This batch has passed Schema, deduplication, source-snapshot, and independent-answer recomputation checks; it is still not marked as human-reviewed.
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-20261008-advance-20261011-r01-natural-number-foundations · 2026-10-08 · 100 questions
This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.
N includes 0. n=0, k=0. List the complete ordered window beginning at n and applying the successor k times.
[0]
Start with n, then add one k times; there are k+1 terms, so k=0 still includes the start. The exact result, retaining the requested order, is [0].
Reviewed step-by-step derivation
Natural number · qf-density11-natural-successor-window · candidate
N includes 0. n=0, k=8. List the complete ordered window beginning at n and applying the successor k times.
[0, 1, 2, 3, 4, 5, 6, 7, 8]
Start with n, then add one k times; there are k+1 terms, so k=0 still includes the start. The exact result, retaining the requested order, is [0, 1, 2, 3, 4, 5, 6, 7, 8].
Reviewed step-by-step derivation
Natural number · qf-density11-natural-successor-window · candidate
N includes 0. n=1, k=2. List the complete ordered window beginning at n and applying the successor k times.
[1, 2, 3]
Start with n, then add one k times; there are k+1 terms, so k=0 still includes the start. The exact result, retaining the requested order, is [1, 2, 3].
Reviewed step-by-step derivation
Natural number · qf-density11-natural-successor-window · candidate
N includes 0. n=2, k=3. List the complete ordered window beginning at n and applying the successor k times.
[2, 3, 4, 5]
Start with n, then add one k times; there are k+1 terms, so k=0 still includes the start. The exact result, retaining the requested order, is [2, 3, 4, 5].
Reviewed step-by-step derivation
Natural number · qf-density11-natural-successor-window · candidate
N includes 0. n=3, k=4. List the complete ordered window beginning at n and applying the successor k times.
[3, 4, 5, 6, 7]
Start with n, then add one k times; there are k+1 terms, so k=0 still includes the start. The exact result, retaining the requested order, is [3, 4, 5, 6, 7].
Reviewed step-by-step derivation
Natural number · qf-density11-natural-successor-window · candidate
N includes 0. n=4, k=5. List the complete ordered window beginning at n and applying the successor k times.
[4, 5, 6, 7, 8, 9]
Start with n, then add one k times; there are k+1 terms, so k=0 still includes the start. The exact result, retaining the requested order, is [4, 5, 6, 7, 8, 9].
Reviewed step-by-step derivation
Natural number · qf-density11-natural-successor-window · candidate
N includes 0. n=5, k=6. List the complete ordered window beginning at n and applying the successor k times.
[5, 6, 7, 8, 9, 10, 11]
Start with n, then add one k times; there are k+1 terms, so k=0 still includes the start. The exact result, retaining the requested order, is [5, 6, 7, 8, 9, 10, 11].
Reviewed step-by-step derivation
Natural number · qf-density11-natural-successor-window · candidate
N includes 0. n=6, k=7. List the complete ordered window beginning at n and applying the successor k times.
[6, 7, 8, 9, 10, 11, 12, 13]
Start with n, then add one k times; there are k+1 terms, so k=0 still includes the start. The exact result, retaining the requested order, is [6, 7, 8, 9, 10, 11, 12, 13].
Reviewed step-by-step derivation
Natural number · qf-density11-natural-successor-window · candidate
N includes 0. n=7, k=8. List the complete ordered window beginning at n and applying the successor k times.
[7, 8, 9, 10, 11, 12, 13, 14, 15]
Start with n, then add one k times; there are k+1 terms, so k=0 still includes the start. The exact result, retaining the requested order, is [7, 8, 9, 10, 11, 12, 13, 14, 15].
Reviewed step-by-step derivation
Natural number · qf-density11-natural-successor-window · candidate
N includes 0. n=8, k=0. List the complete ordered window beginning at n and applying the successor k times.
[8]
Start with n, then add one k times; there are k+1 terms, so k=0 still includes the start. The exact result, retaining the requested order, is [8].
Reviewed step-by-step derivation
Natural number · qf-density11-natural-successor-window · candidate
N includes 0. n=9, k=1. List the complete ordered window beginning at n and applying the successor k times.
[9, 10]
Start with n, then add one k times; there are k+1 terms, so k=0 still includes the start. The exact result, retaining the requested order, is [9, 10].
Reviewed step-by-step derivation
Natural number · qf-density11-natural-successor-window · candidate
N includes 0. n=10, k=2. List the complete ordered window beginning at n and applying the successor k times.
[10, 11, 12]
Start with n, then add one k times; there are k+1 terms, so k=0 still includes the start. The exact result, retaining the requested order, is [10, 11, 12].
Reviewed step-by-step derivation
Natural number · qf-density11-natural-successor-window · candidate
N includes 0. n=11, k=3. List the complete ordered window beginning at n and applying the successor k times.
[11, 12, 13, 14]
Start with n, then add one k times; there are k+1 terms, so k=0 still includes the start. The exact result, retaining the requested order, is [11, 12, 13, 14].
Reviewed step-by-step derivation
Natural number · qf-density11-natural-successor-window · candidate
N includes 0. n=12, k=4. List the complete ordered window beginning at n and applying the successor k times.
[12, 13, 14, 15, 16]
Start with n, then add one k times; there are k+1 terms, so k=0 still includes the start. The exact result, retaining the requested order, is [12, 13, 14, 15, 16].
Reviewed step-by-step derivation
Natural number · qf-density11-natural-successor-window · candidate
N includes 0. n=13, k=5. List the complete ordered window beginning at n and applying the successor k times.
[13, 14, 15, 16, 17, 18]
Start with n, then add one k times; there are k+1 terms, so k=0 still includes the start. The exact result, retaining the requested order, is [13, 14, 15, 16, 17, 18].
Reviewed step-by-step derivation
Natural number · qf-density11-natural-successor-window · candidate
N includes 0. n=14, k=6. List the complete ordered window beginning at n and applying the successor k times.
[14, 15, 16, 17, 18, 19, 20]
Start with n, then add one k times; there are k+1 terms, so k=0 still includes the start. The exact result, retaining the requested order, is [14, 15, 16, 17, 18, 19, 20].
Reviewed step-by-step derivation
Natural number · qf-density11-natural-successor-window · candidate
N includes 0. n=15, k=7. List the complete ordered window beginning at n and applying the successor k times.
[15, 16, 17, 18, 19, 20, 21, 22]
Start with n, then add one k times; there are k+1 terms, so k=0 still includes the start. The exact result, retaining the requested order, is [15, 16, 17, 18, 19, 20, 21, 22].
Reviewed step-by-step derivation
Natural number · qf-density11-natural-successor-window · candidate
N includes 0. n=16, k=8. List the complete ordered window beginning at n and applying the successor k times.
[16, 17, 18, 19, 20, 21, 22, 23, 24]
Start with n, then add one k times; there are k+1 terms, so k=0 still includes the start. The exact result, retaining the requested order, is [16, 17, 18, 19, 20, 21, 22, 23, 24].
Reviewed step-by-step derivation
Natural number · qf-density11-natural-successor-window · candidate
N includes 0. n=17, k=0. List the complete ordered window beginning at n and applying the successor k times.
[17]
Start with n, then add one k times; there are k+1 terms, so k=0 still includes the start. The exact result, retaining the requested order, is [17].
Reviewed step-by-step derivation
Natural number · qf-density11-natural-successor-window · candidate
N includes 0. n=18, k=1. List the complete ordered window beginning at n and applying the successor k times.
[18, 19]
Start with n, then add one k times; there are k+1 terms, so k=0 still includes the start. The exact result, retaining the requested order, is [18, 19].
Reviewed step-by-step derivation
Natural number · qf-density11-natural-successor-window · candidate
N includes 0. n=19, k=2. List the complete ordered window beginning at n and applying the successor k times.
[19, 20, 21]
Start with n, then add one k times; there are k+1 terms, so k=0 still includes the start. The exact result, retaining the requested order, is [19, 20, 21].
Reviewed step-by-step derivation
Natural number · qf-density11-natural-successor-window · candidate
N includes 0. n=20, k=3. List the complete ordered window beginning at n and applying the successor k times.
[20, 21, 22, 23]
Start with n, then add one k times; there are k+1 terms, so k=0 still includes the start. The exact result, retaining the requested order, is [20, 21, 22, 23].
Reviewed step-by-step derivation
Natural number · qf-density11-natural-successor-window · candidate
N includes 0. n=21, k=4. List the complete ordered window beginning at n and applying the successor k times.
[21, 22, 23, 24, 25]
Start with n, then add one k times; there are k+1 terms, so k=0 still includes the start. The exact result, retaining the requested order, is [21, 22, 23, 24, 25].
Reviewed step-by-step derivation
Natural number · qf-density11-natural-successor-window · candidate
N includes 0. n=22, k=5. List the complete ordered window beginning at n and applying the successor k times.
[22, 23, 24, 25, 26, 27]
Start with n, then add one k times; there are k+1 terms, so k=0 still includes the start. The exact result, retaining the requested order, is [22, 23, 24, 25, 26, 27].
Reviewed step-by-step derivation
Natural number · qf-density11-natural-successor-window · candidate
N includes 0. n=23, k=6. List the complete ordered window beginning at n and applying the successor k times.
[23, 24, 25, 26, 27, 28, 29]
Start with n, then add one k times; there are k+1 terms, so k=0 still includes the start. The exact result, retaining the requested order, is [23, 24, 25, 26, 27, 28, 29].
Reviewed step-by-step derivation
Natural number · qf-density11-natural-successor-window · candidate
N includes 0. The ordered values are [0, 0]. Give each predecessor in N, or none if it does not exist, preserving the two fields.
(none, none)
Subtract one only from a positive natural number; zero has no predecessor here, rather than a truncated predecessor equal to zero. The exact result, retaining the requested order, is (none, none).
Reviewed step-by-step derivation
Natural number · qf-density11-natural-predecessor-pair · candidate
N includes 0. The ordered values are [0, 1]. Give each predecessor in N, or none if it does not exist, preserving the two fields.
(none, 0)
Subtract one only from a positive natural number; zero has no predecessor here, rather than a truncated predecessor equal to zero. The exact result, retaining the requested order, is (none, 0).
Reviewed step-by-step derivation
Natural number · qf-density11-natural-predecessor-pair · candidate
N includes 0. The ordered values are [2, 2]. Give each predecessor in N, or none if it does not exist, preserving the two fields.
(1, 1)
Subtract one only from a positive natural number; zero has no predecessor here, rather than a truncated predecessor equal to zero. The exact result, retaining the requested order, is (1, 1).
Reviewed step-by-step derivation
Natural number · qf-density11-natural-predecessor-pair · candidate
N includes 0. The ordered values are [3, 3]. Give each predecessor in N, or none if it does not exist, preserving the two fields.
(2, 2)
Subtract one only from a positive natural number; zero has no predecessor here, rather than a truncated predecessor equal to zero. The exact result, retaining the requested order, is (2, 2).
Reviewed step-by-step derivation
Natural number · qf-density11-natural-predecessor-pair · candidate
N includes 0. The ordered values are [4, 4]. Give each predecessor in N, or none if it does not exist, preserving the two fields.
(3, 3)
Subtract one only from a positive natural number; zero has no predecessor here, rather than a truncated predecessor equal to zero. The exact result, retaining the requested order, is (3, 3).
Reviewed step-by-step derivation
Natural number · qf-density11-natural-predecessor-pair · candidate
N includes 0. The ordered values are [5, 0]. Give each predecessor in N, or none if it does not exist, preserving the two fields.
(4, none)
Subtract one only from a positive natural number; zero has no predecessor here, rather than a truncated predecessor equal to zero. The exact result, retaining the requested order, is (4, none).
Reviewed step-by-step derivation
Natural number · qf-density11-natural-predecessor-pair · candidate
N includes 0. The ordered values are [6, 1]. Give each predecessor in N, or none if it does not exist, preserving the two fields.
(5, 0)
Subtract one only from a positive natural number; zero has no predecessor here, rather than a truncated predecessor equal to zero. The exact result, retaining the requested order, is (5, 0).
Reviewed step-by-step derivation
Natural number · qf-density11-natural-predecessor-pair · candidate
N includes 0. The ordered values are [7, 2]. Give each predecessor in N, or none if it does not exist, preserving the two fields.
(6, 1)
Subtract one only from a positive natural number; zero has no predecessor here, rather than a truncated predecessor equal to zero. The exact result, retaining the requested order, is (6, 1).
Reviewed step-by-step derivation
Natural number · qf-density11-natural-predecessor-pair · candidate
N includes 0. The ordered values are [8, 3]. Give each predecessor in N, or none if it does not exist, preserving the two fields.
(7, 2)
Subtract one only from a positive natural number; zero has no predecessor here, rather than a truncated predecessor equal to zero. The exact result, retaining the requested order, is (7, 2).
Reviewed step-by-step derivation
Natural number · qf-density11-natural-predecessor-pair · candidate
N includes 0. The ordered values are [9, 4]. Give each predecessor in N, or none if it does not exist, preserving the two fields.
(8, 3)
Subtract one only from a positive natural number; zero has no predecessor here, rather than a truncated predecessor equal to zero. The exact result, retaining the requested order, is (8, 3).
Reviewed step-by-step derivation
Natural number · qf-density11-natural-predecessor-pair · candidate
N includes 0. The ordered values are [10, 0]. Give each predecessor in N, or none if it does not exist, preserving the two fields.
(9, none)
Subtract one only from a positive natural number; zero has no predecessor here, rather than a truncated predecessor equal to zero. The exact result, retaining the requested order, is (9, none).
Reviewed step-by-step derivation
Natural number · qf-density11-natural-predecessor-pair · candidate
N includes 0. The ordered values are [11, 1]. Give each predecessor in N, or none if it does not exist, preserving the two fields.
(10, 0)
Subtract one only from a positive natural number; zero has no predecessor here, rather than a truncated predecessor equal to zero. The exact result, retaining the requested order, is (10, 0).
Reviewed step-by-step derivation
Natural number · qf-density11-natural-predecessor-pair · candidate
N includes 0. The ordered values are [12, 2]. Give each predecessor in N, or none if it does not exist, preserving the two fields.
(11, 1)
Subtract one only from a positive natural number; zero has no predecessor here, rather than a truncated predecessor equal to zero. The exact result, retaining the requested order, is (11, 1).
Reviewed step-by-step derivation
Natural number · qf-density11-natural-predecessor-pair · candidate
N includes 0. The ordered values are [13, 3]. Give each predecessor in N, or none if it does not exist, preserving the two fields.
(12, 2)
Subtract one only from a positive natural number; zero has no predecessor here, rather than a truncated predecessor equal to zero. The exact result, retaining the requested order, is (12, 2).
Reviewed step-by-step derivation
Natural number · qf-density11-natural-predecessor-pair · candidate
N includes 0. The ordered values are [14, 4]. Give each predecessor in N, or none if it does not exist, preserving the two fields.
(13, 3)
Subtract one only from a positive natural number; zero has no predecessor here, rather than a truncated predecessor equal to zero. The exact result, retaining the requested order, is (13, 3).
Reviewed step-by-step derivation
Natural number · qf-density11-natural-predecessor-pair · candidate
N includes 0. The ordered values are [15, 0]. Give each predecessor in N, or none if it does not exist, preserving the two fields.
(14, none)
Subtract one only from a positive natural number; zero has no predecessor here, rather than a truncated predecessor equal to zero. The exact result, retaining the requested order, is (14, none).
Reviewed step-by-step derivation
Natural number · qf-density11-natural-predecessor-pair · candidate
N includes 0. The ordered values are [16, 1]. Give each predecessor in N, or none if it does not exist, preserving the two fields.
(15, 0)
Subtract one only from a positive natural number; zero has no predecessor here, rather than a truncated predecessor equal to zero. The exact result, retaining the requested order, is (15, 0).
Reviewed step-by-step derivation
Natural number · qf-density11-natural-predecessor-pair · candidate
N includes 0. The ordered values are [17, 2]. Give each predecessor in N, or none if it does not exist, preserving the two fields.
(16, 1)
Subtract one only from a positive natural number; zero has no predecessor here, rather than a truncated predecessor equal to zero. The exact result, retaining the requested order, is (16, 1).
Reviewed step-by-step derivation
Natural number · qf-density11-natural-predecessor-pair · candidate
N includes 0. The ordered values are [18, 3]. Give each predecessor in N, or none if it does not exist, preserving the two fields.
(17, 2)
Subtract one only from a positive natural number; zero has no predecessor here, rather than a truncated predecessor equal to zero. The exact result, retaining the requested order, is (17, 2).
Reviewed step-by-step derivation
Natural number · qf-density11-natural-predecessor-pair · candidate
N includes 0. The ordered values are [19, 4]. Give each predecessor in N, or none if it does not exist, preserving the two fields.
(18, 3)
Subtract one only from a positive natural number; zero has no predecessor here, rather than a truncated predecessor equal to zero. The exact result, retaining the requested order, is (18, 3).
Reviewed step-by-step derivation
Natural number · qf-density11-natural-predecessor-pair · candidate
N includes 0. The ordered values are [20, 0]. Give each predecessor in N, or none if it does not exist, preserving the two fields.
(19, none)
Subtract one only from a positive natural number; zero has no predecessor here, rather than a truncated predecessor equal to zero. The exact result, retaining the requested order, is (19, none).
Reviewed step-by-step derivation
Natural number · qf-density11-natural-predecessor-pair · candidate
N includes 0. The ordered values are [21, 1]. Give each predecessor in N, or none if it does not exist, preserving the two fields.
(20, 0)
Subtract one only from a positive natural number; zero has no predecessor here, rather than a truncated predecessor equal to zero. The exact result, retaining the requested order, is (20, 0).
Reviewed step-by-step derivation
Natural number · qf-density11-natural-predecessor-pair · candidate
N includes 0. The ordered values are [22, 2]. Give each predecessor in N, or none if it does not exist, preserving the two fields.
(21, 1)
Subtract one only from a positive natural number; zero has no predecessor here, rather than a truncated predecessor equal to zero. The exact result, retaining the requested order, is (21, 1).
Reviewed step-by-step derivation
Natural number · qf-density11-natural-predecessor-pair · candidate
N includes 0. The ordered values are [23, 3]. Give each predecessor in N, or none if it does not exist, preserving the two fields.
(22, 2)
Subtract one only from a positive natural number; zero has no predecessor here, rather than a truncated predecessor equal to zero. The exact result, retaining the requested order, is (22, 2).
Reviewed step-by-step derivation
Natural number · qf-density11-natural-predecessor-pair · candidate
N includes 0. The ordered values are [24, 4]. Give each predecessor in N, or none if it does not exist, preserving the two fields.
(23, 3)
Subtract one only from a positive natural number; zero has no predecessor here, rather than a truncated predecessor equal to zero. The exact result, retaining the requested order, is (23, 3).
Reviewed step-by-step derivation
Natural number · qf-density11-natural-predecessor-pair · candidate
N includes 0. lo=5, hi=4. Find |{n∈N: lo≤n≤hi}|, including endpoints.
0
Count both endpoints of a valid interval; hi<lo gives size zero without swapping the bounds. The exact result, retaining the requested order, is 0.
Reviewed step-by-step derivation
Natural number · qf-density11-natural-interval-size · candidate
N includes 0. lo=0, hi=0. Find |{n∈N: lo≤n≤hi}|, including endpoints.
1
Count both endpoints of a valid interval; hi<lo gives size zero without swapping the bounds. The exact result, retaining the requested order, is 1.
Reviewed step-by-step derivation
Natural number · qf-density11-natural-interval-size · candidate
N includes 0. lo=2, hi=3. Find |{n∈N: lo≤n≤hi}|, including endpoints.
2
Count both endpoints of a valid interval; hi<lo gives size zero without swapping the bounds. The exact result, retaining the requested order, is 2.
Reviewed step-by-step derivation
Natural number · qf-density11-natural-interval-size · candidate
N includes 0. lo=0, hi=2. Find |{n∈N: lo≤n≤hi}|, including endpoints.
3
Count both endpoints of a valid interval; hi<lo gives size zero without swapping the bounds. The exact result, retaining the requested order, is 3.
Reviewed step-by-step derivation
Natural number · qf-density11-natural-interval-size · candidate
N includes 0. lo=1, hi=4. Find |{n∈N: lo≤n≤hi}|, including endpoints.
4
Count both endpoints of a valid interval; hi<lo gives size zero without swapping the bounds. The exact result, retaining the requested order, is 4.
Reviewed step-by-step derivation
Natural number · qf-density11-natural-interval-size · candidate
N includes 0. lo=2, hi=6. Find |{n∈N: lo≤n≤hi}|, including endpoints.
5
Count both endpoints of a valid interval; hi<lo gives size zero without swapping the bounds. The exact result, retaining the requested order, is 5.
Reviewed step-by-step derivation
Natural number · qf-density11-natural-interval-size · candidate
N includes 0. lo=0, hi=5. Find |{n∈N: lo≤n≤hi}|, including endpoints.
6
Count both endpoints of a valid interval; hi<lo gives size zero without swapping the bounds. The exact result, retaining the requested order, is 6.
Reviewed step-by-step derivation
Natural number · qf-density11-natural-interval-size · candidate
N includes 0. lo=1, hi=7. Find |{n∈N: lo≤n≤hi}|, including endpoints.
7
Count both endpoints of a valid interval; hi<lo gives size zero without swapping the bounds. The exact result, retaining the requested order, is 7.
Reviewed step-by-step derivation
Natural number · qf-density11-natural-interval-size · candidate
N includes 0. lo=2, hi=9. Find |{n∈N: lo≤n≤hi}|, including endpoints.
8
Count both endpoints of a valid interval; hi<lo gives size zero without swapping the bounds. The exact result, retaining the requested order, is 8.
Reviewed step-by-step derivation
Natural number · qf-density11-natural-interval-size · candidate
N includes 0. lo=0, hi=8. Find |{n∈N: lo≤n≤hi}|, including endpoints.
9
Count both endpoints of a valid interval; hi<lo gives size zero without swapping the bounds. The exact result, retaining the requested order, is 9.
Reviewed step-by-step derivation
Natural number · qf-density11-natural-interval-size · candidate
N includes 0. lo=1, hi=10. Find |{n∈N: lo≤n≤hi}|, including endpoints.
10
Count both endpoints of a valid interval; hi<lo gives size zero without swapping the bounds. The exact result, retaining the requested order, is 10.
Reviewed step-by-step derivation
Natural number · qf-density11-natural-interval-size · candidate
N includes 0. lo=2, hi=12. Find |{n∈N: lo≤n≤hi}|, including endpoints.
11
Count both endpoints of a valid interval; hi<lo gives size zero without swapping the bounds. The exact result, retaining the requested order, is 11.
Reviewed step-by-step derivation
Natural number · qf-density11-natural-interval-size · candidate
N includes 0. lo=0, hi=11. Find |{n∈N: lo≤n≤hi}|, including endpoints.
12
Count both endpoints of a valid interval; hi<lo gives size zero without swapping the bounds. The exact result, retaining the requested order, is 12.
Reviewed step-by-step derivation
Natural number · qf-density11-natural-interval-size · candidate
N includes 0. lo=1, hi=13. Find |{n∈N: lo≤n≤hi}|, including endpoints.
13
Count both endpoints of a valid interval; hi<lo gives size zero without swapping the bounds. The exact result, retaining the requested order, is 13.
Reviewed step-by-step derivation
Natural number · qf-density11-natural-interval-size · candidate
N includes 0. lo=2, hi=15. Find |{n∈N: lo≤n≤hi}|, including endpoints.
14
Count both endpoints of a valid interval; hi<lo gives size zero without swapping the bounds. The exact result, retaining the requested order, is 14.
Reviewed step-by-step derivation
Natural number · qf-density11-natural-interval-size · candidate
N includes 0. lo=0, hi=14. Find |{n∈N: lo≤n≤hi}|, including endpoints.
15
Count both endpoints of a valid interval; hi<lo gives size zero without swapping the bounds. The exact result, retaining the requested order, is 15.
Reviewed step-by-step derivation
Natural number · qf-density11-natural-interval-size · candidate
N includes 0. lo=1, hi=16. Find |{n∈N: lo≤n≤hi}|, including endpoints.
16
Count both endpoints of a valid interval; hi<lo gives size zero without swapping the bounds. The exact result, retaining the requested order, is 16.
Reviewed step-by-step derivation
Natural number · qf-density11-natural-interval-size · candidate
N includes 0. lo=2, hi=18. Find |{n∈N: lo≤n≤hi}|, including endpoints.
17
Count both endpoints of a valid interval; hi<lo gives size zero without swapping the bounds. The exact result, retaining the requested order, is 17.
Reviewed step-by-step derivation
Natural number · qf-density11-natural-interval-size · candidate
N includes 0. lo=0, hi=17. Find |{n∈N: lo≤n≤hi}|, including endpoints.
18
Count both endpoints of a valid interval; hi<lo gives size zero without swapping the bounds. The exact result, retaining the requested order, is 18.
Reviewed step-by-step derivation
Natural number · qf-density11-natural-interval-size · candidate
N includes 0. lo=1, hi=19. Find |{n∈N: lo≤n≤hi}|, including endpoints.
19
Count both endpoints of a valid interval; hi<lo gives size zero without swapping the bounds. The exact result, retaining the requested order, is 19.
Reviewed step-by-step derivation
Natural number · qf-density11-natural-interval-size · candidate
N includes 0. lo=2, hi=21. Find |{n∈N: lo≤n≤hi}|, including endpoints.
20
Count both endpoints of a valid interval; hi<lo gives size zero without swapping the bounds. The exact result, retaining the requested order, is 20.
Reviewed step-by-step derivation
Natural number · qf-density11-natural-interval-size · candidate
N includes 0. lo=0, hi=20. Find |{n∈N: lo≤n≤hi}|, including endpoints.
21
Count both endpoints of a valid interval; hi<lo gives size zero without swapping the bounds. The exact result, retaining the requested order, is 21.
Reviewed step-by-step derivation
Natural number · qf-density11-natural-interval-size · candidate
N includes 0. lo=1, hi=22. Find |{n∈N: lo≤n≤hi}|, including endpoints.
22
Count both endpoints of a valid interval; hi<lo gives size zero without swapping the bounds. The exact result, retaining the requested order, is 22.
Reviewed step-by-step derivation
Natural number · qf-density11-natural-interval-size · candidate
N includes 0. lo=2, hi=24. Find |{n∈N: lo≤n≤hi}|, including endpoints.
23
Count both endpoints of a valid interval; hi<lo gives size zero without swapping the bounds. The exact result, retaining the requested order, is 23.
Reviewed step-by-step derivation
Natural number · qf-density11-natural-interval-size · candidate
N includes 0. lo=0, hi=23. Find |{n∈N: lo≤n≤hi}|, including endpoints.
24
Count both endpoints of a valid interval; hi<lo gives size zero without swapping the bounds. The exact result, retaining the requested order, is 24.
Reviewed step-by-step derivation
Natural number · qf-density11-natural-interval-size · candidate
N includes 0. Ordered rational candidates=[0/1, -1/1, 1/2]. Give the Yes/No membership list for N.
[Yes, No, No]
Reduce each rational exactly, then require a nonnegative value and denominator one; zero passes, but negative integers and nonintegers do not. The exact result, retaining the requested order, is [Yes, No, No].
Reviewed step-by-step derivation
Natural number · qf-density11-natural-rational-membership-mask · candidate
N includes 0. Ordered rational candidates=[2/2, 0/3, -1/2]. Give the Yes/No membership list for N.
[Yes, Yes, No]
Reduce each rational exactly, then require a nonnegative value and denominator one; zero passes, but negative integers and nonintegers do not. The exact result, retaining the requested order, is [Yes, Yes, No].
Reviewed step-by-step derivation
Natural number · qf-density11-natural-rational-membership-mask · candidate
N includes 0. Ordered rational candidates=[2/1, -2/1, 3/5]. Give the Yes/No membership list for N.
[Yes, No, No]
Reduce each rational exactly, then require a nonnegative value and denominator one; zero passes, but negative integers and nonintegers do not. The exact result, retaining the requested order, is [Yes, No, No].
Reviewed step-by-step derivation
Natural number · qf-density11-natural-rational-membership-mask · candidate
N includes 0. Ordered rational candidates=[3/1, -3/1, 4/7]. Give the Yes/No membership list for N.
[Yes, No, No]
Reduce each rational exactly, then require a nonnegative value and denominator one; zero passes, but negative integers and nonintegers do not. The exact result, retaining the requested order, is [Yes, No, No].
Reviewed step-by-step derivation
Natural number · qf-density11-natural-rational-membership-mask · candidate
N includes 0. Ordered rational candidates=[4/1, -4/1, 5/9]. Give the Yes/No membership list for N.
[Yes, No, No]
Reduce each rational exactly, then require a nonnegative value and denominator one; zero passes, but negative integers and nonintegers do not. The exact result, retaining the requested order, is [Yes, No, No].
Reviewed step-by-step derivation
Natural number · qf-density11-natural-rational-membership-mask · candidate
N includes 0. Ordered rational candidates=[5/1, -5/1, 6/11]. Give the Yes/No membership list for N.
[Yes, No, No]
Reduce each rational exactly, then require a nonnegative value and denominator one; zero passes, but negative integers and nonintegers do not. The exact result, retaining the requested order, is [Yes, No, No].
Reviewed step-by-step derivation
Natural number · qf-density11-natural-rational-membership-mask · candidate
N includes 0. Ordered rational candidates=[6/1, -6/1, 7/13]. Give the Yes/No membership list for N.
[Yes, No, No]
Reduce each rational exactly, then require a nonnegative value and denominator one; zero passes, but negative integers and nonintegers do not. The exact result, retaining the requested order, is [Yes, No, No].
Reviewed step-by-step derivation
Natural number · qf-density11-natural-rational-membership-mask · candidate
N includes 0. Ordered rational candidates=[7/1, -7/1, 8/15]. Give the Yes/No membership list for N.
[Yes, No, No]
Reduce each rational exactly, then require a nonnegative value and denominator one; zero passes, but negative integers and nonintegers do not. The exact result, retaining the requested order, is [Yes, No, No].
Reviewed step-by-step derivation
Natural number · qf-density11-natural-rational-membership-mask · candidate
N includes 0. Ordered rational candidates=[8/1, -8/1, 9/17]. Give the Yes/No membership list for N.
[Yes, No, No]
Reduce each rational exactly, then require a nonnegative value and denominator one; zero passes, but negative integers and nonintegers do not. The exact result, retaining the requested order, is [Yes, No, No].
Reviewed step-by-step derivation
Natural number · qf-density11-natural-rational-membership-mask · candidate
N includes 0. Ordered rational candidates=[9/1, -9/1, 10/19]. Give the Yes/No membership list for N.
[Yes, No, No]
Reduce each rational exactly, then require a nonnegative value and denominator one; zero passes, but negative integers and nonintegers do not. The exact result, retaining the requested order, is [Yes, No, No].
Reviewed step-by-step derivation
Natural number · qf-density11-natural-rational-membership-mask · candidate
N includes 0. Ordered rational candidates=[10/1, -10/1, 11/21]. Give the Yes/No membership list for N.
[Yes, No, No]
Reduce each rational exactly, then require a nonnegative value and denominator one; zero passes, but negative integers and nonintegers do not. The exact result, retaining the requested order, is [Yes, No, No].
Reviewed step-by-step derivation
Natural number · qf-density11-natural-rational-membership-mask · candidate
N includes 0. Ordered rational candidates=[11/1, -11/1, 12/23]. Give the Yes/No membership list for N.
[Yes, No, No]
Reduce each rational exactly, then require a nonnegative value and denominator one; zero passes, but negative integers and nonintegers do not. The exact result, retaining the requested order, is [Yes, No, No].
Reviewed step-by-step derivation
Natural number · qf-density11-natural-rational-membership-mask · candidate
N includes 0. Ordered rational candidates=[12/1, -12/1, 13/25]. Give the Yes/No membership list for N.
[Yes, No, No]
Reduce each rational exactly, then require a nonnegative value and denominator one; zero passes, but negative integers and nonintegers do not. The exact result, retaining the requested order, is [Yes, No, No].
Reviewed step-by-step derivation
Natural number · qf-density11-natural-rational-membership-mask · candidate
N includes 0. Ordered rational candidates=[13/1, -13/1, 14/27]. Give the Yes/No membership list for N.
[Yes, No, No]
Reduce each rational exactly, then require a nonnegative value and denominator one; zero passes, but negative integers and nonintegers do not. The exact result, retaining the requested order, is [Yes, No, No].
Reviewed step-by-step derivation
Natural number · qf-density11-natural-rational-membership-mask · candidate
N includes 0. Ordered rational candidates=[14/1, -14/1, 15/29]. Give the Yes/No membership list for N.
[Yes, No, No]
Reduce each rational exactly, then require a nonnegative value and denominator one; zero passes, but negative integers and nonintegers do not. The exact result, retaining the requested order, is [Yes, No, No].
Reviewed step-by-step derivation
Natural number · qf-density11-natural-rational-membership-mask · candidate
N includes 0. Ordered rational candidates=[15/1, -15/1, 16/31]. Give the Yes/No membership list for N.
[Yes, No, No]
Reduce each rational exactly, then require a nonnegative value and denominator one; zero passes, but negative integers and nonintegers do not. The exact result, retaining the requested order, is [Yes, No, No].
Reviewed step-by-step derivation
Natural number · qf-density11-natural-rational-membership-mask · candidate
N includes 0. Ordered rational candidates=[16/1, -16/1, 17/33]. Give the Yes/No membership list for N.
[Yes, No, No]
Reduce each rational exactly, then require a nonnegative value and denominator one; zero passes, but negative integers and nonintegers do not. The exact result, retaining the requested order, is [Yes, No, No].
Reviewed step-by-step derivation
Natural number · qf-density11-natural-rational-membership-mask · candidate
N includes 0. Ordered rational candidates=[17/1, -17/1, 18/35]. Give the Yes/No membership list for N.
[Yes, No, No]
Reduce each rational exactly, then require a nonnegative value and denominator one; zero passes, but negative integers and nonintegers do not. The exact result, retaining the requested order, is [Yes, No, No].
Reviewed step-by-step derivation
Natural number · qf-density11-natural-rational-membership-mask · candidate
N includes 0. Ordered rational candidates=[18/1, -18/1, 19/37]. Give the Yes/No membership list for N.
[Yes, No, No]
Reduce each rational exactly, then require a nonnegative value and denominator one; zero passes, but negative integers and nonintegers do not. The exact result, retaining the requested order, is [Yes, No, No].
Reviewed step-by-step derivation
Natural number · qf-density11-natural-rational-membership-mask · candidate
N includes 0. Ordered rational candidates=[19/1, -19/1, 20/39]. Give the Yes/No membership list for N.
[Yes, No, No]
Reduce each rational exactly, then require a nonnegative value and denominator one; zero passes, but negative integers and nonintegers do not. The exact result, retaining the requested order, is [Yes, No, No].
Reviewed step-by-step derivation
Natural number · qf-density11-natural-rational-membership-mask · candidate
N includes 0. Ordered rational candidates=[20/1, -20/1, 21/41]. Give the Yes/No membership list for N.
[Yes, No, No]
Reduce each rational exactly, then require a nonnegative value and denominator one; zero passes, but negative integers and nonintegers do not. The exact result, retaining the requested order, is [Yes, No, No].
Reviewed step-by-step derivation
Natural number · qf-density11-natural-rational-membership-mask · candidate
N includes 0. Ordered rational candidates=[21/1, -21/1, 22/43]. Give the Yes/No membership list for N.
[Yes, No, No]
Reduce each rational exactly, then require a nonnegative value and denominator one; zero passes, but negative integers and nonintegers do not. The exact result, retaining the requested order, is [Yes, No, No].
Reviewed step-by-step derivation
Natural number · qf-density11-natural-rational-membership-mask · candidate
N includes 0. Ordered rational candidates=[22/1, -22/1, 23/45]. Give the Yes/No membership list for N.
[Yes, No, No]
Reduce each rational exactly, then require a nonnegative value and denominator one; zero passes, but negative integers and nonintegers do not. The exact result, retaining the requested order, is [Yes, No, No].
Reviewed step-by-step derivation
Natural number · qf-density11-natural-rational-membership-mask · candidate
N includes 0. Ordered rational candidates=[23/1, -23/1, 24/47]. Give the Yes/No membership list for N.
[Yes, No, No]
Reduce each rational exactly, then require a nonnegative value and denominator one; zero passes, but negative integers and nonintegers do not. The exact result, retaining the requested order, is [Yes, No, No].
Reviewed step-by-step derivation
Natural number · qf-density11-natural-rational-membership-mask · candidate
N includes 0. Ordered rational candidates=[24/1, -24/1, 25/49]. Give the Yes/No membership list for N.
[Yes, No, No]
Reduce each rational exactly, then require a nonnegative value and denominator one; zero passes, but negative integers and nonintegers do not. The exact result, retaining the requested order, is [Yes, No, No].
Reviewed step-by-step derivation
Natural number · qf-density11-natural-rational-membership-mask · candidate
Canonical question JSONL · English JSONL · Validation report