5/3, 1/2, -1/4], [2, 1, 0], [-1/4, 1/2, 5/3], [<, <]).

Reviewed step-by-step derivation

Usual order on the real numbers · qf-density17-order-three-rational-sort · candidate

19. Stable Rational Sort with Original Index Witnesses

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. Ordered inputs=[6/3, 1/2, -1/4]. Give (normalized inputs, stable sorted original indices, sorted values, adjacent comparison symbols). Equal values retain their original index order.

View answer and explanation

([2/1, 1/2, -1/4], [2, 1, 0], [-1/4, 1/2, 2/1], [<, <])

Compare exact rational cross-products and use original indices to break equal-value ties. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([2/1, 1/2, -1/4], [2, 1, 0], [-1/4, 1/2, 2/1], [<, <]).

Reviewed step-by-step derivation

Usual order on the real numbers · qf-density17-order-three-rational-sort · candidate

20. Stable Rational Sort with Original Index Witnesses

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. Ordered inputs=[7/3, 1/2, -1/4]. Give (normalized inputs, stable sorted original indices, sorted values, adjacent comparison symbols). Equal values retain their original index order.

View answer and explanation

([7/3, 1/2, -1/4], [2, 1, 0], [-1/4, 1/2, 7/3], [<, <])

Compare exact rational cross-products and use original indices to break equal-value ties. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([7/3, 1/2, -1/4], [2, 1, 0], [-1/4, 1/2, 7/3], [<, <]).

Reviewed step-by-step derivation

Usual order on the real numbers · qf-density17-order-three-rational-sort · candidate

21. Stable Rational Sort with Original Index Witnesses

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. Ordered inputs=[8/3, 1/2, -1/4]. Give (normalized inputs, stable sorted original indices, sorted values, adjacent comparison symbols). Equal values retain their original index order.

View answer and explanation

([8/3, 1/2, -1/4], [2, 1, 0], [-1/4, 1/2, 8/3], [<, <])

Compare exact rational cross-products and use original indices to break equal-value ties. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([8/3, 1/2, -1/4], [2, 1, 0], [-1/4, 1/2, 8/3], [<, <]).

Reviewed step-by-step derivation

Usual order on the real numbers · qf-density17-order-three-rational-sort · candidate

22. Stable Rational Sort with Original Index Witnesses

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. Ordered inputs=[9/3, 1/2, -1/4]. Give (normalized inputs, stable sorted original indices, sorted values, adjacent comparison symbols). Equal values retain their original index order.

View answer and explanation

([3/1, 1/2, -1/4], [2, 1, 0], [-1/4, 1/2, 3/1], [<, <])

Compare exact rational cross-products and use original indices to break equal-value ties. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([3/1, 1/2, -1/4], [2, 1, 0], [-1/4, 1/2, 3/1], [<, <]).

Reviewed step-by-step derivation

Usual order on the real numbers · qf-density17-order-three-rational-sort · candidate

23. Stable Rational Sort with Original Index Witnesses

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. Ordered inputs=[10/3, 1/2, -1/4]. Give (normalized inputs, stable sorted original indices, sorted values, adjacent comparison symbols). Equal values retain their original index order.

View answer and explanation

([10/3, 1/2, -1/4], [2, 1, 0], [-1/4, 1/2, 10/3], [<, <])

Compare exact rational cross-products and use original indices to break equal-value ties. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([10/3, 1/2, -1/4], [2, 1, 0], [-1/4, 1/2, 10/3], [<, <]).

Reviewed step-by-step derivation

Usual order on the real numbers · qf-density17-order-three-rational-sort · candidate

24. Stable Rational Sort with Original Index Witnesses

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. Ordered inputs=[11/3, 1/2, -1/4]. Give (normalized inputs, stable sorted original indices, sorted values, adjacent comparison symbols). Equal values retain their original index order.

View answer and explanation

([11/3, 1/2, -1/4], [2, 1, 0], [-1/4, 1/2, 11/3], [<, <])

Compare exact rational cross-products and use original indices to break equal-value ties. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([11/3, 1/2, -1/4], [2, 1, 0], [-1/4, 1/2, 11/3], [<, <]).

Reviewed step-by-step derivation

Usual order on the real numbers · qf-density17-order-three-rational-sort · candidate

25. Stable Rational Sort with Original Index Witnesses

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. Ordered inputs=[12/3, 1/2, -1/4]. Give (normalized inputs, stable sorted original indices, sorted values, adjacent comparison symbols). Equal values retain their original index order.

View answer and explanation

([4/1, 1/2, -1/4], [2, 1, 0], [-1/4, 1/2, 4/1], [<, <])

Compare exact rational cross-products and use original indices to break equal-value ties. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([4/1, 1/2, -1/4], [2, 1, 0], [-1/4, 1/2, 4/1], [<, <]).

Reviewed step-by-step derivation

Usual order on the real numbers · qf-density17-order-three-rational-sort · candidate

26. Closed-Interval Intersection Including Touching Endpoints

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 [0/1, 1/1] and [1/1, 2/1]. Give (first endpoints, second endpoints, maximum lower bound, minimum upper bound, nonempty?, intersection endpoints or empty-set).

View answer and explanation

([0/1, 1/1], [1/1, 2/1], 1/1, 1/1, Yes, [1/1, 1/1])

Use inclusive endpoint comparisons; equality of the two inner bounds gives a singleton, not an empty intersection. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0/1, 1/1], [1/1, 2/1], 1/1, 1/1, Yes, [1/1, 1/1]).

Reviewed step-by-step derivation

Usual order on the real numbers · qf-density17-order-interval-intersection · candidate

27. Closed-Interval Intersection Including Touching Endpoints

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 [-2/1, -1/1] and [0/1, 1/1]. Give (first endpoints, second endpoints, maximum lower bound, minimum upper bound, nonempty?, intersection endpoints or empty-set).

View answer and explanation

([-2/1, -1/1], [0/1, 1/1], 0/1, -1/1, No, empty-set)

Use inclusive endpoint comparisons; equality of the two inner bounds gives a singleton, not an empty intersection. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([-2/1, -1/1], [0/1, 1/1], 0/1, -1/1, No, empty-set).

Reviewed step-by-step derivation

Usual order on the real numbers · qf-density17-order-interval-intersection · candidate

28. Closed-Interval Intersection Including Touching Endpoints

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 [0/1, 0/1] and [0/1, 0/1]. Give (first endpoints, second endpoints, maximum lower bound, minimum upper bound, nonempty?, intersection endpoints or empty-set).

View answer and explanation

([0/1, 0/1], [0/1, 0/1], 0/1, 0/1, Yes, [0/1, 0/1])

Use inclusive endpoint comparisons; equality of the two inner bounds gives a singleton, not an empty intersection. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0/1, 0/1], [0/1, 0/1], 0/1, 0/1, Yes, [0/1, 0/1]).

Reviewed step-by-step derivation

Usual order on the real numbers · qf-density17-order-interval-intersection · candidate

29. Closed-Interval Intersection Including Touching Endpoints

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 [-2/1, 2/1] and [-1/1, 3/1]. Give (first endpoints, second endpoints, maximum lower bound, minimum upper bound, nonempty?, intersection endpoints or empty-set).

View answer and explanation

([-2/1, 2/1], [-1/1, 3/1], -1/1, 2/1, Yes, [-1/1, 2/1])

Use inclusive endpoint comparisons; equality of the two inner bounds gives a singleton, not an empty intersection. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([-2/1, 2/1], [-1/1, 3/1], -1/1, 2/1, Yes, [-1/1, 2/1]).

Reviewed step-by-step derivation

Usual order on the real numbers · qf-density17-order-interval-intersection · candidate

30. Closed-Interval Intersection Including Touching Endpoints

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 [-8/2, -2/2] and [-5/2, 1/2]. Give (first endpoints, second endpoints, maximum lower bound, minimum upper bound, nonempty?, intersection endpoints or empty-set).

View answer and explanation

([-4/1, -1/1], [-5/2, 1/2], -5/2, -1/1, Yes, [-5/2, -1/1])

Use inclusive endpoint comparisons; equality of the two inner bounds gives a singleton, not an empty intersection. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([-4/1, -1/1], [-5/2, 1/2], -5/2, -1/1, Yes, [-5/2, -1/1]).

Reviewed step-by-step derivation

Usual order on the real numbers · qf-density17-order-interval-intersection · candidate

31. Closed-Interval Intersection Including Touching Endpoints

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

View answer and explanation

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

Use inclusive endpoint comparisons; equality of the two inner bounds gives a singleton, not an empty intersection. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([-7/2, -1/2], [-2/1, 1/1], -2/1, -1/2, Yes, [-2/1, -1/2]).

Reviewed step-by-step derivation

Usual order on the real numbers · qf-density17-order-interval-intersection · candidate

32. Closed-Interval Intersection Including Touching Endpoints

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 [-6/2, 0/2] and [-3/2, 3/2]. Give (first endpoints, second endpoints, maximum lower bound, minimum upper bound, nonempty?, intersection endpoints or empty-set).

View answer and explanation

([-3/1, 0/1], [-3/2, 3/2], -3/2, 0/1, Yes, [-3/2, 0/1])

Use inclusive endpoint comparisons; equality of the two inner bounds gives a singleton, not an empty intersection. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([-3/1, 0/1], [-3/2, 3/2], -3/2, 0/1, Yes, [-3/2, 0/1]).

Reviewed step-by-step derivation

Usual order on the real numbers · qf-density17-order-interval-intersection · candidate

33. Closed-Interval Intersection Including Touching Endpoints

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 [-5/2, 1/2] and [-2/2, 4/2]. Give (first endpoints, second endpoints, maximum lower bound, minimum upper bound, nonempty?, intersection endpoints or empty-set).

View answer and explanation

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

Use inclusive endpoint comparisons; equality of the two inner bounds gives a singleton, not an empty intersection. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([-5/2, 1/2], [-1/1, 2/1], -1/1, 1/2, Yes, [-1/1, 1/2]).

Reviewed step-by-step derivation

Usual order on the real numbers · qf-density17-order-interval-intersection · candidate

34. Closed-Interval Intersection Including Touching Endpoints

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 [-4/2, 2/2] and [-1/2, 5/2]. Give (first endpoints, second endpoints, maximum lower bound, minimum upper bound, nonempty?, intersection endpoints or empty-set).

View answer and explanation

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

Use inclusive endpoint comparisons; equality of the two inner bounds gives a singleton, not an empty intersection. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([-2/1, 1/1], [-1/2, 5/2], -1/2, 1/1, Yes, [-1/2, 1/1]).

Reviewed step-by-step derivation

Usual order on the real numbers · qf-density17-order-interval-intersection · candidate

35. Closed-Interval Intersection Including Touching Endpoints

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 [-3/2, 3/2] and [0/2, 6/2]. Give (first endpoints, second endpoints, maximum lower bound, minimum upper bound, nonempty?, intersection endpoints or empty-set).

View answer and explanation

([-3/2, 3/2], [0/1, 3/1], 0/1, 3/2, Yes, [0/1, 3/2])

Use inclusive endpoint comparisons; equality of the two inner bounds gives a singleton, not an empty intersection. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([-3/2, 3/2], [0/1, 3/1], 0/1, 3/2, Yes, [0/1, 3/2]).

Reviewed step-by-step derivation

Usual order on the real numbers · qf-density17-order-interval-intersection · candidate

36. Closed-Interval Intersection Including Touching Endpoints

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

View answer and explanation

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

Use inclusive endpoint comparisons; equality of the two inner bounds gives a singleton, not an empty intersection. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([-1/1, 2/1], [1/2, 7/2], 1/2, 2/1, Yes, [1/2, 2/1]).

Reviewed step-by-step derivation

Usual order on the real numbers · qf-density17-order-interval-intersection · candidate

37. Closed-Interval Intersection Including Touching Endpoints

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, 5/2] and [2/2, 8/2]. Give (first endpoints, second endpoints, maximum lower bound, minimum upper bound, nonempty?, intersection endpoints or empty-set).

View answer and explanation

([-1/2, 5/2], [1/1, 4/1], 1/1, 5/2, Yes, [1/1, 5/2])

Use inclusive endpoint comparisons; equality of the two inner bounds gives a singleton, not an empty intersection. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([-1/2, 5/2], [1/1, 4/1], 1/1, 5/2, Yes, [1/1, 5/2]).

Reviewed step-by-step derivation

Usual order on the real numbers · qf-density17-order-interval-intersection · candidate

38. Closed-Interval Intersection Including Touching Endpoints

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 [0/2, 6/2] and [3/2, 9/2]. Give (first endpoints, second endpoints, maximum lower bound, minimum upper bound, nonempty?, intersection endpoints or empty-set).

View answer and explanation

([0/1, 3/1], [3/2, 9/2], 3/2, 3/1, Yes, [3/2, 3/1])

Use inclusive endpoint comparisons; equality of the two inner bounds gives a singleton, not an empty intersection. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0/1, 3/1], [3/2, 9/2], 3/2, 3/1, Yes, [3/2, 3/1]).

Reviewed step-by-step derivation

Usual order on the real numbers · qf-density17-order-interval-intersection · candidate

39. Closed-Interval Intersection Including Touching Endpoints

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).

View answer and explanation

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

Use inclusive endpoint comparisons; equality of the two inner bounds gives a singleton, not an empty intersection. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1/2, 7/2], [2/1, 5/1], 2/1, 7/2, Yes, [2/1, 7/2]).

Reviewed step-by-step derivation

Usual order on the real numbers · qf-density17-order-interval-intersection · candidate

40. Closed-Interval Intersection Including Touching Endpoints

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 [2/2, 8/2] and [5/2, 11/2]. Give (first endpoints, second endpoints, maximum lower bound, minimum upper bound, nonempty?, intersection endpoints or empty-set).

View answer and explanation

([1/1, 4/1], [5/2, 11/2], 5/2, 4/1, Yes, [5/2, 4/1])

Use inclusive endpoint comparisons; equality of the two inner bounds gives a singleton, not an empty intersection. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([1/1, 4/1], [5/2, 11/2], 5/2, 4/1, Yes, [5/2, 4/1]).

Reviewed step-by-step derivation

Usual order on the real numbers · qf-density17-order-interval-intersection · candidate

41. Closed-Interval Intersection Including Touching Endpoints

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 [3/2, 9/2] and [6/2, 12/2]. Give (first endpoints, second endpoints, maximum lower bound, minimum upper bound, nonempty?, intersection endpoints or empty-set).

View answer and explanation

([3/2, 9/2], [3/1, 6/1], 3/1, 9/2, Yes, [3/1, 9/2])

Use inclusive endpoint comparisons; equality of the two inner bounds gives a singleton, not an empty intersection. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([3/2, 9/2], [3/1, 6/1], 3/1, 9/2, Yes, [3/1, 9/2]).

Reviewed step-by-step derivation

Usual order on the real numbers · qf-density17-order-interval-intersection · candidate

42. Closed-Interval Intersection Including Touching Endpoints

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 [4/2, 10/2] and [7/2, 13/2]. Give (first endpoints, second endpoints, maximum lower bound, minimum upper bound, nonempty?, intersection endpoints or empty-set).

View answer and explanation

([2/1, 5/1], [7/2, 13/2], 7/2, 5/1, Yes, [7/2, 5/1])

Use inclusive endpoint comparisons; equality of the two inner bounds gives a singleton, not an empty intersection. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([2/1, 5/1], [7/2, 13/2], 7/2, 5/1, Yes, [7/2, 5/1]).

Reviewed step-by-step derivation

Usual order on the real numbers · qf-density17-order-interval-intersection · candidate

43. Closed-Interval Intersection Including Touching Endpoints

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 [5/2, 11/2] and [8/2, 14/2]. Give (first endpoints, second endpoints, maximum lower bound, minimum upper bound, nonempty?, intersection endpoints or empty-set).

View answer and explanation

([5/2, 11/2], [4/1, 7/1], 4/1, 11/2, Yes, [4/1, 11/2])

Use inclusive endpoint comparisons; equality of the two inner bounds gives a singleton, not an empty intersection. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([5/2, 11/2], [4/1, 7/1], 4/1, 11/2, Yes, [4/1, 11/2]).

Reviewed step-by-step derivation

Usual order on the real numbers · qf-density17-order-interval-intersection · candidate

44. Closed-Interval Intersection Including Touching Endpoints

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 [6/2, 12/2] and [9/2, 15/2]. Give (first endpoints, second endpoints, maximum lower bound, minimum upper bound, nonempty?, intersection endpoints or empty-set).

View answer and explanation

([3/1, 6/1], [9/2, 15/2], 9/2, 6/1, Yes, [9/2, 6/1])

Use inclusive endpoint comparisons; equality of the two inner bounds gives a singleton, not an empty intersection. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([3/1, 6/1], [9/2, 15/2], 9/2, 6/1, Yes, [9/2, 6/1]).

Reviewed step-by-step derivation

Usual order on the real numbers · qf-density17-order-interval-intersection · candidate

45. Closed-Interval Intersection Including Touching Endpoints

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 [7/2, 13/2] and [10/2, 16/2]. Give (first endpoints, second endpoints, maximum lower bound, minimum upper bound, nonempty?, intersection endpoints or empty-set).

View answer and explanation

([7/2, 13/2], [5/1, 8/1], 5/1, 13/2, Yes, [5/1, 13/2])

Use inclusive endpoint comparisons; equality of the two inner bounds gives a singleton, not an empty intersection. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([7/2, 13/2], [5/1, 8/1], 5/1, 13/2, Yes, [5/1, 13/2]).

Reviewed step-by-step derivation

Usual order on the real numbers · qf-density17-order-interval-intersection · candidate

46. Closed-Interval Intersection Including Touching Endpoints

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 [8/2, 14/2] and [11/2, 17/2]. Give (first endpoints, second endpoints, maximum lower bound, minimum upper bound, nonempty?, intersection endpoints or empty-set).

View answer and explanation

([4/1, 7/1], [11/2, 17/2], 11/2, 7/1, Yes, [11/2, 7/1])

Use inclusive endpoint comparisons; equality of the two inner bounds gives a singleton, not an empty intersection. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([4/1, 7/1], [11/2, 17/2], 11/2, 7/1, Yes, [11/2, 7/1]).

Reviewed step-by-step derivation

Usual order on the real numbers · qf-density17-order-interval-intersection · candidate

47. Closed-Interval Intersection Including Touching Endpoints

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 [9/2, 15/2] and [12/2, 18/2]. Give (first endpoints, second endpoints, maximum lower bound, minimum upper bound, nonempty?, intersection endpoints or empty-set).

View answer and explanation

([9/2, 15/2], [6/1, 9/1], 6/1, 15/2, Yes, [6/1, 15/2])

Use inclusive endpoint comparisons; equality of the two inner bounds gives a singleton, not an empty intersection. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([9/2, 15/2], [6/1, 9/1], 6/1, 15/2, Yes, [6/1, 15/2]).

Reviewed step-by-step derivation

Usual order on the real numbers · qf-density17-order-interval-intersection · candidate

48. Closed-Interval Intersection Including Touching Endpoints

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 [10/2, 16/2] and [13/2, 19/2]. Give (first endpoints, second endpoints, maximum lower bound, minimum upper bound, nonempty?, intersection endpoints or empty-set).

View answer and explanation

([5/1, 8/1], [13/2, 19/2], 13/2, 8/1, Yes, [13/2, 8/1])

Use inclusive endpoint comparisons; equality of the two inner bounds gives a singleton, not an empty intersection. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([5/1, 8/1], [13/2, 19/2], 13/2, 8/1, Yes, [13/2, 8/1]).

Reviewed step-by-step derivation

Usual order on the real numbers · qf-density17-order-interval-intersection · candidate

49. Closed-Interval Intersection Including Touching Endpoints

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 [11/2, 17/2] and [14/2, 20/2]. Give (first endpoints, second endpoints, maximum lower bound, minimum upper bound, nonempty?, intersection endpoints or empty-set).

View answer and explanation

([11/2, 17/2], [7/1, 10/1], 7/1, 17/2, Yes, [7/1, 17/2])

Use inclusive endpoint comparisons; equality of the two inner bounds gives a singleton, not an empty intersection. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([11/2, 17/2], [7/1, 10/1], 7/1, 17/2, Yes, [7/1, 17/2]).

Reviewed step-by-step derivation

Usual order on the real numbers · qf-density17-order-interval-intersection · candidate

50. Closed-Interval Intersection Including Touching Endpoints

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 [12/2, 18/2] and [15/2, 21/2]. Give (first endpoints, second endpoints, maximum lower bound, minimum upper bound, nonempty?, intersection endpoints or empty-set).

View answer and explanation

([6/1, 9/1], [15/2, 21/2], 15/2, 9/1, Yes, [15/2, 9/1])

Use inclusive endpoint comparisons; equality of the two inner bounds gives a singleton, not an empty intersection. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([6/1, 9/1], [15/2, 21/2], 15/2, 9/1, Yes, [15/2, 9/1]).

Reviewed step-by-step derivation

Usual order on the real numbers · qf-density17-order-interval-intersection · candidate

51. Three Closed-Interval Membership Tests and Signed Slacks

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=[0/1, 0/1], queries=[-1/1, 0/1, 1/1]. Give (interval endpoints, normalized queries, three membership flags, each [query−lower, upper−query] signed slack pair).

View answer and explanation

([0/1, 0/1], [-1/1, 0/1, 1/1], [No, Yes, No], [[-1/1, 1/1], [0/1, 0/1], [1/1, -1/1]])

Membership requires both signed slacks to be nonnegative, including zero at the endpoints. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0/1, 0/1], [-1/1, 0/1, 1/1], [No, Yes, No], [[-1/1, 1/1], [0/1, 0/1], [1/1, -1/1]]).

Reviewed step-by-step derivation

Usual order on the real numbers · qf-density17-order-three-membership-slacks · candidate

52. Three Closed-Interval Membership Tests and Signed Slacks

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=[-1/2, 1/2], queries=[-1/2, 1/2, 1/1]. Give (interval endpoints, normalized queries, three membership flags, each [query−lower, upper−query] signed slack pair).

View answer and explanation

([-1/2, 1/2], [-1/2, 1/2, 1/1], [Yes, Yes, No], [[0/1, 1/1], [1/1, 0/1], [3/2, -1/2]])

Membership requires both signed slacks to be nonnegative, including zero at the endpoints. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([-1/2, 1/2], [-1/2, 1/2, 1/1], [Yes, Yes, No], [[0/1, 1/1], [1/1, 0/1], [3/2, -1/2]]).

Reviewed step-by-step derivation

Usual order on the real numbers · qf-density17-order-three-membership-slacks · candidate

53. Three Closed-Interval Membership Tests and Signed Slacks

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=[-2/3, 2/2], queries=[0/1, -2/3, 3/2]. Give (interval endpoints, normalized queries, three membership flags, each [query−lower, upper−query] signed slack pair).

View answer and explanation

([-2/3, 1/1], [0/1, -2/3, 3/2], [Yes, Yes, No], [[2/3, 1/1], [0/1, 5/3], [13/6, -1/2]])

Membership requires both signed slacks to be nonnegative, including zero at the endpoints. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([-2/3, 1/1], [0/1, -2/3, 3/2], [Yes, Yes, No], [[2/3, 1/1], [0/1, 5/3], [13/6, -1/2]]).

Reviewed step-by-step derivation

Usual order on the real numbers · qf-density17-order-three-membership-slacks · candidate

54. Three Closed-Interval Membership Tests and Signed Slacks

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=[-3/3, 3/2], queries=[0/1, -3/3, 4/2]. Give (interval endpoints, normalized queries, three membership flags, each [query−lower, upper−query] signed slack pair).

View answer and explanation

([-1/1, 3/2], [0/1, -1/1, 2/1], [Yes, Yes, No], [[1/1, 3/2], [0/1, 5/2], [3/1, -1/2]])

Membership requires both signed slacks to be nonnegative, including zero at the endpoints. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([-1/1, 3/2], [0/1, -1/1, 2/1], [Yes, Yes, No], [[1/1, 3/2], [0/1, 5/2], [3/1, -1/2]]).

Reviewed step-by-step derivation

Usual order on the real numbers · qf-density17-order-three-membership-slacks · candidate

55. Three Closed-Interval Membership Tests and Signed Slacks

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=[-4/3, 4/2], queries=[0/1, -4/3, 5/2]. Give (interval endpoints, normalized queries, three membership flags, each [query−lower, upper−query] signed slack pair).

View answer and explanation

([-4/3, 2/1], [0/1, -4/3, 5/2], [Yes, Yes, No], [[4/3, 2/1], [0/1, 10/3], [23/6, -1/2]])

Membership requires both signed slacks to be nonnegative, including zero at the endpoints. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([-4/3, 2/1], [0/1, -4/3, 5/2], [Yes, Yes, No], [[4/3, 2/1], [0/1, 10/3], [23/6, -1/2]]).

Reviewed step-by-step derivation

Usual order on the real numbers · qf-density17-order-three-membership-slacks · candidate

56. Three Closed-Interval Membership Tests and Signed Slacks

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=[-5/3, 5/2], queries=[0/1, -5/3, 6/2]. Give (interval endpoints, normalized queries, three membership flags, each [query−lower, upper−query] signed slack pair).

View answer and explanation

([-5/3, 5/2], [0/1, -5/3, 3/1], [Yes, Yes, No], [[5/3, 5/2], [0/1, 25/6], [14/3, -1/2]])

Membership requires both signed slacks to be nonnegative, including zero at the endpoints. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([-5/3, 5/2], [0/1, -5/3, 3/1], [Yes, Yes, No], [[5/3, 5/2], [0/1, 25/6], [14/3, -1/2]]).

Reviewed step-by-step derivation

Usual order on the real numbers · qf-density17-order-three-membership-slacks · candidate

57. Three Closed-Interval Membership Tests and Signed Slacks

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=[-6/3, 6/2], queries=[0/1, -6/3, 7/2]. Give (interval endpoints, normalized queries, three membership flags, each [query−lower, upper−query] signed slack pair).

View answer and explanation

([-2/1, 3/1], [0/1, -2/1, 7/2], [Yes, Yes, No], [[2/1, 3/1], [0/1, 5/1], [11/2, -1/2]])

Membership requires both signed slacks to be nonnegative, including zero at the endpoints. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([-2/1, 3/1], [0/1, -2/1, 7/2], [Yes, Yes, No], [[2/1, 3/1], [0/1, 5/1], [11/2, -1/2]]).

Reviewed step-by-step derivation

Usual order on the real numbers · qf-density17-order-three-membership-slacks · candidate

58. Three Closed-Interval Membership Tests and Signed Slacks

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).

View answer and explanation

([-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]])

Membership requires both signed slacks to be nonnegative, including zero at the endpoints. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([-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]]).

Reviewed step-by-step derivation

Usual order on the real numbers · qf-density17-order-three-membership-slacks · candidate

59. Three Closed-Interval Membership Tests and Signed Slacks

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=[-8/3, 8/2], queries=[0/1, -8/3, 9/2]. Give (interval endpoints, normalized queries, three membership flags, each [query−lower, upper−query] signed slack pair).

View answer and explanation

([-8/3, 4/1], [0/1, -8/3, 9/2], [Yes, Yes, No], [[8/3, 4/1], [0/1, 20/3], [43/6, -1/2]])

Membership requires both signed slacks to be nonnegative, including zero at the endpoints. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([-8/3, 4/1], [0/1, -8/3, 9/2], [Yes, Yes, No], [[8/3, 4/1], [0/1, 20/3], [43/6, -1/2]]).

Reviewed step-by-step derivation

Usual order on the real numbers · qf-density17-order-three-membership-slacks · candidate

60. Three Closed-Interval Membership Tests and Signed Slacks

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=[-9/3, 9/2], queries=[0/1, -9/3, 10/2]. Give (interval endpoints, normalized queries, three membership flags, each [query−lower, upper−query] signed slack pair).

View answer and explanation

([-3/1, 9/2], [0/1, -3/1, 5/1], [Yes, Yes, No], [[3/1, 9/2], [0/1, 15/2], [8/1, -1/2]])

Membership requires both signed slacks to be nonnegative, including zero at the endpoints. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([-3/1, 9/2], [0/1, -3/1, 5/1], [Yes, Yes, No], [[3/1, 9/2], [0/1, 15/2], [8/1, -1/2]]).

Reviewed step-by-step derivation

Usual order on the real numbers · qf-density17-order-three-membership-slacks · candidate

61. Three Closed-Interval Membership Tests and Signed Slacks

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=[-10/3, 10/2], queries=[0/1, -10/3, 11/2]. Give (interval endpoints, normalized queries, three membership flags, each [query−lower, upper−query] signed slack pair).

View answer and explanation

([-10/3, 5/1], [0/1, -10/3, 11/2], [Yes, Yes, No], [[10/3, 5/1], [0/1, 25/3], [53/6, -1/2]])

Membership requires both signed slacks to be nonnegative, including zero at the endpoints. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([-10/3, 5/1], [0/1, -10/3, 11/2], [Yes, Yes, No], [[10/3, 5/1], [0/1, 25/3], [53/6, -1/2]]).

Reviewed step-by-step derivation

Usual order on the real numbers · qf-density17-order-three-membership-slacks · candidate

62. Three Closed-Interval Membership Tests and Signed Slacks

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=[-11/3, 11/2], queries=[0/1, -11/3, 12/2]. Give (interval endpoints, normalized queries, three membership flags, each [query−lower, upper−query] signed slack pair).

View answer and explanation

([-11/3, 11/2], [0/1, -11/3, 6/1], [Yes, Yes, No], [[11/3, 11/2], [0/1, 55/6], [29/3, -1/2]])

Membership requires both signed slacks to be nonnegative, including zero at the endpoints. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([-11/3, 11/2], [0/1, -11/3, 6/1], [Yes, Yes, No], [[11/3, 11/2], [0/1, 55/6], [29/3, -1/2]]).

Reviewed step-by-step derivation

Usual order on the real numbers · qf-density17-order-three-membership-slacks · candidate

63. Three Closed-Interval Membership Tests and Signed Slacks

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=[-12/3, 12/2], queries=[0/1, -12/3, 13/2]. Give (interval endpoints, normalized queries, three membership flags, each [query−lower, upper−query] signed slack pair).

View answer and explanation

([-4/1, 6/1], [0/1, -4/1, 13/2], [Yes, Yes, No], [[4/1, 6/1], [0/1, 10/1], [21/2, -1/2]])

Membership requires both signed slacks to be nonnegative, including zero at the endpoints. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([-4/1, 6/1], [0/1, -4/1, 13/2], [Yes, Yes, No], [[4/1, 6/1], [0/1, 10/1], [21/2, -1/2]]).

Reviewed step-by-step derivation

Usual order on the real numbers · qf-density17-order-three-membership-slacks · candidate

64. Three Closed-Interval Membership Tests and Signed Slacks

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=[-13/3, 13/2], queries=[0/1, -13/3, 14/2]. Give (interval endpoints, normalized queries, three membership flags, each [query−lower, upper−query] signed slack pair).

View answer and explanation

([-13/3, 13/2], [0/1, -13/3, 7/1], [Yes, Yes, No], [[13/3, 13/2], [0/1, 65/6], [34/3, -1/2]])

Membership requires both signed slacks to be nonnegative, including zero at the endpoints. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([-13/3, 13/2], [0/1, -13/3, 7/1], [Yes, Yes, No], [[13/3, 13/2], [0/1, 65/6], [34/3, -1/2]]).

Reviewed step-by-step derivation

Usual order on the real numbers · qf-density17-order-three-membership-slacks · candidate

65. Three Closed-Interval Membership Tests and Signed Slacks

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=[-14/3, 14/2], queries=[0/1, -14/3, 15/2]. Give (interval endpoints, normalized queries, three membership flags, each [query−lower, upper−query] signed slack pair).

View answer and explanation

([-14/3, 7/1], [0/1, -14/3, 15/2], [Yes, Yes, No], [[14/3, 7/1], [0/1, 35/3], [73/6, -1/2]])

Membership requires both signed slacks to be nonnegative, including zero at the endpoints. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([-14/3, 7/1], [0/1, -14/3, 15/2], [Yes, Yes, No], [[14/3, 7/1], [0/1, 35/3], [73/6, -1/2]]).

Reviewed step-by-step derivation

Usual order on the real numbers · qf-density17-order-three-membership-slacks · candidate

66. Three Closed-Interval Membership Tests and Signed Slacks

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=[-15/3, 15/2], queries=[0/1, -15/3, 16/2]. Give (interval endpoints, normalized queries, three membership flags, each [query−lower, upper−query] signed slack pair).

View answer and explanation

([-5/1, 15/2], [0/1, -5/1, 8/1], [Yes, Yes, No], [[5/1, 15/2], [0/1, 25/2], [13/1, -1/2]])

Membership requires both signed slacks to be nonnegative, including zero at the endpoints. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([-5/1, 15/2], [0/1, -5/1, 8/1], [Yes, Yes, No], [[5/1, 15/2], [0/1, 25/2], [13/1, -1/2]]).

Reviewed step-by-step derivation

Usual order on the real numbers · qf-density17-order-three-membership-slacks · candidate

67. Three Closed-Interval Membership Tests and Signed Slacks

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=[-16/3, 16/2], queries=[0/1, -16/3, 17/2]. Give (interval endpoints, normalized queries, three membership flags, each [query−lower, upper−query] signed slack pair).

View answer and explanation

([-16/3, 8/1], [0/1, -16/3, 17/2], [Yes, Yes, No], [[16/3, 8/1], [0/1, 40/3], [83/6, -1/2]])

Membership requires both signed slacks to be nonnegative, including zero at the endpoints. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([-16/3, 8/1], [0/1, -16/3, 17/2], [Yes, Yes, No], [[16/3, 8/1], [0/1, 40/3], [83/6, -1/2]]).

Reviewed step-by-step derivation

Usual order on the real numbers · qf-density17-order-three-membership-slacks · candidate

68. Three Closed-Interval Membership Tests and Signed Slacks

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=[-17/3, 17/2], queries=[0/1, -17/3, 18/2]. Give (interval endpoints, normalized queries, three membership flags, each [query−lower, upper−query] signed slack pair).

View answer and explanation

([-17/3, 17/2], [0/1, -17/3, 9/1], [Yes, Yes, No], [[17/3, 17/2], [0/1, 85/6], [44/3, -1/2]])

Membership requires both signed slacks to be nonnegative, including zero at the endpoints. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([-17/3, 17/2], [0/1, -17/3, 9/1], [Yes, Yes, No], [[17/3, 17/2], [0/1, 85/6], [44/3, -1/2]]).

Reviewed step-by-step derivation

Usual order on the real numbers · qf-density17-order-three-membership-slacks · candidate

69. Three Closed-Interval Membership Tests and Signed Slacks

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=[-18/3, 18/2], queries=[0/1, -18/3, 19/2]. Give (interval endpoints, normalized queries, three membership flags, each [query−lower, upper−query] signed slack pair).

View answer and explanation

([-6/1, 9/1], [0/1, -6/1, 19/2], [Yes, Yes, No], [[6/1, 9/1], [0/1, 15/1], [31/2, -1/2]])

Membership requires both signed slacks to be nonnegative, including zero at the endpoints. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([-6/1, 9/1], [0/1, -6/1, 19/2], [Yes, Yes, No], [[6/1, 9/1], [0/1, 15/1], [31/2, -1/2]]).

Reviewed step-by-step derivation

Usual order on the real numbers · qf-density17-order-three-membership-slacks · candidate

70. Three Closed-Interval Membership Tests and Signed Slacks

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=[-19/3, 19/2], queries=[0/1, -19/3, 20/2]. Give (interval endpoints, normalized queries, three membership flags, each [query−lower, upper−query] signed slack pair).

View answer and explanation

([-19/3, 19/2], [0/1, -19/3, 10/1], [Yes, Yes, No], [[19/3, 19/2], [0/1, 95/6], [49/3, -1/2]])

Membership requires both signed slacks to be nonnegative, including zero at the endpoints. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([-19/3, 19/2], [0/1, -19/3, 10/1], [Yes, Yes, No], [[19/3, 19/2], [0/1, 95/6], [49/3, -1/2]]).

Reviewed step-by-step derivation

Usual order on the real numbers · qf-density17-order-three-membership-slacks · candidate

71. Three Closed-Interval Membership Tests and Signed Slacks

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=[-20/3, 20/2], queries=[0/1, -20/3, 21/2]. Give (interval endpoints, normalized queries, three membership flags, each [query−lower, upper−query] signed slack pair).

View answer and explanation

([-20/3, 10/1], [0/1, -20/3, 21/2], [Yes, Yes, No], [[20/3, 10/1], [0/1, 50/3], [103/6, -1/2]])

Membership requires both signed slacks to be nonnegative, including zero at the endpoints. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([-20/3, 10/1], [0/1, -20/3, 21/2], [Yes, Yes, No], [[20/3, 10/1], [0/1, 50/3], [103/6, -1/2]]).

Reviewed step-by-step derivation

Usual order on the real numbers · qf-density17-order-three-membership-slacks · candidate

72. Three Closed-Interval Membership Tests and Signed Slacks

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=[-21/3, 21/2], queries=[0/1, -21/3, 22/2]. Give (interval endpoints, normalized queries, three membership flags, each [query−lower, upper−query] signed slack pair).

View answer and explanation

([-7/1, 21/2], [0/1, -7/1, 11/1], [Yes, Yes, No], [[7/1, 21/2], [0/1, 35/2], [18/1, -1/2]])

Membership requires both signed slacks to be nonnegative, including zero at the endpoints. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([-7/1, 21/2], [0/1, -7/1, 11/1], [Yes, Yes, No], [[7/1, 21/2], [0/1, 35/2], [18/1, -1/2]]).

Reviewed step-by-step derivation

Usual order on the real numbers · qf-density17-order-three-membership-slacks · candidate

73. Three Closed-Interval Membership Tests and Signed Slacks

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=[-22/3, 22/2], queries=[0/1, -22/3, 23/2]. Give (interval endpoints, normalized queries, three membership flags, each [query−lower, upper−query] signed slack pair).

View answer and explanation

([-22/3, 11/1], [0/1, -22/3, 23/2], [Yes, Yes, No], [[22/3, 11/1], [0/1, 55/3], [113/6, -1/2]])

Membership requires both signed slacks to be nonnegative, including zero at the endpoints. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([-22/3, 11/1], [0/1, -22/3, 23/2], [Yes, Yes, No], [[22/3, 11/1], [0/1, 55/3], [113/6, -1/2]]).

Reviewed step-by-step derivation

Usual order on the real numbers · qf-density17-order-three-membership-slacks · candidate

74. Three Closed-Interval Membership Tests and Signed Slacks

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=[-23/3, 23/2], queries=[0/1, -23/3, 24/2]. Give (interval endpoints, normalized queries, three membership flags, each [query−lower, upper−query] signed slack pair).

View answer and explanation

([-23/3, 23/2], [0/1, -23/3, 12/1], [Yes, Yes, No], [[23/3, 23/2], [0/1, 115/6], [59/3, -1/2]])

Membership requires both signed slacks to be nonnegative, including zero at the endpoints. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([-23/3, 23/2], [0/1, -23/3, 12/1], [Yes, Yes, No], [[23/3, 23/2], [0/1, 115/6], [59/3, -1/2]]).

Reviewed step-by-step derivation

Usual order on the real numbers · qf-density17-order-three-membership-slacks · candidate

75. Three Closed-Interval Membership Tests and Signed Slacks

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).

View answer and explanation

([-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]])

Membership requires both signed slacks to be nonnegative, including zero at the endpoints. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([-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]]).

Reviewed step-by-step derivation

Usual order on the real numbers · qf-density17-order-three-membership-slacks · candidate

76. Scaled Rational Values and Order Direction

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. a=-1/2, b=1/3, t=0/1. Give (a, b, t, t*a, t*b, original comparison, scaled comparison, order-preserved/order-reversed/collapsed according to the sign of t).

View answer and explanation

(-1/2, 1/3, 0/1, 0/1, 0/1, <, =, collapsed)

Calculate both scaled values: positive scaling preserves order, negative scaling reverses it, and zero scaling collapses both values. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (-1/2, 1/3, 0/1, 0/1, 0/1, <, =, collapsed).

Reviewed step-by-step derivation

Usual order on the real numbers · qf-density17-order-scaling-direction · candidate

77. Scaled Rational Values and Order Direction

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. a=0/1, b=0/1, t=-2/1. Give (a, b, t, t*a, t*b, original comparison, scaled comparison, order-preserved/order-reversed/collapsed according to the sign of t).

View answer and explanation

(0/1, 0/1, -2/1, 0/1, 0/1, =, =, order-reversed)

Calculate both scaled values: positive scaling preserves order, negative scaling reverses it, and zero scaling collapses both values. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (0/1, 0/1, -2/1, 0/1, 0/1, =, =, order-reversed).

Reviewed step-by-step derivation

Usual order on the real numbers · qf-density17-order-scaling-direction · candidate

78. Scaled Rational Values and Order Direction

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. a=-10/3, b=-9/2, t=3/3. Give (a, b, t, t*a, t*b, original comparison, scaled comparison, order-preserved/order-reversed/collapsed according to the sign of t).

View answer and explanation

(-10/3, -9/2, 1/1, -10/3, -9/2, >, >, order-preserved)

Calculate both scaled values: positive scaling preserves order, negative scaling reverses it, and zero scaling collapses both values. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (-10/3, -9/2, 1/1, -10/3, -9/2, >, >, order-preserved).

Reviewed step-by-step derivation

Usual order on the real numbers · qf-density17-order-scaling-direction · candidate

79. Scaled Rational Values and Order Direction

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. a=-9/3, b=-8/2, t=0/4. Give (a, b, t, t*a, t*b, original comparison, scaled comparison, order-preserved/order-reversed/collapsed according to the sign of t).

View answer and explanation

(-3/1, -4/1, 0/1, 0/1, 0/1, >, =, collapsed)

Calculate both scaled values: positive scaling preserves order, negative scaling reverses it, and zero scaling collapses both values. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (-3/1, -4/1, 0/1, 0/1, 0/1, >, =, collapsed).

Reviewed step-by-step derivation

Usual order on the real numbers · qf-density17-order-scaling-direction · candidate

80. Scaled Rational Values and Order Direction

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. a=-8/3, b=-7/2, t=3/1. Give (a, b, t, t*a, t*b, original comparison, scaled comparison, order-preserved/order-reversed/collapsed according to the sign of t).

View answer and explanation

(-8/3, -7/2, 3/1, -8/1, -21/2, >, >, order-preserved)

Calculate both scaled values: positive scaling preserves order, negative scaling reverses it, and zero scaling collapses both values. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (-8/3, -7/2, 3/1, -8/1, -21/2, >, >, order-preserved).

Reviewed step-by-step derivation

Usual order on the real numbers · qf-density17-order-scaling-direction · candidate

81. Scaled Rational Values and Order Direction

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. a=-7/3, b=-6/2, t=-2/2. Give (a, b, t, t*a, t*b, original comparison, scaled comparison, order-preserved/order-reversed/collapsed according to the sign of t).

View answer and explanation

(-7/3, -3/1, -1/1, 7/3, 3/1, >, <, order-reversed)

Calculate both scaled values: positive scaling preserves order, negative scaling reverses it, and zero scaling collapses both values. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (-7/3, -3/1, -1/1, 7/3, 3/1, >, <, order-reversed).

Reviewed step-by-step derivation

Usual order on the real numbers · qf-density17-order-scaling-direction · candidate

82. Scaled Rational Values and Order Direction

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. a=-6/3, b=-5/2, t=0/3. Give (a, b, t, t*a, t*b, original comparison, scaled comparison, order-preserved/order-reversed/collapsed according to the sign of t).

View answer and explanation

(-2/1, -5/2, 0/1, 0/1, 0/1, >, =, collapsed)

Calculate both scaled values: positive scaling preserves order, negative scaling reverses it, and zero scaling collapses both values. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (-2/1, -5/2, 0/1, 0/1, 0/1, >, =, collapsed).

Reviewed step-by-step derivation

Usual order on the real numbers · qf-density17-order-scaling-direction · candidate

83. Scaled Rational Values and Order Direction

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. a=-5/3, b=-4/2, t=-2/4. Give (a, b, t, t*a, t*b, original comparison, scaled comparison, order-preserved/order-reversed/collapsed according to the sign of t).

View answer and explanation

(-5/3, -2/1, -1/2, 5/6, 1/1, >, <, order-reversed)

Calculate both scaled values: positive scaling preserves order, negative scaling reverses it, and zero scaling collapses both values. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (-5/3, -2/1, -1/2, 5/6, 1/1, >, <, order-reversed).

Reviewed step-by-step derivation

Usual order on the real numbers · qf-density17-order-scaling-direction · candidate

84. Scaled Rational Values and Order Direction

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. a=-4/3, b=-3/2, t=3/1. Give (a, b, t, t*a, t*b, original comparison, scaled comparison, order-preserved/order-reversed/collapsed according to the sign of t).

View answer and explanation

(-4/3, -3/2, 3/1, -4/1, -9/2, >, >, order-preserved)

Calculate both scaled values: positive scaling preserves order, negative scaling reverses it, and zero scaling collapses both values. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (-4/3, -3/2, 3/1, -4/1, -9/2, >, >, order-preserved).

Reviewed step-by-step derivation

Usual order on the real numbers · qf-density17-order-scaling-direction · candidate

85. Scaled Rational Values and Order Direction

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. a=-3/3, b=-2/2, t=0/2. Give (a, b, t, t*a, t*b, original comparison, scaled comparison, order-preserved/order-reversed/collapsed according to the sign of t).

View answer and explanation

(-1/1, -1/1, 0/1, 0/1, 0/1, =, =, collapsed)

Calculate both scaled values: positive scaling preserves order, negative scaling reverses it, and zero scaling collapses both values. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (-1/1, -1/1, 0/1, 0/1, 0/1, =, =, collapsed).

Reviewed step-by-step derivation

Usual order on the real numbers · qf-density17-order-scaling-direction · candidate

86. Scaled Rational Values and Order Direction

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. a=-2/3, b=-1/2, t=3/3. Give (a, b, t, t*a, t*b, original comparison, scaled comparison, order-preserved/order-reversed/collapsed according to the sign of t).

View answer and explanation

(-2/3, -1/2, 1/1, -2/3, -1/2, <, <, order-preserved)

Calculate both scaled values: positive scaling preserves order, negative scaling reverses it, and zero scaling collapses both values. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (-2/3, -1/2, 1/1, -2/3, -1/2, <, <, order-preserved).

Reviewed step-by-step derivation

Usual order on the real numbers · qf-density17-order-scaling-direction · candidate

87. Scaled Rational Values and Order Direction

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. a=-1/3, b=0/2, t=-2/4. Give (a, b, t, t*a, t*b, original comparison, scaled comparison, order-preserved/order-reversed/collapsed according to the sign of t).

View answer and explanation

(-1/3, 0/1, -1/2, 1/6, 0/1, <, >, order-reversed)

Calculate both scaled values: positive scaling preserves order, negative scaling reverses it, and zero scaling collapses both values. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (-1/3, 0/1, -1/2, 1/6, 0/1, <, >, order-reversed).

Reviewed step-by-step derivation

Usual order on the real numbers · qf-density17-order-scaling-direction · candidate

88. Scaled Rational Values and Order Direction

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. a=0/3, b=1/2, t=0/1. Give (a, b, t, t*a, t*b, original comparison, scaled comparison, order-preserved/order-reversed/collapsed according to the sign of t).

View answer and explanation

(0/1, 1/2, 0/1, 0/1, 0/1, <, =, collapsed)

Calculate both scaled values: positive scaling preserves order, negative scaling reverses it, and zero scaling collapses both values. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (0/1, 1/2, 0/1, 0/1, 0/1, <, =, collapsed).

Reviewed step-by-step derivation

Usual order on the real numbers · qf-density17-order-scaling-direction · candidate

89. Scaled Rational Values and Order Direction

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. a=1/3, b=2/2, t=-2/2. Give (a, b, t, t*a, t*b, original comparison, scaled comparison, order-preserved/order-reversed/collapsed according to the sign of t).

View answer and explanation

(1/3, 1/1, -1/1, -1/3, -1/1, <, >, order-reversed)

Calculate both scaled values: positive scaling preserves order, negative scaling reverses it, and zero scaling collapses both values. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (1/3, 1/1, -1/1, -1/3, -1/1, <, >, order-reversed).

Reviewed step-by-step derivation

Usual order on the real numbers · qf-density17-order-scaling-direction · candidate

90. Scaled Rational Values and Order Direction

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. a=2/3, b=3/2, t=3/3. Give (a, b, t, t*a, t*b, original comparison, scaled comparison, order-preserved/order-reversed/collapsed according to the sign of t).

View answer and explanation

(2/3, 3/2, 1/1, 2/3, 3/2, <, <, order-preserved)

Calculate both scaled values: positive scaling preserves order, negative scaling reverses it, and zero scaling collapses both values. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (2/3, 3/2, 1/1, 2/3, 3/2, <, <, order-preserved).

Reviewed step-by-step derivation

Usual order on the real numbers · qf-density17-order-scaling-direction · candidate

91. Scaled Rational Values and Order Direction

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. a=3/3, b=4/2, t=0/4. Give (a, b, t, t*a, t*b, original comparison, scaled comparison, order-preserved/order-reversed/collapsed according to the sign of t).

View answer and explanation

(1/1, 2/1, 0/1, 0/1, 0/1, <, =, collapsed)

Calculate both scaled values: positive scaling preserves order, negative scaling reverses it, and zero scaling collapses both values. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (1/1, 2/1, 0/1, 0/1, 0/1, <, =, collapsed).

Reviewed step-by-step derivation

Usual order on the real numbers · qf-density17-order-scaling-direction · candidate

92. Scaled Rational Values and Order Direction

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. a=4/3, b=5/2, t=3/1. Give (a, b, t, t*a, t*b, original comparison, scaled comparison, order-preserved/order-reversed/collapsed according to the sign of t).

View answer and explanation

(4/3, 5/2, 3/1, 4/1, 15/2, <, <, order-preserved)

Calculate both scaled values: positive scaling preserves order, negative scaling reverses it, and zero scaling collapses both values. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (4/3, 5/2, 3/1, 4/1, 15/2, <, <, order-preserved).

Reviewed step-by-step derivation

Usual order on the real numbers · qf-density17-order-scaling-direction · candidate

93. Scaled Rational Values and Order Direction

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. a=5/3, b=6/2, t=-2/2. Give (a, b, t, t*a, t*b, original comparison, scaled comparison, order-preserved/order-reversed/collapsed according to the sign of t).

View answer and explanation

(5/3, 3/1, -1/1, -5/3, -3/1, <, >, order-reversed)

Calculate both scaled values: positive scaling preserves order, negative scaling reverses it, and zero scaling collapses both values. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (5/3, 3/1, -1/1, -5/3, -3/1, <, >, order-reversed).

Reviewed step-by-step derivation

Usual order on the real numbers · qf-density17-order-scaling-direction · candidate

94. Scaled Rational Values and Order Direction

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. a=6/3, b=7/2, t=0/3. Give (a, b, t, t*a, t*b, original comparison, scaled comparison, order-preserved/order-reversed/collapsed according to the sign of t).

View answer and explanation

(2/1, 7/2, 0/1, 0/1, 0/1, <, =, collapsed)

Calculate both scaled values: positive scaling preserves order, negative scaling reverses it, and zero scaling collapses both values. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (2/1, 7/2, 0/1, 0/1, 0/1, <, =, collapsed).

Reviewed step-by-step derivation

Usual order on the real numbers · qf-density17-order-scaling-direction · candidate

95. Scaled Rational Values and Order Direction

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. a=7/3, b=8/2, t=-2/4. Give (a, b, t, t*a, t*b, original comparison, scaled comparison, order-preserved/order-reversed/collapsed according to the sign of t).

View answer and explanation

(7/3, 4/1, -1/2, -7/6, -2/1, <, >, order-reversed)

Calculate both scaled values: positive scaling preserves order, negative scaling reverses it, and zero scaling collapses both values. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (7/3, 4/1, -1/2, -7/6, -2/1, <, >, order-reversed).

Reviewed step-by-step derivation

Usual order on the real numbers · qf-density17-order-scaling-direction · candidate

96. Scaled Rational Values and Order Direction

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. a=8/3, b=9/2, t=3/1. Give (a, b, t, t*a, t*b, original comparison, scaled comparison, order-preserved/order-reversed/collapsed according to the sign of t).

View answer and explanation

(8/3, 9/2, 3/1, 8/1, 27/2, <, <, order-preserved)

Calculate both scaled values: positive scaling preserves order, negative scaling reverses it, and zero scaling collapses both values. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (8/3, 9/2, 3/1, 8/1, 27/2, <, <, order-preserved).

Reviewed step-by-step derivation

Usual order on the real numbers · qf-density17-order-scaling-direction · candidate

97. Scaled Rational Values and Order Direction

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. a=9/3, b=10/2, t=0/2. Give (a, b, t, t*a, t*b, original comparison, scaled comparison, order-preserved/order-reversed/collapsed according to the sign of t).

View answer and explanation

(3/1, 5/1, 0/1, 0/1, 0/1, <, =, collapsed)

Calculate both scaled values: positive scaling preserves order, negative scaling reverses it, and zero scaling collapses both values. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (3/1, 5/1, 0/1, 0/1, 0/1, <, =, collapsed).

Reviewed step-by-step derivation

Usual order on the real numbers · qf-density17-order-scaling-direction · candidate

98. Scaled Rational Values and Order Direction

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. a=10/3, b=11/2, t=3/3. Give (a, b, t, t*a, t*b, original comparison, scaled comparison, order-preserved/order-reversed/collapsed according to the sign of t).

View answer and explanation

(10/3, 11/2, 1/1, 10/3, 11/2, <, <, order-preserved)

Calculate both scaled values: positive scaling preserves order, negative scaling reverses it, and zero scaling collapses both values. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (10/3, 11/2, 1/1, 10/3, 11/2, <, <, order-preserved).

Reviewed step-by-step derivation

Usual order on the real numbers · qf-density17-order-scaling-direction · candidate

99. Scaled Rational Values and Order Direction

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. a=11/3, b=12/2, t=-2/4. Give (a, b, t, t*a, t*b, original comparison, scaled comparison, order-preserved/order-reversed/collapsed according to the sign of t).

View answer and explanation

(11/3, 6/1, -1/2, -11/6, -3/1, <, >, order-reversed)

Calculate both scaled values: positive scaling preserves order, negative scaling reverses it, and zero scaling collapses both values. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (11/3, 6/1, -1/2, -11/6, -3/1, <, >, order-reversed).

Reviewed step-by-step derivation

Usual order on the real numbers · qf-density17-order-scaling-direction · candidate

100. Scaled Rational Values and Order Direction

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. a=12/3, b=13/2, t=0/1. Give (a, b, t, t*a, t*b, original comparison, scaled comparison, order-preserved/order-reversed/collapsed according to the sign of t).

View answer and explanation

(4/1, 13/2, 0/1, 0/1, 0/1, <, =, collapsed)

Calculate both scaled values: positive scaling preserves order, negative scaling reverses it, and zero scaling collapses both values. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is (4/1, 13/2, 0/1, 0/1, 0/1, <, =, collapsed).

Reviewed step-by-step derivation

Usual order on the real numbers · qf-density17-order-scaling-direction · candidate

Canonical question JSONL · English JSONL · Validation report