Exact Rational Midpoints and Closed Intervals
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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-20261010-advance-20261014-r02-real-number-foundations · 2026-10-10 · 100 questions
This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.
1. Ordered Rational Midpoint and Endpoint Gaps
Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. a=0/1≤b=0/1. Give (midpoint, midpoint−a, b−midpoint). Use exact reduced fractions with positive denominators, retaining /1 for integers.
View answer and explanation
(0/1, 0/1, 0/1)
Add and divide by two exactly, then subtract each endpoint separately; no irrational classification or real-completeness proof is claimed. The exact result, retaining the requested order, is (0/1, 0/1, 0/1).
Reviewed step-by-step derivation
Real number · qf-density14-real-ordered-midpoint · candidate
2. Ordered Rational Midpoint and Endpoint Gaps
Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. a=-1/2≤b=1/2. Give (midpoint, midpoint−a, b−midpoint). Use exact reduced fractions with positive denominators, retaining /1 for integers.
View answer and explanation
(0/1, 1/2, 1/2)
Add and divide by two exactly, then subtract each endpoint separately; no irrational classification or real-completeness proof is claimed. The exact result, retaining the requested order, is (0/1, 1/2, 1/2).
Reviewed step-by-step derivation
Real number · qf-density14-real-ordered-midpoint · candidate
3. Ordered Rational Midpoint and Endpoint Gaps
Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. a=-2/2≤b=3/3. Give (midpoint, midpoint−a, b−midpoint). Use exact reduced fractions with positive denominators, retaining /1 for integers.
View answer and explanation
(0/1, 1/1, 1/1)
Add and divide by two exactly, then subtract each endpoint separately; no irrational classification or real-completeness proof is claimed. The exact result, retaining the requested order, is (0/1, 1/1, 1/1).
Reviewed step-by-step derivation
Real number · qf-density14-real-ordered-midpoint · candidate
4. Ordered Rational Midpoint and Endpoint Gaps
Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. a=-3/2≤b=4/3. Give (midpoint, midpoint−a, b−midpoint). Use exact reduced fractions with positive denominators, retaining /1 for integers.
View answer and explanation
(-1/12, 17/12, 17/12)
Add and divide by two exactly, then subtract each endpoint separately; no irrational classification or real-completeness proof is claimed. The exact result, retaining the requested order, is (-1/12, 17/12, 17/12).
Reviewed step-by-step derivation
Real number · qf-density14-real-ordered-midpoint · candidate
5. Ordered Rational Midpoint and Endpoint Gaps
Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. a=-4/2≤b=5/3. Give (midpoint, midpoint−a, b−midpoint). Use exact reduced fractions with positive denominators, retaining /1 for integers.
View answer and explanation
(-1/6, 11/6, 11/6)
Add and divide by two exactly, then subtract each endpoint separately; no irrational classification or real-completeness proof is claimed. The exact result, retaining the requested order, is (-1/6, 11/6, 11/6).
Reviewed step-by-step derivation
Real number · qf-density14-real-ordered-midpoint · candidate
6. Ordered Rational Midpoint and Endpoint Gaps
Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. a=-5/2≤b=6/3. Give (midpoint, midpoint−a, b−midpoint). Use exact reduced fractions with positive denominators, retaining /1 for integers.
View answer and explanation
(-1/4, 9/4, 9/4)
Add and divide by two exactly, then subtract each endpoint separately; no irrational classification or real-completeness proof is claimed. The exact result, retaining the requested order, is (-1/4, 9/4, 9/4).
Reviewed step-by-step derivation
Real number · qf-density14-real-ordered-midpoint · candidate
7. Ordered Rational Midpoint and Endpoint Gaps
Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. a=-6/2≤b=7/3. Give (midpoint, midpoint−a, b−midpoint). Use exact reduced fractions with positive denominators, retaining /1 for integers.
View answer and explanation
(-1/3, 8/3, 8/3)
Add and divide by two exactly, then subtract each endpoint separately; no irrational classification or real-completeness proof is claimed. The exact result, retaining the requested order, is (-1/3, 8/3, 8/3).
Reviewed step-by-step derivation
Real number · qf-density14-real-ordered-midpoint · candidate
8. Ordered Rational Midpoint and Endpoint Gaps
Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. a=-7/2≤b=8/3. Give (midpoint, midpoint−a, b−midpoint). Use exact reduced fractions with positive denominators, retaining /1 for integers.
View answer and explanation
(-5/12, 37/12, 37/12)
Add and divide by two exactly, then subtract each endpoint separately; no irrational classification or real-completeness proof is claimed. The exact result, retaining the requested order, is (-5/12, 37/12, 37/12).
Reviewed step-by-step derivation
Real number · qf-density14-real-ordered-midpoint · candidate
9. Ordered Rational Midpoint and Endpoint Gaps
Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. a=-8/2≤b=9/3. Give (midpoint, midpoint−a, b−midpoint). Use exact reduced fractions with positive denominators, retaining /1 for integers.
View answer and explanation
(-1/2, 7/2, 7/2)
Add and divide by two exactly, then subtract each endpoint separately; no irrational classification or real-completeness proof is claimed. The exact result, retaining the requested order, is (-1/2, 7/2, 7/2).
Reviewed step-by-step derivation
Real number · qf-density14-real-ordered-midpoint · candidate
10. Ordered Rational Midpoint and Endpoint Gaps
Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. a=-9/2≤b=10/3. Give (midpoint, midpoint−a, b−midpoint). Use exact reduced fractions with positive denominators, retaining /1 for integers.
View answer and explanation
(-7/12, 47/12, 47/12)
Add and divide by two exactly, then subtract each endpoint separately; no irrational classification or real-completeness proof is claimed. The exact result, retaining the requested order, is (-7/12, 47/12, 47/12).
Reviewed step-by-step derivation
Real number · qf-density14-real-ordered-midpoint · candidate
11. Ordered Rational Midpoint and Endpoint Gaps
Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. a=-10/2≤b=11/3. Give (midpoint, midpoint−a, b−midpoint). Use exact reduced fractions with positive denominators, retaining /1 for integers.
View answer and explanation
(-2/3, 13/3, 13/3)
Add and divide by two exactly, then subtract each endpoint separately; no irrational classification or real-completeness proof is claimed. The exact result, retaining the requested order, is (-2/3, 13/3, 13/3).
Reviewed step-by-step derivation
Real number · qf-density14-real-ordered-midpoint · candidate
12. Ordered Rational Midpoint and Endpoint Gaps
Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. a=-11/2≤b=12/3. Give (midpoint, midpoint−a, b−midpoint). Use exact reduced fractions with positive denominators, retaining /1 for integers.
View answer and explanation
(-3/4, 19/4, 19/4)
Add and divide by two exactly, then subtract each endpoint separately; no irrational classification or real-completeness proof is claimed. The exact result, retaining the requested order, is (-3/4, 19/4, 19/4).
Reviewed step-by-step derivation
Real number · qf-density14-real-ordered-midpoint · candidate
13. Ordered Rational Midpoint and Endpoint Gaps
Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. a=-12/2≤b=13/3. Give (midpoint, midpoint−a, b−midpoint). Use exact reduced fractions with positive denominators, retaining /1 for integers.
View answer and explanation
(-5/6, 31/6, 31/6)
Add and divide by two exactly, then subtract each endpoint separately; no irrational classification or real-completeness proof is claimed. The exact result, retaining the requested order, is (-5/6, 31/6, 31/6).
Reviewed step-by-step derivation
Real number · qf-density14-real-ordered-midpoint · candidate
14. Ordered Rational Midpoint and Endpoint Gaps
Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. a=-13/2≤b=14/3. Give (midpoint, midpoint−a, b−midpoint). Use exact reduced fractions with positive denominators, retaining /1 for integers.
View answer and explanation
(-11/12, 67/12, 67/12)
Add and divide by two exactly, then subtract each endpoint separately; no irrational classification or real-completeness proof is claimed. The exact result, retaining the requested order, is (-11/12, 67/12, 67/12).
Reviewed step-by-step derivation
Real number · qf-density14-real-ordered-midpoint · candidate
15. Ordered Rational Midpoint and Endpoint Gaps
Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. a=-14/2≤b=15/3. Give (midpoint, midpoint−a, b−midpoint). Use exact reduced fractions with positive denominators, retaining /1 for integers.
View answer and explanation
(-1/1, 6/1, 6/1)
Add and divide by two exactly, then subtract each endpoint separately; no irrational classification or real-completeness proof is claimed. The exact result, retaining the requested order, is (-1/1, 6/1, 6/1).
Reviewed step-by-step derivation
Real number · qf-density14-real-ordered-midpoint · candidate
16. Ordered Rational Midpoint and Endpoint Gaps
Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. a=-15/2≤b=16/3. Give (midpoint, midpoint−a, b−midpoint). Use exact reduced fractions with positive denominators, retaining /1 for integers.
View answer and explanation
(-13/12, 77/12, 77/12)
Add and divide by two exactly, then subtract each endpoint separately; no irrational classification or real-completeness proof is claimed. The exact result, retaining the requested order, is (-13/12, 77/12, 77/12).
Reviewed step-by-step derivation
Real number · qf-density14-real-ordered-midpoint · candidate
17. Ordered Rational Midpoint and Endpoint Gaps
Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. a=-16/2≤b=17/3. Give (midpoint, midpoint−a, b−midpoint). Use exact reduced fractions with positive denominators, retaining /1 for integers.
View answer and explanation
(-7/6, 41/6, 41/6)
Add and divide by two exactly, then subtract each endpoint separately; no irrational classification or real-completeness proof is claimed. The exact result, retaining the requested order, is (-7/6, 41/6, 41/6).
Reviewed step-by-step derivation
Real number · qf-density14-real-ordered-midpoint · candidate
18. Ordered Rational Midpoint and Endpoint Gaps
Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. a=-17/2≤b=18/3. Give (midpoint, midpoint−a, b−midpoint). Use exact reduced fractions with positive denominators, retaining /1 for integers.
View answer and explanation
(-5/4, 29/4, 29/4)
Add and divide by two exactly, then subtract each endpoint separately; no irrational classification or real-completeness proof is claimed. The exact result, retaining the requested order, is (-5/4, 29/4, 29/4).
Reviewed step-by-step derivation
Real number · qf-density14-real-ordered-midpoint · candidate
19. Ordered Rational Midpoint and Endpoint Gaps
Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. a=-18/2≤b=19/3. Give (midpoint, midpoint−a, b−midpoint). Use exact reduced fractions with positive denominators, retaining /1 for integers.
View answer and explanation
(-4/3, 23/3, 23/3)
Add and divide by two exactly, then subtract each endpoint separately; no irrational classification or real-completeness proof is claimed. The exact result, retaining the requested order, is (-4/3, 23/3, 23/3).
Reviewed step-by-step derivation
Real number · qf-density14-real-ordered-midpoint · candidate
20. Ordered Rational Midpoint and Endpoint Gaps
Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. a=-19/2≤b=20/3. Give (midpoint, midpoint−a, b−midpoint). Use exact reduced fractions with positive denominators, retaining /1 for integers.
View answer and explanation
(-17/12, 97/12, 97/12)
Add and divide by two exactly, then subtract each endpoint separately; no irrational classification or real-completeness proof is claimed. The exact result, retaining the requested order, is (-17/12, 97/12, 97/12).
Reviewed step-by-step derivation
Real number · qf-density14-real-ordered-midpoint · candidate
21. Ordered Rational Midpoint and Endpoint Gaps
Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. a=-20/2≤b=21/3. Give (midpoint, midpoint−a, b−midpoint). Use exact reduced fractions with positive denominators, retaining /1 for integers.
View answer and explanation
(-3/2, 17/2, 17/2)
Add and divide by two exactly, then subtract each endpoint separately; no irrational classification or real-completeness proof is claimed. The exact result, retaining the requested order, is (-3/2, 17/2, 17/2).
Reviewed step-by-step derivation
Real number · qf-density14-real-ordered-midpoint · candidate
22. Ordered Rational Midpoint and Endpoint Gaps
Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. a=-21/2≤b=22/3. Give (midpoint, midpoint−a, b−midpoint). Use exact reduced fractions with positive denominators, retaining /1 for integers.
View answer and explanation
(-19/12, 107/12, 107/12)
Add and divide by two exactly, then subtract each endpoint separately; no irrational classification or real-completeness proof is claimed. The exact result, retaining the requested order, is (-19/12, 107/12, 107/12).
Reviewed step-by-step derivation
Real number · qf-density14-real-ordered-midpoint · candidate
23. Ordered Rational Midpoint and Endpoint Gaps
Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. a=-22/2≤b=23/3. Give (midpoint, midpoint−a, b−midpoint). Use exact reduced fractions with positive denominators, retaining /1 for integers.
View answer and explanation
(-5/3, 28/3, 28/3)
Add and divide by two exactly, then subtract each endpoint separately; no irrational classification or real-completeness proof is claimed. The exact result, retaining the requested order, is (-5/3, 28/3, 28/3).
Reviewed step-by-step derivation
Real number · qf-density14-real-ordered-midpoint · candidate
24. Ordered Rational Midpoint and Endpoint Gaps
Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. a=-23/2≤b=24/3. Give (midpoint, midpoint−a, b−midpoint). Use exact reduced fractions with positive denominators, retaining /1 for integers.
View answer and explanation
(-7/4, 39/4, 39/4)
Add and divide by two exactly, then subtract each endpoint separately; no irrational classification or real-completeness proof is claimed. The exact result, retaining the requested order, is (-7/4, 39/4, 39/4).
Reviewed step-by-step derivation
Real number · qf-density14-real-ordered-midpoint · candidate
25. Ordered Rational Midpoint and Endpoint Gaps
Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. a=-24/2≤b=25/3. Give (midpoint, midpoint−a, b−midpoint). Use exact reduced fractions with positive denominators, retaining /1 for integers.
View answer and explanation
(-11/6, 61/6, 61/6)
Add and divide by two exactly, then subtract each endpoint separately; no irrational classification or real-completeness proof is claimed. The exact result, retaining the requested order, is (-11/6, 61/6, 61/6).
Reviewed step-by-step derivation
Real number · qf-density14-real-ordered-midpoint · candidate
26. Finite Distinct Points and Least Containing Closed Interval
Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. Ordered points=[0/1, 0/-2, 0/3]. Give (increasing distinct points, minimum lo, maximum hi, hi−lo); the containing interval is [lo,hi], not the point set’s topological closure.
View answer and explanation
([0/1], 0/1, 0/1, 0/1)
Sort and deduplicate by exact rational value, then take the extreme endpoints; the least containing closed interval differs from topological closure of a finite point set. The exact result, retaining the requested order, is ([0/1], 0/1, 0/1, 0/1).
Reviewed step-by-step derivation
Real number · qf-density14-real-point-closed-hull · candidate
27. Finite Distinct Points and Least Containing Closed Interval
Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. Ordered points=[1/2, -1/2, 1/2]. Give (increasing distinct points, minimum lo, maximum hi, hi−lo); the containing interval is [lo,hi], not the point set’s topological closure.
View answer and explanation
([-1/2, 1/2], -1/2, 1/2, 1/1)
Sort and deduplicate by exact rational value, then take the extreme endpoints; the least containing closed interval differs from topological closure of a finite point set. The exact result, retaining the requested order, is ([-1/2, 1/2], -1/2, 1/2, 1/1).
Reviewed step-by-step derivation
Real number · qf-density14-real-point-closed-hull · candidate
28. Finite Distinct Points and Least Containing Closed Interval
Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. Ordered points=[-2/2, 0/1, 3/3]. Give (increasing distinct points, minimum lo, maximum hi, hi−lo); the containing interval is [lo,hi], not the point set’s topological closure.
View answer and explanation
([-1/1, 0/1, 1/1], -1/1, 1/1, 2/1)
Sort and deduplicate by exact rational value, then take the extreme endpoints; the least containing closed interval differs from topological closure of a finite point set. The exact result, retaining the requested order, is ([-1/1, 0/1, 1/1], -1/1, 1/1, 2/1).
Reviewed step-by-step derivation
Real number · qf-density14-real-point-closed-hull · candidate
29. Finite Distinct Points and Least Containing Closed Interval
Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. Ordered points=[-3/2, 0/1, 4/3]. Give (increasing distinct points, minimum lo, maximum hi, hi−lo); the containing interval is [lo,hi], not the point set’s topological closure.
View answer and explanation
([-3/2, 0/1, 4/3], -3/2, 4/3, 17/6)
Sort and deduplicate by exact rational value, then take the extreme endpoints; the least containing closed interval differs from topological closure of a finite point set. The exact result, retaining the requested order, is ([-3/2, 0/1, 4/3], -3/2, 4/3, 17/6).
Reviewed step-by-step derivation
Real number · qf-density14-real-point-closed-hull · candidate
30. Finite Distinct Points and Least Containing Closed Interval
Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. Ordered points=[-4/2, 0/1, 5/3]. Give (increasing distinct points, minimum lo, maximum hi, hi−lo); the containing interval is [lo,hi], not the point set’s topological closure.
View answer and explanation
([-2/1, 0/1, 5/3], -2/1, 5/3, 11/3)
Sort and deduplicate by exact rational value, then take the extreme endpoints; the least containing closed interval differs from topological closure of a finite point set. The exact result, retaining the requested order, is ([-2/1, 0/1, 5/3], -2/1, 5/3, 11/3).
Reviewed step-by-step derivation
Real number · qf-density14-real-point-closed-hull · candidate
31. Finite Distinct Points and Least Containing Closed Interval
Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. Ordered points=[-5/2, 0/1, 6/3]. Give (increasing distinct points, minimum lo, maximum hi, hi−lo); the containing interval is [lo,hi], not the point set’s topological closure.
View answer and explanation
([-5/2, 0/1, 2/1], -5/2, 2/1, 9/2)
Sort and deduplicate by exact rational value, then take the extreme endpoints; the least containing closed interval differs from topological closure of a finite point set. The exact result, retaining the requested order, is ([-5/2, 0/1, 2/1], -5/2, 2/1, 9/2).
Reviewed step-by-step derivation
Real number · qf-density14-real-point-closed-hull · candidate
32. Finite Distinct Points and Least Containing Closed Interval
Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. Ordered points=[-6/2, 0/1, 7/3]. Give (increasing distinct points, minimum lo, maximum hi, hi−lo); the containing interval is [lo,hi], not the point set’s topological closure.
View answer and explanation
([-3/1, 0/1, 7/3], -3/1, 7/3, 16/3)
Sort and deduplicate by exact rational value, then take the extreme endpoints; the least containing closed interval differs from topological closure of a finite point set. The exact result, retaining the requested order, is ([-3/1, 0/1, 7/3], -3/1, 7/3, 16/3).
Reviewed step-by-step derivation
Real number · qf-density14-real-point-closed-hull · candidate
33. Finite Distinct Points and Least Containing Closed Interval
Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. Ordered points=[-7/2, 0/1, 8/3]. Give (increasing distinct points, minimum lo, maximum hi, hi−lo); the containing interval is [lo,hi], not the point set’s topological closure.
View answer and explanation
([-7/2, 0/1, 8/3], -7/2, 8/3, 37/6)
Sort and deduplicate by exact rational value, then take the extreme endpoints; the least containing closed interval differs from topological closure of a finite point set. The exact result, retaining the requested order, is ([-7/2, 0/1, 8/3], -7/2, 8/3, 37/6).
Reviewed step-by-step derivation
Real number · qf-density14-real-point-closed-hull · candidate
34. Finite Distinct Points and Least Containing Closed Interval
Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. Ordered points=[-8/2, 0/1, 9/3]. Give (increasing distinct points, minimum lo, maximum hi, hi−lo); the containing interval is [lo,hi], not the point set’s topological closure.
View answer and explanation
([-4/1, 0/1, 3/1], -4/1, 3/1, 7/1)
Sort and deduplicate by exact rational value, then take the extreme endpoints; the least containing closed interval differs from topological closure of a finite point set. The exact result, retaining the requested order, is ([-4/1, 0/1, 3/1], -4/1, 3/1, 7/1).
Reviewed step-by-step derivation
Real number · qf-density14-real-point-closed-hull · candidate
35. Finite Distinct Points and Least Containing Closed Interval
Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. Ordered points=[-9/2, 0/1, 10/3]. Give (increasing distinct points, minimum lo, maximum hi, hi−lo); the containing interval is [lo,hi], not the point set’s topological closure.
View answer and explanation
([-9/2, 0/1, 10/3], -9/2, 10/3, 47/6)
Sort and deduplicate by exact rational value, then take the extreme endpoints; the least containing closed interval differs from topological closure of a finite point set. The exact result, retaining the requested order, is ([-9/2, 0/1, 10/3], -9/2, 10/3, 47/6).
Reviewed step-by-step derivation
Real number · qf-density14-real-point-closed-hull · candidate
36. Finite Distinct Points and Least Containing Closed Interval
Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. Ordered points=[-10/2, 0/1, 11/3]. Give (increasing distinct points, minimum lo, maximum hi, hi−lo); the containing interval is [lo,hi], not the point set’s topological closure.
View answer and explanation
([-5/1, 0/1, 11/3], -5/1, 11/3, 26/3)
Sort and deduplicate by exact rational value, then take the extreme endpoints; the least containing closed interval differs from topological closure of a finite point set. The exact result, retaining the requested order, is ([-5/1, 0/1, 11/3], -5/1, 11/3, 26/3).
Reviewed step-by-step derivation
Real number · qf-density14-real-point-closed-hull · candidate
37. Finite Distinct Points and Least Containing Closed Interval
Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. Ordered points=[-11/2, 0/1, 12/3]. Give (increasing distinct points, minimum lo, maximum hi, hi−lo); the containing interval is [lo,hi], not the point set’s topological closure.
View answer and explanation
([-11/2, 0/1, 4/1], -11/2, 4/1, 19/2)
Sort and deduplicate by exact rational value, then take the extreme endpoints; the least containing closed interval differs from topological closure of a finite point set. The exact result, retaining the requested order, is ([-11/2, 0/1, 4/1], -11/2, 4/1, 19/2).
Reviewed step-by-step derivation
Real number · qf-density14-real-point-closed-hull · candidate
38. Finite Distinct Points and Least Containing Closed Interval
Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. Ordered points=[-12/2, 0/1, 13/3]. Give (increasing distinct points, minimum lo, maximum hi, hi−lo); the containing interval is [lo,hi], not the point set’s topological closure.
View answer and explanation
([-6/1, 0/1, 13/3], -6/1, 13/3, 31/3)
Sort and deduplicate by exact rational value, then take the extreme endpoints; the least containing closed interval differs from topological closure of a finite point set. The exact result, retaining the requested order, is ([-6/1, 0/1, 13/3], -6/1, 13/3, 31/3).
Reviewed step-by-step derivation
Real number · qf-density14-real-point-closed-hull · candidate
39. Finite Distinct Points and Least Containing Closed Interval
Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. Ordered points=[-13/2, 0/1, 14/3]. Give (increasing distinct points, minimum lo, maximum hi, hi−lo); the containing interval is [lo,hi], not the point set’s topological closure.
View answer and explanation
([-13/2, 0/1, 14/3], -13/2, 14/3, 67/6)
Sort and deduplicate by exact rational value, then take the extreme endpoints; the least containing closed interval differs from topological closure of a finite point set. The exact result, retaining the requested order, is ([-13/2, 0/1, 14/3], -13/2, 14/3, 67/6).
Reviewed step-by-step derivation
Real number · qf-density14-real-point-closed-hull · candidate
40. Finite Distinct Points and Least Containing Closed Interval
Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. Ordered points=[-14/2, 0/1, 15/3]. Give (increasing distinct points, minimum lo, maximum hi, hi−lo); the containing interval is [lo,hi], not the point set’s topological closure.
View answer and explanation
([-7/1, 0/1, 5/1], -7/1, 5/1, 12/1)
Sort and deduplicate by exact rational value, then take the extreme endpoints; the least containing closed interval differs from topological closure of a finite point set. The exact result, retaining the requested order, is ([-7/1, 0/1, 5/1], -7/1, 5/1, 12/1).
Reviewed step-by-step derivation
Real number · qf-density14-real-point-closed-hull · candidate
41. Finite Distinct Points and Least Containing Closed Interval
Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. Ordered points=[-15/2, 0/1, 16/3]. Give (increasing distinct points, minimum lo, maximum hi, hi−lo); the containing interval is [lo,hi], not the point set’s topological closure.
View answer and explanation
([-15/2, 0/1, 16/3], -15/2, 16/3, 77/6)
Sort and deduplicate by exact rational value, then take the extreme endpoints; the least containing closed interval differs from topological closure of a finite point set. The exact result, retaining the requested order, is ([-15/2, 0/1, 16/3], -15/2, 16/3, 77/6).
Reviewed step-by-step derivation
Real number · qf-density14-real-point-closed-hull · candidate
42. Finite Distinct Points and Least Containing Closed Interval
Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. Ordered points=[-16/2, 0/1, 17/3]. Give (increasing distinct points, minimum lo, maximum hi, hi−lo); the containing interval is [lo,hi], not the point set’s topological closure.
View answer and explanation
([-8/1, 0/1, 17/3], -8/1, 17/3, 41/3)
Sort and deduplicate by exact rational value, then take the extreme endpoints; the least containing closed interval differs from topological closure of a finite point set. The exact result, retaining the requested order, is ([-8/1, 0/1, 17/3], -8/1, 17/3, 41/3).
Reviewed step-by-step derivation
Real number · qf-density14-real-point-closed-hull · candidate
43. Finite Distinct Points and Least Containing Closed Interval
Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. Ordered points=[-17/2, 0/1, 18/3]. Give (increasing distinct points, minimum lo, maximum hi, hi−lo); the containing interval is [lo,hi], not the point set’s topological closure.
View answer and explanation
([-17/2, 0/1, 6/1], -17/2, 6/1, 29/2)
Sort and deduplicate by exact rational value, then take the extreme endpoints; the least containing closed interval differs from topological closure of a finite point set. The exact result, retaining the requested order, is ([-17/2, 0/1, 6/1], -17/2, 6/1, 29/2).
Reviewed step-by-step derivation
Real number · qf-density14-real-point-closed-hull · candidate
44. Finite Distinct Points and Least Containing Closed Interval
Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. Ordered points=[-18/2, 0/1, 19/3]. Give (increasing distinct points, minimum lo, maximum hi, hi−lo); the containing interval is [lo,hi], not the point set’s topological closure.
View answer and explanation
([-9/1, 0/1, 19/3], -9/1, 19/3, 46/3)
Sort and deduplicate by exact rational value, then take the extreme endpoints; the least containing closed interval differs from topological closure of a finite point set. The exact result, retaining the requested order, is ([-9/1, 0/1, 19/3], -9/1, 19/3, 46/3).
Reviewed step-by-step derivation
Real number · qf-density14-real-point-closed-hull · candidate
45. Finite Distinct Points and Least Containing Closed Interval
Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. Ordered points=[-19/2, 0/1, 20/3]. Give (increasing distinct points, minimum lo, maximum hi, hi−lo); the containing interval is [lo,hi], not the point set’s topological closure.
View answer and explanation
([-19/2, 0/1, 20/3], -19/2, 20/3, 97/6)
Sort and deduplicate by exact rational value, then take the extreme endpoints; the least containing closed interval differs from topological closure of a finite point set. The exact result, retaining the requested order, is ([-19/2, 0/1, 20/3], -19/2, 20/3, 97/6).
Reviewed step-by-step derivation
Real number · qf-density14-real-point-closed-hull · candidate
46. Finite Distinct Points and Least Containing Closed Interval
Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. Ordered points=[-20/2, 0/1, 21/3]. Give (increasing distinct points, minimum lo, maximum hi, hi−lo); the containing interval is [lo,hi], not the point set’s topological closure.
View answer and explanation
([-10/1, 0/1, 7/1], -10/1, 7/1, 17/1)
Sort and deduplicate by exact rational value, then take the extreme endpoints; the least containing closed interval differs from topological closure of a finite point set. The exact result, retaining the requested order, is ([-10/1, 0/1, 7/1], -10/1, 7/1, 17/1).
Reviewed step-by-step derivation
Real number · qf-density14-real-point-closed-hull · candidate
47. Finite Distinct Points and Least Containing Closed Interval
Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. Ordered points=[-21/2, 0/1, 22/3]. Give (increasing distinct points, minimum lo, maximum hi, hi−lo); the containing interval is [lo,hi], not the point set’s topological closure.
View answer and explanation
([-21/2, 0/1, 22/3], -21/2, 22/3, 107/6)
Sort and deduplicate by exact rational value, then take the extreme endpoints; the least containing closed interval differs from topological closure of a finite point set. The exact result, retaining the requested order, is ([-21/2, 0/1, 22/3], -21/2, 22/3, 107/6).
Reviewed step-by-step derivation
Real number · qf-density14-real-point-closed-hull · candidate
48. Finite Distinct Points and Least Containing Closed Interval
Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. Ordered points=[-22/2, 0/1, 23/3]. Give (increasing distinct points, minimum lo, maximum hi, hi−lo); the containing interval is [lo,hi], not the point set’s topological closure.
View answer and explanation
([-11/1, 0/1, 23/3], -11/1, 23/3, 56/3)
Sort and deduplicate by exact rational value, then take the extreme endpoints; the least containing closed interval differs from topological closure of a finite point set. The exact result, retaining the requested order, is ([-11/1, 0/1, 23/3], -11/1, 23/3, 56/3).
Reviewed step-by-step derivation
Real number · qf-density14-real-point-closed-hull · candidate
49. Finite Distinct Points and Least Containing Closed Interval
Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. Ordered points=[-23/2, 0/1, 24/3]. Give (increasing distinct points, minimum lo, maximum hi, hi−lo); the containing interval is [lo,hi], not the point set’s topological closure.
View answer and explanation
([-23/2, 0/1, 8/1], -23/2, 8/1, 39/2)
Sort and deduplicate by exact rational value, then take the extreme endpoints; the least containing closed interval differs from topological closure of a finite point set. The exact result, retaining the requested order, is ([-23/2, 0/1, 8/1], -23/2, 8/1, 39/2).
Reviewed step-by-step derivation
Real number · qf-density14-real-point-closed-hull · candidate
50. Finite Distinct Points and Least Containing Closed Interval
Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. Ordered points=[-24/2, 0/1, 25/3]. Give (increasing distinct points, minimum lo, maximum hi, hi−lo); the containing interval is [lo,hi], not the point set’s topological closure.
View answer and explanation
([-12/1, 0/1, 25/3], -12/1, 25/3, 61/3)
Sort and deduplicate by exact rational value, then take the extreme endpoints; the least containing closed interval differs from topological closure of a finite point set. The exact result, retaining the requested order, is ([-12/1, 0/1, 25/3], -12/1, 25/3, 61/3).
Reviewed step-by-step derivation
Real number · qf-density14-real-point-closed-hull · candidate
51. Closed Interval Intersection and Separation Gap
Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. I=[0/1, 1/1], J=[1/1, 2/1] have ordered inclusive endpoints. Give (is the intersection nonempty?, intersection [lo,hi] or [], separation gap). Use exact reduced fractions with positive denominators, retaining /1 for integers.
View answer and explanation
(Yes, [1/1, 1/1], 0/1)
Take the larger lower endpoint and smaller upper endpoint; intersecting intervals have gap 0/1, and touching at one included point is not an empty intersection. The exact result, retaining the requested order, is (Yes, [1/1, 1/1], 0/1).
Reviewed step-by-step derivation
Real number · qf-density14-real-overlap-closed-intervals · candidate
52. Closed Interval Intersection and Separation Gap
Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. I=[0/1, 1/1], J=[2/1, 3/1] have ordered inclusive endpoints. Give (is the intersection nonempty?, intersection [lo,hi] or [], separation gap). Use exact reduced fractions with positive denominators, retaining /1 for integers.
View answer and explanation
(No, [], 1/1)
Take the larger lower endpoint and smaller upper endpoint; intersecting intervals have gap 0/1, and touching at one included point is not an empty intersection. The exact result, retaining the requested order, is (No, [], 1/1).
Reviewed step-by-step derivation
Real number · qf-density14-real-overlap-closed-intervals · candidate
53. Closed Interval Intersection and Separation Gap
Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. I=[0/1, 0/1], J=[0/1, 1/1] have ordered inclusive endpoints. Give (is the intersection nonempty?, intersection [lo,hi] or [], separation gap). Use exact reduced fractions with positive denominators, retaining /1 for integers.
View answer and explanation
(Yes, [0/1, 0/1], 0/1)
Take the larger lower endpoint and smaller upper endpoint; intersecting intervals have gap 0/1, and touching at one included point is not an empty intersection. The exact result, retaining the requested order, is (Yes, [0/1, 0/1], 0/1).
Reviewed step-by-step derivation
Real number · qf-density14-real-overlap-closed-intervals · candidate
54. Closed Interval Intersection and Separation Gap
Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. I=[-3/2, 3/2], J=[0/1, 5/2] have ordered inclusive endpoints. Give (is the intersection nonempty?, intersection [lo,hi] or [], separation gap). Use exact reduced fractions with positive denominators, retaining /1 for integers.
View answer and explanation
(Yes, [0/1, 3/2], 0/1)
Take the larger lower endpoint and smaller upper endpoint; intersecting intervals have gap 0/1, and touching at one included point is not an empty intersection. The exact result, retaining the requested order, is (Yes, [0/1, 3/2], 0/1).
Reviewed step-by-step derivation
Real number · qf-density14-real-overlap-closed-intervals · candidate
55. Closed Interval Intersection and Separation Gap
Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. I=[-4/2, 4/2], J=[0/1, 6/2] have ordered inclusive endpoints. Give (is the intersection nonempty?, intersection [lo,hi] or [], separation gap). Use exact reduced fractions with positive denominators, retaining /1 for integers.
View answer and explanation
(Yes, [0/1, 2/1], 0/1)
Take the larger lower endpoint and smaller upper endpoint; intersecting intervals have gap 0/1, and touching at one included point is not an empty intersection. The exact result, retaining the requested order, is (Yes, [0/1, 2/1], 0/1).
Reviewed step-by-step derivation
Real number · qf-density14-real-overlap-closed-intervals · candidate
56. Closed Interval Intersection and Separation Gap
Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. I=[-5/2, 5/2], J=[0/1, 7/2] have ordered inclusive endpoints. Give (is the intersection nonempty?, intersection [lo,hi] or [], separation gap). Use exact reduced fractions with positive denominators, retaining /1 for integers.
View answer and explanation
(Yes, [0/1, 5/2], 0/1)
Take the larger lower endpoint and smaller upper endpoint; intersecting intervals have gap 0/1, and touching at one included point is not an empty intersection. The exact result, retaining the requested order, is (Yes, [0/1, 5/2], 0/1).
Reviewed step-by-step derivation
Real number · qf-density14-real-overlap-closed-intervals · candidate
57. Closed Interval Intersection and Separation Gap
Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. I=[-6/2, 6/2], J=[0/1, 8/2] have ordered inclusive endpoints. Give (is the intersection nonempty?, intersection [lo,hi] or [], separation gap). Use exact reduced fractions with positive denominators, retaining /1 for integers.
View answer and explanation
(Yes, [0/1, 3/1], 0/1)
Take the larger lower endpoint and smaller upper endpoint; intersecting intervals have gap 0/1, and touching at one included point is not an empty intersection. The exact result, retaining the requested order, is (Yes, [0/1, 3/1], 0/1).
Reviewed step-by-step derivation
Real number · qf-density14-real-overlap-closed-intervals · candidate
58. Closed Interval Intersection and Separation Gap
Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. I=[-7/2, 7/2], J=[0/1, 9/2] have ordered inclusive endpoints. Give (is the intersection nonempty?, intersection [lo,hi] or [], separation gap). Use exact reduced fractions with positive denominators, retaining /1 for integers.
View answer and explanation
(Yes, [0/1, 7/2], 0/1)
Take the larger lower endpoint and smaller upper endpoint; intersecting intervals have gap 0/1, and touching at one included point is not an empty intersection. The exact result, retaining the requested order, is (Yes, [0/1, 7/2], 0/1).
Reviewed step-by-step derivation
Real number · qf-density14-real-overlap-closed-intervals · candidate
59. Closed Interval Intersection and Separation Gap
Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. I=[-8/2, 8/2], J=[0/1, 10/2] have ordered inclusive endpoints. Give (is the intersection nonempty?, intersection [lo,hi] or [], separation gap). Use exact reduced fractions with positive denominators, retaining /1 for integers.
View answer and explanation
(Yes, [0/1, 4/1], 0/1)
Take the larger lower endpoint and smaller upper endpoint; intersecting intervals have gap 0/1, and touching at one included point is not an empty intersection. The exact result, retaining the requested order, is (Yes, [0/1, 4/1], 0/1).
Reviewed step-by-step derivation
Real number · qf-density14-real-overlap-closed-intervals · candidate
60. Closed Interval Intersection and Separation Gap
Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. I=[-9/2, 9/2], J=[0/1, 11/2] have ordered inclusive endpoints. Give (is the intersection nonempty?, intersection [lo,hi] or [], separation gap). Use exact reduced fractions with positive denominators, retaining /1 for integers.
View answer and explanation
(Yes, [0/1, 9/2], 0/1)
Take the larger lower endpoint and smaller upper endpoint; intersecting intervals have gap 0/1, and touching at one included point is not an empty intersection. The exact result, retaining the requested order, is (Yes, [0/1, 9/2], 0/1).
Reviewed step-by-step derivation
Real number · qf-density14-real-overlap-closed-intervals · candidate
61. Closed Interval Intersection and Separation Gap
Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. I=[-10/2, 10/2], J=[0/1, 12/2] have ordered inclusive endpoints. Give (is the intersection nonempty?, intersection [lo,hi] or [], separation gap). Use exact reduced fractions with positive denominators, retaining /1 for integers.
View answer and explanation
(Yes, [0/1, 5/1], 0/1)
Take the larger lower endpoint and smaller upper endpoint; intersecting intervals have gap 0/1, and touching at one included point is not an empty intersection. The exact result, retaining the requested order, is (Yes, [0/1, 5/1], 0/1).
Reviewed step-by-step derivation
Real number · qf-density14-real-overlap-closed-intervals · candidate
62. Closed Interval Intersection and Separation Gap
Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. I=[-11/2, 11/2], J=[0/1, 13/2] have ordered inclusive endpoints. Give (is the intersection nonempty?, intersection [lo,hi] or [], separation gap). Use exact reduced fractions with positive denominators, retaining /1 for integers.
View answer and explanation
(Yes, [0/1, 11/2], 0/1)
Take the larger lower endpoint and smaller upper endpoint; intersecting intervals have gap 0/1, and touching at one included point is not an empty intersection. The exact result, retaining the requested order, is (Yes, [0/1, 11/2], 0/1).
Reviewed step-by-step derivation
Real number · qf-density14-real-overlap-closed-intervals · candidate
63. Closed Interval Intersection and Separation Gap
Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. I=[-12/2, 12/2], J=[0/1, 14/2] have ordered inclusive endpoints. Give (is the intersection nonempty?, intersection [lo,hi] or [], separation gap). Use exact reduced fractions with positive denominators, retaining /1 for integers.
View answer and explanation
(Yes, [0/1, 6/1], 0/1)
Take the larger lower endpoint and smaller upper endpoint; intersecting intervals have gap 0/1, and touching at one included point is not an empty intersection. The exact result, retaining the requested order, is (Yes, [0/1, 6/1], 0/1).
Reviewed step-by-step derivation
Real number · qf-density14-real-overlap-closed-intervals · candidate
64. Closed Interval Intersection and Separation Gap
Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. I=[-13/2, 13/2], J=[0/1, 15/2] have ordered inclusive endpoints. Give (is the intersection nonempty?, intersection [lo,hi] or [], separation gap). Use exact reduced fractions with positive denominators, retaining /1 for integers.
View answer and explanation
(Yes, [0/1, 13/2], 0/1)
Take the larger lower endpoint and smaller upper endpoint; intersecting intervals have gap 0/1, and touching at one included point is not an empty intersection. The exact result, retaining the requested order, is (Yes, [0/1, 13/2], 0/1).
Reviewed step-by-step derivation
Real number · qf-density14-real-overlap-closed-intervals · candidate
65. Closed Interval Intersection and Separation Gap
Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. I=[-14/2, 14/2], J=[0/1, 16/2] have ordered inclusive endpoints. Give (is the intersection nonempty?, intersection [lo,hi] or [], separation gap). Use exact reduced fractions with positive denominators, retaining /1 for integers.
View answer and explanation
(Yes, [0/1, 7/1], 0/1)
Take the larger lower endpoint and smaller upper endpoint; intersecting intervals have gap 0/1, and touching at one included point is not an empty intersection. The exact result, retaining the requested order, is (Yes, [0/1, 7/1], 0/1).
Reviewed step-by-step derivation
Real number · qf-density14-real-overlap-closed-intervals · candidate
66. Closed Interval Intersection and Separation Gap
Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. I=[-15/2, 15/2], J=[0/1, 17/2] have ordered inclusive endpoints. Give (is the intersection nonempty?, intersection [lo,hi] or [], separation gap). Use exact reduced fractions with positive denominators, retaining /1 for integers.
View answer and explanation
(Yes, [0/1, 15/2], 0/1)
Take the larger lower endpoint and smaller upper endpoint; intersecting intervals have gap 0/1, and touching at one included point is not an empty intersection. The exact result, retaining the requested order, is (Yes, [0/1, 15/2], 0/1).
Reviewed step-by-step derivation
Real number · qf-density14-real-overlap-closed-intervals · candidate
67. Closed Interval Intersection and Separation Gap
Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. I=[-16/2, 16/2], J=[0/1, 18/2] have ordered inclusive endpoints. Give (is the intersection nonempty?, intersection [lo,hi] or [], separation gap). Use exact reduced fractions with positive denominators, retaining /1 for integers.
View answer and explanation
(Yes, [0/1, 8/1], 0/1)
Take the larger lower endpoint and smaller upper endpoint; intersecting intervals have gap 0/1, and touching at one included point is not an empty intersection. The exact result, retaining the requested order, is (Yes, [0/1, 8/1], 0/1).
Reviewed step-by-step derivation
Real number · qf-density14-real-overlap-closed-intervals · candidate
68. Closed Interval Intersection and Separation Gap
Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. I=[-17/2, 17/2], J=[0/1, 19/2] have ordered inclusive endpoints. Give (is the intersection nonempty?, intersection [lo,hi] or [], separation gap). Use exact reduced fractions with positive denominators, retaining /1 for integers.
View answer and explanation
(Yes, [0/1, 17/2], 0/1)
Take the larger lower endpoint and smaller upper endpoint; intersecting intervals have gap 0/1, and touching at one included point is not an empty intersection. The exact result, retaining the requested order, is (Yes, [0/1, 17/2], 0/1).
Reviewed step-by-step derivation
Real number · qf-density14-real-overlap-closed-intervals · candidate
69. Closed Interval Intersection and Separation Gap
Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. I=[-18/2, 18/2], J=[0/1, 20/2] have ordered inclusive endpoints. Give (is the intersection nonempty?, intersection [lo,hi] or [], separation gap). Use exact reduced fractions with positive denominators, retaining /1 for integers.
View answer and explanation
(Yes, [0/1, 9/1], 0/1)
Take the larger lower endpoint and smaller upper endpoint; intersecting intervals have gap 0/1, and touching at one included point is not an empty intersection. The exact result, retaining the requested order, is (Yes, [0/1, 9/1], 0/1).
Reviewed step-by-step derivation
Real number · qf-density14-real-overlap-closed-intervals · candidate
70. Closed Interval Intersection and Separation Gap
Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. I=[-19/2, 19/2], J=[0/1, 21/2] have ordered inclusive endpoints. Give (is the intersection nonempty?, intersection [lo,hi] or [], separation gap). Use exact reduced fractions with positive denominators, retaining /1 for integers.
View answer and explanation
(Yes, [0/1, 19/2], 0/1)
Take the larger lower endpoint and smaller upper endpoint; intersecting intervals have gap 0/1, and touching at one included point is not an empty intersection. The exact result, retaining the requested order, is (Yes, [0/1, 19/2], 0/1).
Reviewed step-by-step derivation
Real number · qf-density14-real-overlap-closed-intervals · candidate
71. Closed Interval Intersection and Separation Gap
Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. I=[-20/2, 20/2], J=[0/1, 22/2] have ordered inclusive endpoints. Give (is the intersection nonempty?, intersection [lo,hi] or [], separation gap). Use exact reduced fractions with positive denominators, retaining /1 for integers.
View answer and explanation
(Yes, [0/1, 10/1], 0/1)
Take the larger lower endpoint and smaller upper endpoint; intersecting intervals have gap 0/1, and touching at one included point is not an empty intersection. The exact result, retaining the requested order, is (Yes, [0/1, 10/1], 0/1).
Reviewed step-by-step derivation
Real number · qf-density14-real-overlap-closed-intervals · candidate
72. Closed Interval Intersection and Separation Gap
Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. I=[-21/2, 21/2], J=[0/1, 23/2] have ordered inclusive endpoints. Give (is the intersection nonempty?, intersection [lo,hi] or [], separation gap). Use exact reduced fractions with positive denominators, retaining /1 for integers.
View answer and explanation
(Yes, [0/1, 21/2], 0/1)
Take the larger lower endpoint and smaller upper endpoint; intersecting intervals have gap 0/1, and touching at one included point is not an empty intersection. The exact result, retaining the requested order, is (Yes, [0/1, 21/2], 0/1).
Reviewed step-by-step derivation
Real number · qf-density14-real-overlap-closed-intervals · candidate
73. Closed Interval Intersection and Separation Gap
Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. I=[-22/2, 22/2], J=[0/1, 24/2] have ordered inclusive endpoints. Give (is the intersection nonempty?, intersection [lo,hi] or [], separation gap). Use exact reduced fractions with positive denominators, retaining /1 for integers.
View answer and explanation
(Yes, [0/1, 11/1], 0/1)
Take the larger lower endpoint and smaller upper endpoint; intersecting intervals have gap 0/1, and touching at one included point is not an empty intersection. The exact result, retaining the requested order, is (Yes, [0/1, 11/1], 0/1).
Reviewed step-by-step derivation
Real number · qf-density14-real-overlap-closed-intervals · candidate
74. Closed Interval Intersection and Separation Gap
Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. I=[-23/2, 23/2], J=[0/1, 25/2] have ordered inclusive endpoints. Give (is the intersection nonempty?, intersection [lo,hi] or [], separation gap). Use exact reduced fractions with positive denominators, retaining /1 for integers.
View answer and explanation
(Yes, [0/1, 23/2], 0/1)
Take the larger lower endpoint and smaller upper endpoint; intersecting intervals have gap 0/1, and touching at one included point is not an empty intersection. The exact result, retaining the requested order, is (Yes, [0/1, 23/2], 0/1).
Reviewed step-by-step derivation
Real number · qf-density14-real-overlap-closed-intervals · candidate
75. Closed Interval Intersection and Separation Gap
Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. I=[-24/2, 24/2], J=[0/1, 26/2] have ordered inclusive endpoints. Give (is the intersection nonempty?, intersection [lo,hi] or [], separation gap). Use exact reduced fractions with positive denominators, retaining /1 for integers.
View answer and explanation
(Yes, [0/1, 12/1], 0/1)
Take the larger lower endpoint and smaller upper endpoint; intersecting intervals have gap 0/1, and touching at one included point is not an empty intersection. The exact result, retaining the requested order, is (Yes, [0/1, 12/1], 0/1).
Reviewed step-by-step derivation
Real number · qf-density14-real-overlap-closed-intervals · candidate
76. Ordered Rational Offsets and Absolute Distances
Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. origin=-12/2, ordered points=[-10/1, -13/2, -6/1, -16/3]. Give (each point−origin, each offset’s absolute value), retaining the four-field order.
View answer and explanation
([-4/1, -1/2, 0/1, 2/3], [4/1, 1/2, 0/1, 2/3])
Subtract exactly to retain direction, then take the nonnegative absolute value; directed offsets and distances are separate. The exact result, retaining the requested order, is ([-4/1, -1/2, 0/1, 2/3], [4/1, 1/2, 0/1, 2/3]).
Reviewed step-by-step derivation
Real number · qf-density14-real-point-offset-distances · candidate
77. Ordered Rational Offsets and Absolute Distances
Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. origin=-11/2, ordered points=[-55/6, -6/1, -11/2, -29/6]. Give (each point−origin, each offset’s absolute value), retaining the four-field order.
View answer and explanation
([-11/3, -1/2, 0/1, 2/3], [11/3, 1/2, 0/1, 2/3])
Subtract exactly to retain direction, then take the nonnegative absolute value; directed offsets and distances are separate. The exact result, retaining the requested order, is ([-11/3, -1/2, 0/1, 2/3], [11/3, 1/2, 0/1, 2/3]).
Reviewed step-by-step derivation
Real number · qf-density14-real-point-offset-distances · candidate
78. Ordered Rational Offsets and Absolute Distances
Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. origin=-10/2, ordered points=[-25/3, -11/2, -5/1, -13/3]. Give (each point−origin, each offset’s absolute value), retaining the four-field order.
View answer and explanation
([-10/3, -1/2, 0/1, 2/3], [10/3, 1/2, 0/1, 2/3])
Subtract exactly to retain direction, then take the nonnegative absolute value; directed offsets and distances are separate. The exact result, retaining the requested order, is ([-10/3, -1/2, 0/1, 2/3], [10/3, 1/2, 0/1, 2/3]).
Reviewed step-by-step derivation
Real number · qf-density14-real-point-offset-distances · candidate
79. Ordered Rational Offsets and Absolute Distances
Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. origin=-9/2, ordered points=[-15/2, -5/1, -9/2, -23/6]. Give (each point−origin, each offset’s absolute value), retaining the four-field order.
View answer and explanation
([-3/1, -1/2, 0/1, 2/3], [3/1, 1/2, 0/1, 2/3])
Subtract exactly to retain direction, then take the nonnegative absolute value; directed offsets and distances are separate. The exact result, retaining the requested order, is ([-3/1, -1/2, 0/1, 2/3], [3/1, 1/2, 0/1, 2/3]).
Reviewed step-by-step derivation
Real number · qf-density14-real-point-offset-distances · candidate
80. Ordered Rational Offsets and Absolute Distances
Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. origin=-8/2, ordered points=[-20/3, -9/2, -4/1, -10/3]. Give (each point−origin, each offset’s absolute value), retaining the four-field order.
View answer and explanation
([-8/3, -1/2, 0/1, 2/3], [8/3, 1/2, 0/1, 2/3])
Subtract exactly to retain direction, then take the nonnegative absolute value; directed offsets and distances are separate. The exact result, retaining the requested order, is ([-8/3, -1/2, 0/1, 2/3], [8/3, 1/2, 0/1, 2/3]).
Reviewed step-by-step derivation
Real number · qf-density14-real-point-offset-distances · candidate
81. Ordered Rational Offsets and Absolute Distances
Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. origin=-7/2, ordered points=[-35/6, -4/1, -7/2, -17/6]. Give (each point−origin, each offset’s absolute value), retaining the four-field order.
View answer and explanation
([-7/3, -1/2, 0/1, 2/3], [7/3, 1/2, 0/1, 2/3])
Subtract exactly to retain direction, then take the nonnegative absolute value; directed offsets and distances are separate. The exact result, retaining the requested order, is ([-7/3, -1/2, 0/1, 2/3], [7/3, 1/2, 0/1, 2/3]).
Reviewed step-by-step derivation
Real number · qf-density14-real-point-offset-distances · candidate
82. Ordered Rational Offsets and Absolute Distances
Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. origin=-6/2, ordered points=[-5/1, -7/2, -3/1, -7/3]. Give (each point−origin, each offset’s absolute value), retaining the four-field order.
View answer and explanation
([-2/1, -1/2, 0/1, 2/3], [2/1, 1/2, 0/1, 2/3])
Subtract exactly to retain direction, then take the nonnegative absolute value; directed offsets and distances are separate. The exact result, retaining the requested order, is ([-2/1, -1/2, 0/1, 2/3], [2/1, 1/2, 0/1, 2/3]).
Reviewed step-by-step derivation
Real number · qf-density14-real-point-offset-distances · candidate
83. Ordered Rational Offsets and Absolute Distances
Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. origin=-5/2, ordered points=[-25/6, -3/1, -5/2, -11/6]. Give (each point−origin, each offset’s absolute value), retaining the four-field order.
View answer and explanation
([-5/3, -1/2, 0/1, 2/3], [5/3, 1/2, 0/1, 2/3])
Subtract exactly to retain direction, then take the nonnegative absolute value; directed offsets and distances are separate. The exact result, retaining the requested order, is ([-5/3, -1/2, 0/1, 2/3], [5/3, 1/2, 0/1, 2/3]).
Reviewed step-by-step derivation
Real number · qf-density14-real-point-offset-distances · candidate
84. Ordered Rational Offsets and Absolute Distances
Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. origin=-4/2, ordered points=[-10/3, -5/2, -2/1, -4/3]. Give (each point−origin, each offset’s absolute value), retaining the four-field order.
View answer and explanation
([-4/3, -1/2, 0/1, 2/3], [4/3, 1/2, 0/1, 2/3])
Subtract exactly to retain direction, then take the nonnegative absolute value; directed offsets and distances are separate. The exact result, retaining the requested order, is ([-4/3, -1/2, 0/1, 2/3], [4/3, 1/2, 0/1, 2/3]).
Reviewed step-by-step derivation
Real number · qf-density14-real-point-offset-distances · candidate
85. Ordered Rational Offsets and Absolute Distances
Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. origin=-3/2, ordered points=[-5/2, -2/1, -3/2, -5/6]. Give (each point−origin, each offset’s absolute value), retaining the four-field order.
View answer and explanation
([-1/1, -1/2, 0/1, 2/3], [1/1, 1/2, 0/1, 2/3])
Subtract exactly to retain direction, then take the nonnegative absolute value; directed offsets and distances are separate. The exact result, retaining the requested order, is ([-1/1, -1/2, 0/1, 2/3], [1/1, 1/2, 0/1, 2/3]).
Reviewed step-by-step derivation
Real number · qf-density14-real-point-offset-distances · candidate
86. Ordered Rational Offsets and Absolute Distances
Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. origin=-2/2, ordered points=[-5/3, -3/2, -1/1, -1/3]. Give (each point−origin, each offset’s absolute value), retaining the four-field order.
View answer and explanation
([-2/3, -1/2, 0/1, 2/3], [2/3, 1/2, 0/1, 2/3])
Subtract exactly to retain direction, then take the nonnegative absolute value; directed offsets and distances are separate. The exact result, retaining the requested order, is ([-2/3, -1/2, 0/1, 2/3], [2/3, 1/2, 0/1, 2/3]).
Reviewed step-by-step derivation
Real number · qf-density14-real-point-offset-distances · candidate
87. Ordered Rational Offsets and Absolute Distances
Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. origin=-1/2, ordered points=[-5/6, -1/1, -1/2, 1/6]. Give (each point−origin, each offset’s absolute value), retaining the four-field order.
View answer and explanation
([-1/3, -1/2, 0/1, 2/3], [1/3, 1/2, 0/1, 2/3])
Subtract exactly to retain direction, then take the nonnegative absolute value; directed offsets and distances are separate. The exact result, retaining the requested order, is ([-1/3, -1/2, 0/1, 2/3], [1/3, 1/2, 0/1, 2/3]).
Reviewed step-by-step derivation
Real number · qf-density14-real-point-offset-distances · candidate
88. Ordered Rational Offsets and Absolute Distances
Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. origin=0/2, ordered points=[0/1, -1/2, 0/1, 2/3]. Give (each point−origin, each offset’s absolute value), retaining the four-field order.
View answer and explanation
([0/1, -1/2, 0/1, 2/3], [0/1, 1/2, 0/1, 2/3])
Subtract exactly to retain direction, then take the nonnegative absolute value; directed offsets and distances are separate. The exact result, retaining the requested order, is ([0/1, -1/2, 0/1, 2/3], [0/1, 1/2, 0/1, 2/3]).
Reviewed step-by-step derivation
Real number · qf-density14-real-point-offset-distances · candidate
89. Ordered Rational Offsets and Absolute Distances
Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. origin=1/2, ordered points=[5/6, 0/1, 1/2, 7/6]. Give (each point−origin, each offset’s absolute value), retaining the four-field order.
View answer and explanation
([1/3, -1/2, 0/1, 2/3], [1/3, 1/2, 0/1, 2/3])
Subtract exactly to retain direction, then take the nonnegative absolute value; directed offsets and distances are separate. The exact result, retaining the requested order, is ([1/3, -1/2, 0/1, 2/3], [1/3, 1/2, 0/1, 2/3]).
Reviewed step-by-step derivation
Real number · qf-density14-real-point-offset-distances · candidate
90. Ordered Rational Offsets and Absolute Distances
Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. origin=2/2, ordered points=[5/3, 1/2, 1/1, 5/3]. Give (each point−origin, each offset’s absolute value), retaining the four-field order.
View answer and explanation
([2/3, -1/2, 0/1, 2/3], [2/3, 1/2, 0/1, 2/3])
Subtract exactly to retain direction, then take the nonnegative absolute value; directed offsets and distances are separate. The exact result, retaining the requested order, is ([2/3, -1/2, 0/1, 2/3], [2/3, 1/2, 0/1, 2/3]).
Reviewed step-by-step derivation
Real number · qf-density14-real-point-offset-distances · candidate
91. Ordered Rational Offsets and Absolute Distances
Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. origin=3/2, ordered points=[5/2, 1/1, 3/2, 13/6]. Give (each point−origin, each offset’s absolute value), retaining the four-field order.
View answer and explanation
([1/1, -1/2, 0/1, 2/3], [1/1, 1/2, 0/1, 2/3])
Subtract exactly to retain direction, then take the nonnegative absolute value; directed offsets and distances are separate. The exact result, retaining the requested order, is ([1/1, -1/2, 0/1, 2/3], [1/1, 1/2, 0/1, 2/3]).
Reviewed step-by-step derivation
Real number · qf-density14-real-point-offset-distances · candidate
92. Ordered Rational Offsets and Absolute Distances
Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. origin=4/2, ordered points=[10/3, 3/2, 2/1, 8/3]. Give (each point−origin, each offset’s absolute value), retaining the four-field order.
View answer and explanation
([4/3, -1/2, 0/1, 2/3], [4/3, 1/2, 0/1, 2/3])
Subtract exactly to retain direction, then take the nonnegative absolute value; directed offsets and distances are separate. The exact result, retaining the requested order, is ([4/3, -1/2, 0/1, 2/3], [4/3, 1/2, 0/1, 2/3]).
Reviewed step-by-step derivation
Real number · qf-density14-real-point-offset-distances · candidate
93. Ordered Rational Offsets and Absolute Distances
Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. origin=5/2, ordered points=[25/6, 2/1, 5/2, 19/6]. Give (each point−origin, each offset’s absolute value), retaining the four-field order.
View answer and explanation
([5/3, -1/2, 0/1, 2/3], [5/3, 1/2, 0/1, 2/3])
Subtract exactly to retain direction, then take the nonnegative absolute value; directed offsets and distances are separate. The exact result, retaining the requested order, is ([5/3, -1/2, 0/1, 2/3], [5/3, 1/2, 0/1, 2/3]).
Reviewed step-by-step derivation
Real number · qf-density14-real-point-offset-distances · candidate
94. Ordered Rational Offsets and Absolute Distances
Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. origin=6/2, ordered points=[5/1, 5/2, 3/1, 11/3]. Give (each point−origin, each offset’s absolute value), retaining the four-field order.
View answer and explanation
([2/1, -1/2, 0/1, 2/3], [2/1, 1/2, 0/1, 2/3])
Subtract exactly to retain direction, then take the nonnegative absolute value; directed offsets and distances are separate. The exact result, retaining the requested order, is ([2/1, -1/2, 0/1, 2/3], [2/1, 1/2, 0/1, 2/3]).
Reviewed step-by-step derivation
Real number · qf-density14-real-point-offset-distances · candidate
95. Ordered Rational Offsets and Absolute Distances
Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. origin=7/2, ordered points=[35/6, 3/1, 7/2, 25/6]. Give (each point−origin, each offset’s absolute value), retaining the four-field order.
View answer and explanation
([7/3, -1/2, 0/1, 2/3], [7/3, 1/2, 0/1, 2/3])
Subtract exactly to retain direction, then take the nonnegative absolute value; directed offsets and distances are separate. The exact result, retaining the requested order, is ([7/3, -1/2, 0/1, 2/3], [7/3, 1/2, 0/1, 2/3]).
Reviewed step-by-step derivation
Real number · qf-density14-real-point-offset-distances · candidate
96. Ordered Rational Offsets and Absolute Distances
Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. origin=8/2, ordered points=[20/3, 7/2, 4/1, 14/3]. Give (each point−origin, each offset’s absolute value), retaining the four-field order.
View answer and explanation
([8/3, -1/2, 0/1, 2/3], [8/3, 1/2, 0/1, 2/3])
Subtract exactly to retain direction, then take the nonnegative absolute value; directed offsets and distances are separate. The exact result, retaining the requested order, is ([8/3, -1/2, 0/1, 2/3], [8/3, 1/2, 0/1, 2/3]).
Reviewed step-by-step derivation
Real number · qf-density14-real-point-offset-distances · candidate
97. Ordered Rational Offsets and Absolute Distances
Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. origin=9/2, ordered points=[15/2, 4/1, 9/2, 31/6]. Give (each point−origin, each offset’s absolute value), retaining the four-field order.
View answer and explanation
([3/1, -1/2, 0/1, 2/3], [3/1, 1/2, 0/1, 2/3])
Subtract exactly to retain direction, then take the nonnegative absolute value; directed offsets and distances are separate. The exact result, retaining the requested order, is ([3/1, -1/2, 0/1, 2/3], [3/1, 1/2, 0/1, 2/3]).
Reviewed step-by-step derivation
Real number · qf-density14-real-point-offset-distances · candidate
98. Ordered Rational Offsets and Absolute Distances
Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. origin=10/2, ordered points=[25/3, 9/2, 5/1, 17/3]. Give (each point−origin, each offset’s absolute value), retaining the four-field order.
View answer and explanation
([10/3, -1/2, 0/1, 2/3], [10/3, 1/2, 0/1, 2/3])
Subtract exactly to retain direction, then take the nonnegative absolute value; directed offsets and distances are separate. The exact result, retaining the requested order, is ([10/3, -1/2, 0/1, 2/3], [10/3, 1/2, 0/1, 2/3]).
Reviewed step-by-step derivation
Real number · qf-density14-real-point-offset-distances · candidate
99. Ordered Rational Offsets and Absolute Distances
Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. origin=11/2, ordered points=[55/6, 5/1, 11/2, 37/6]. Give (each point−origin, each offset’s absolute value), retaining the four-field order.
View answer and explanation
([11/3, -1/2, 0/1, 2/3], [11/3, 1/2, 0/1, 2/3])
Subtract exactly to retain direction, then take the nonnegative absolute value; directed offsets and distances are separate. The exact result, retaining the requested order, is ([11/3, -1/2, 0/1, 2/3], [11/3, 1/2, 0/1, 2/3]).
Reviewed step-by-step derivation
Real number · qf-density14-real-point-offset-distances · candidate
100. Ordered Rational Offsets and Absolute Distances
Use an exact finite rational-point slice embedded in R, without approximate decimals or a completeness claim. origin=12/2, ordered points=[10/1, 11/2, 6/1, 20/3]. Give (each point−origin, each offset’s absolute value), retaining the four-field order.
View answer and explanation
([4/1, -1/2, 0/1, 2/3], [4/1, 1/2, 0/1, 2/3])
Subtract exactly to retain direction, then take the nonnegative absolute value; directed offsets and distances are separate. The exact result, retaining the requested order, is ([4/1, -1/2, 0/1, 2/3], [4/1, 1/2, 0/1, 2/3]).
Reviewed step-by-step derivation
Real number · qf-density14-real-point-offset-distances · candidate
Canonical question JSONL · English JSONL · Validation report