# Zero-Inclusive Naturals and Partial Predecessors

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.



## 1. Inclusive Natural Successor Window

N includes 0. n=0, k=0. List the complete ordered window beginning at n and applying the successor k times.

View answer and explanation[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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r01-natural-number-0001/)

Natural number · qf-density11-natural-successor-window · candidate





## 2. Inclusive Natural Successor Window

N includes 0. n=0, k=8. List the complete ordered window beginning at n and applying the successor k times.

View answer and explanation[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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r01-natural-number-0002/)

Natural number · qf-density11-natural-successor-window · candidate





## 3. Inclusive Natural Successor Window

N includes 0. n=1, k=2. List the complete ordered window beginning at n and applying the successor k times.

View answer and explanation[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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r01-natural-number-0003/)

Natural number · qf-density11-natural-successor-window · candidate





## 4. Inclusive Natural Successor Window

N includes 0. n=2, k=3. List the complete ordered window beginning at n and applying the successor k times.

View answer and explanation[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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r01-natural-number-0004/)

Natural number · qf-density11-natural-successor-window · candidate





## 5. Inclusive Natural Successor Window

N includes 0. n=3, k=4. List the complete ordered window beginning at n and applying the successor k times.

View answer and explanation[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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r01-natural-number-0005/)

Natural number · qf-density11-natural-successor-window · candidate





## 6. Inclusive Natural Successor Window

N includes 0. n=4, k=5. List the complete ordered window beginning at n and applying the successor k times.

View answer and explanation[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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r01-natural-number-0006/)

Natural number · qf-density11-natural-successor-window · candidate





## 7. Inclusive Natural Successor Window

N includes 0. n=5, k=6. List the complete ordered window beginning at n and applying the successor k times.

View answer and explanation[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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r01-natural-number-0007/)

Natural number · qf-density11-natural-successor-window · candidate





## 8. Inclusive Natural Successor Window

N includes 0. n=6, k=7. List the complete ordered window beginning at n and applying the successor k times.

View answer and explanation[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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r01-natural-number-0008/)

Natural number · qf-density11-natural-successor-window · candidate





## 9. Inclusive Natural Successor Window

N includes 0. n=7, k=8. List the complete ordered window beginning at n and applying the successor k times.

View answer and explanation[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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r01-natural-number-0009/)

Natural number · qf-density11-natural-successor-window · candidate





## 10. Inclusive Natural Successor Window

N includes 0. n=8, k=0. List the complete ordered window beginning at n and applying the successor k times.

View answer and explanation[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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r01-natural-number-0010/)

Natural number · qf-density11-natural-successor-window · candidate





## 11. Inclusive Natural Successor Window

N includes 0. n=9, k=1. List the complete ordered window beginning at n and applying the successor k times.

View answer and explanation[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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r01-natural-number-0011/)

Natural number · qf-density11-natural-successor-window · candidate





## 12. Inclusive Natural Successor Window

N includes 0. n=10, k=2. List the complete ordered window beginning at n and applying the successor k times.

View answer and explanation[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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r01-natural-number-0012/)

Natural number · qf-density11-natural-successor-window · candidate





## 13. Inclusive Natural Successor Window

N includes 0. n=11, k=3. List the complete ordered window beginning at n and applying the successor k times.

View answer and explanation[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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r01-natural-number-0013/)

Natural number · qf-density11-natural-successor-window · candidate





## 14. Inclusive Natural Successor Window

N includes 0. n=12, k=4. List the complete ordered window beginning at n and applying the successor k times.

View answer and explanation[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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r01-natural-number-0014/)

Natural number · qf-density11-natural-successor-window · candidate





## 15. Inclusive Natural Successor Window

N includes 0. n=13, k=5. List the complete ordered window beginning at n and applying the successor k times.

View answer and explanation[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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r01-natural-number-0015/)

Natural number · qf-density11-natural-successor-window · candidate





## 16. Inclusive Natural Successor Window

N includes 0. n=14, k=6. List the complete ordered window beginning at n and applying the successor k times.

View answer and explanation[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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r01-natural-number-0016/)

Natural number · qf-density11-natural-successor-window · candidate





## 17. Inclusive Natural Successor Window

N includes 0. n=15, k=7. List the complete ordered window beginning at n and applying the successor k times.

View answer and explanation[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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r01-natural-number-0017/)

Natural number · qf-density11-natural-successor-window · candidate





## 18. Inclusive Natural Successor Window

N includes 0. n=16, k=8. List the complete ordered window beginning at n and applying the successor k times.

View answer and explanation[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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r01-natural-number-0018/)

Natural number · qf-density11-natural-successor-window · candidate





## 19. Inclusive Natural Successor Window

N includes 0. n=17, k=0. List the complete ordered window beginning at n and applying the successor k times.

View answer and explanation[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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r01-natural-number-0019/)

Natural number · qf-density11-natural-successor-window · candidate





## 20. Inclusive Natural Successor Window

N includes 0. n=18, k=1. List the complete ordered window beginning at n and applying the successor k times.

View answer and explanation[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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r01-natural-number-0020/)

Natural number · qf-density11-natural-successor-window · candidate





## 21. Inclusive Natural Successor Window

N includes 0. n=19, k=2. List the complete ordered window beginning at n and applying the successor k times.

View answer and explanation[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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r01-natural-number-0021/)

Natural number · qf-density11-natural-successor-window · candidate





## 22. Inclusive Natural Successor Window

N includes 0. n=20, k=3. List the complete ordered window beginning at n and applying the successor k times.

View answer and explanation[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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r01-natural-number-0022/)

Natural number · qf-density11-natural-successor-window · candidate





## 23. Inclusive Natural Successor Window

N includes 0. n=21, k=4. List the complete ordered window beginning at n and applying the successor k times.

View answer and explanation[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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r01-natural-number-0023/)

Natural number · qf-density11-natural-successor-window · candidate





## 24. Inclusive Natural Successor Window

N includes 0. n=22, k=5. List the complete ordered window beginning at n and applying the successor k times.

View answer and explanation[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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r01-natural-number-0024/)

Natural number · qf-density11-natural-successor-window · candidate





## 25. Inclusive Natural Successor Window

N includes 0. n=23, k=6. List the complete ordered window beginning at n and applying the successor k times.

View answer and explanation[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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r01-natural-number-0025/)

Natural number · qf-density11-natural-successor-window · candidate





## 26. Partial Natural Predecessors

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.

View answer and explanation(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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r01-natural-number-0026/)

Natural number · qf-density11-natural-predecessor-pair · candidate





## 27. Partial Natural Predecessors

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.

View answer and explanation(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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r01-natural-number-0027/)

Natural number · qf-density11-natural-predecessor-pair · candidate





## 28. Partial Natural Predecessors

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.

View answer and explanation(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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r01-natural-number-0028/)

Natural number · qf-density11-natural-predecessor-pair · candidate





## 29. Partial Natural Predecessors

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.

View answer and explanation(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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r01-natural-number-0029/)

Natural number · qf-density11-natural-predecessor-pair · candidate





## 30. Partial Natural Predecessors

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.

View answer and explanation(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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r01-natural-number-0030/)

Natural number · qf-density11-natural-predecessor-pair · candidate





## 31. Partial Natural Predecessors

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.

View answer and explanation(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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r01-natural-number-0031/)

Natural number · qf-density11-natural-predecessor-pair · candidate





## 32. Partial Natural Predecessors

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.

View answer and explanation(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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r01-natural-number-0032/)

Natural number · qf-density11-natural-predecessor-pair · candidate





## 33. Partial Natural Predecessors

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.

View answer and explanation(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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r01-natural-number-0033/)

Natural number · qf-density11-natural-predecessor-pair · candidate





## 34. Partial Natural Predecessors

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.

View answer and explanation(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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r01-natural-number-0034/)

Natural number · qf-density11-natural-predecessor-pair · candidate





## 35. Partial Natural Predecessors

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.

View answer and explanation(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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r01-natural-number-0035/)

Natural number · qf-density11-natural-predecessor-pair · candidate





## 36. Partial Natural Predecessors

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.

View answer and explanation(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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r01-natural-number-0036/)

Natural number · qf-density11-natural-predecessor-pair · candidate





## 37. Partial Natural Predecessors

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.

View answer and explanation(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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r01-natural-number-0037/)

Natural number · qf-density11-natural-predecessor-pair · candidate





## 38. Partial Natural Predecessors

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.

View answer and explanation(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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r01-natural-number-0038/)

Natural number · qf-density11-natural-predecessor-pair · candidate





## 39. Partial Natural Predecessors

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.

View answer and explanation(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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r01-natural-number-0039/)

Natural number · qf-density11-natural-predecessor-pair · candidate





## 40. Partial Natural Predecessors

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.

View answer and explanation(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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r01-natural-number-0040/)

Natural number · qf-density11-natural-predecessor-pair · candidate





## 41. Partial Natural Predecessors

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.

View answer and explanation(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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r01-natural-number-0041/)

Natural number · qf-density11-natural-predecessor-pair · candidate





## 42. Partial Natural Predecessors

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.

View answer and explanation(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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r01-natural-number-0042/)

Natural number · qf-density11-natural-predecessor-pair · candidate





## 43. Partial Natural Predecessors

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.

View answer and explanation(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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r01-natural-number-0043/)

Natural number · qf-density11-natural-predecessor-pair · candidate





## 44. Partial Natural Predecessors

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.

View answer and explanation(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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r01-natural-number-0044/)

Natural number · qf-density11-natural-predecessor-pair · candidate





## 45. Partial Natural Predecessors

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.

View answer and explanation(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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r01-natural-number-0045/)

Natural number · qf-density11-natural-predecessor-pair · candidate





## 46. Partial Natural Predecessors

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.

View answer and explanation(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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r01-natural-number-0046/)

Natural number · qf-density11-natural-predecessor-pair · candidate





## 47. Partial Natural Predecessors

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.

View answer and explanation(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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r01-natural-number-0047/)

Natural number · qf-density11-natural-predecessor-pair · candidate





## 48. Partial Natural Predecessors

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.

View answer and explanation(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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r01-natural-number-0048/)

Natural number · qf-density11-natural-predecessor-pair · candidate





## 49. Partial Natural Predecessors

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.

View answer and explanation(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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r01-natural-number-0049/)

Natural number · qf-density11-natural-predecessor-pair · candidate





## 50. Partial Natural Predecessors

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.

View answer and explanation(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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r01-natural-number-0050/)

Natural number · qf-density11-natural-predecessor-pair · candidate





## 51. Natural Closed-Interval Cardinality

N includes 0. lo=5, hi=4. Find |{n∈N: lo≤n≤hi}|, including endpoints.

View answer and explanation0

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r01-natural-number-0051/)

Natural number · qf-density11-natural-interval-size · candidate





## 52. Natural Closed-Interval Cardinality

N includes 0. lo=0, hi=0. Find |{n∈N: lo≤n≤hi}|, including endpoints.

View answer and explanation1

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r01-natural-number-0052/)

Natural number · qf-density11-natural-interval-size · candidate





## 53. Natural Closed-Interval Cardinality

N includes 0. lo=2, hi=3. Find |{n∈N: lo≤n≤hi}|, including endpoints.

View answer and explanation2

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r01-natural-number-0053/)

Natural number · qf-density11-natural-interval-size · candidate





## 54. Natural Closed-Interval Cardinality

N includes 0. lo=0, hi=2. Find |{n∈N: lo≤n≤hi}|, including endpoints.

View answer and explanation3

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r01-natural-number-0054/)

Natural number · qf-density11-natural-interval-size · candidate





## 55. Natural Closed-Interval Cardinality

N includes 0. lo=1, hi=4. Find |{n∈N: lo≤n≤hi}|, including endpoints.

View answer and explanation4

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r01-natural-number-0055/)

Natural number · qf-density11-natural-interval-size · candidate





## 56. Natural Closed-Interval Cardinality

N includes 0. lo=2, hi=6. Find |{n∈N: lo≤n≤hi}|, including endpoints.

View answer and explanation5

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r01-natural-number-0056/)

Natural number · qf-density11-natural-interval-size · candidate





## 57. Natural Closed-Interval Cardinality

N includes 0. lo=0, hi=5. Find |{n∈N: lo≤n≤hi}|, including endpoints.

View answer and explanation6

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r01-natural-number-0057/)

Natural number · qf-density11-natural-interval-size · candidate





## 58. Natural Closed-Interval Cardinality

N includes 0. lo=1, hi=7. Find |{n∈N: lo≤n≤hi}|, including endpoints.

View answer and explanation7

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r01-natural-number-0058/)

Natural number · qf-density11-natural-interval-size · candidate





## 59. Natural Closed-Interval Cardinality

N includes 0. lo=2, hi=9. Find |{n∈N: lo≤n≤hi}|, including endpoints.

View answer and explanation8

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r01-natural-number-0059/)

Natural number · qf-density11-natural-interval-size · candidate





## 60. Natural Closed-Interval Cardinality

N includes 0. lo=0, hi=8. Find |{n∈N: lo≤n≤hi}|, including endpoints.

View answer and explanation9

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r01-natural-number-0060/)

Natural number · qf-density11-natural-interval-size · candidate





## 61. Natural Closed-Interval Cardinality

N includes 0. lo=1, hi=10. Find |{n∈N: lo≤n≤hi}|, including endpoints.

View answer and explanation10

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r01-natural-number-0061/)

Natural number · qf-density11-natural-interval-size · candidate





## 62. Natural Closed-Interval Cardinality

N includes 0. lo=2, hi=12. Find |{n∈N: lo≤n≤hi}|, including endpoints.

View answer and explanation11

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r01-natural-number-0062/)

Natural number · qf-density11-natural-interval-size · candidate





## 63. Natural Closed-Interval Cardinality

N includes 0. lo=0, hi=11. Find |{n∈N: lo≤n≤hi}|, including endpoints.

View answer and explanation12

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r01-natural-number-0063/)

Natural number · qf-density11-natural-interval-size · candidate





## 64. Natural Closed-Interval Cardinality

N includes 0. lo=1, hi=13. Find |{n∈N: lo≤n≤hi}|, including endpoints.

View answer and explanation13

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r01-natural-number-0064/)

Natural number · qf-density11-natural-interval-size · candidate





## 65. Natural Closed-Interval Cardinality

N includes 0. lo=2, hi=15. Find |{n∈N: lo≤n≤hi}|, including endpoints.

View answer and explanation14

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r01-natural-number-0065/)

Natural number · qf-density11-natural-interval-size · candidate





## 66. Natural Closed-Interval Cardinality

N includes 0. lo=0, hi=14. Find |{n∈N: lo≤n≤hi}|, including endpoints.

View answer and explanation15

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r01-natural-number-0066/)

Natural number · qf-density11-natural-interval-size · candidate





## 67. Natural Closed-Interval Cardinality

N includes 0. lo=1, hi=16. Find |{n∈N: lo≤n≤hi}|, including endpoints.

View answer and explanation16

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r01-natural-number-0067/)

Natural number · qf-density11-natural-interval-size · candidate





## 68. Natural Closed-Interval Cardinality

N includes 0. lo=2, hi=18. Find |{n∈N: lo≤n≤hi}|, including endpoints.

View answer and explanation17

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r01-natural-number-0068/)

Natural number · qf-density11-natural-interval-size · candidate





## 69. Natural Closed-Interval Cardinality

N includes 0. lo=0, hi=17. Find |{n∈N: lo≤n≤hi}|, including endpoints.

View answer and explanation18

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r01-natural-number-0069/)

Natural number · qf-density11-natural-interval-size · candidate





## 70. Natural Closed-Interval Cardinality

N includes 0. lo=1, hi=19. Find |{n∈N: lo≤n≤hi}|, including endpoints.

View answer and explanation19

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r01-natural-number-0070/)

Natural number · qf-density11-natural-interval-size · candidate





## 71. Natural Closed-Interval Cardinality

N includes 0. lo=2, hi=21. Find |{n∈N: lo≤n≤hi}|, including endpoints.

View answer and explanation20

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r01-natural-number-0071/)

Natural number · qf-density11-natural-interval-size · candidate





## 72. Natural Closed-Interval Cardinality

N includes 0. lo=0, hi=20. Find |{n∈N: lo≤n≤hi}|, including endpoints.

View answer and explanation21

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r01-natural-number-0072/)

Natural number · qf-density11-natural-interval-size · candidate





## 73. Natural Closed-Interval Cardinality

N includes 0. lo=1, hi=22. Find |{n∈N: lo≤n≤hi}|, including endpoints.

View answer and explanation22

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r01-natural-number-0073/)

Natural number · qf-density11-natural-interval-size · candidate





## 74. Natural Closed-Interval Cardinality

N includes 0. lo=2, hi=24. Find |{n∈N: lo≤n≤hi}|, including endpoints.

View answer and explanation23

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r01-natural-number-0074/)

Natural number · qf-density11-natural-interval-size · candidate





## 75. Natural Closed-Interval Cardinality

N includes 0. lo=0, hi=23. Find |{n∈N: lo≤n≤hi}|, including endpoints.

View answer and explanation24

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r01-natural-number-0075/)

Natural number · qf-density11-natural-interval-size · candidate





## 76. Exact Rational Candidates for N

N includes 0. Ordered rational candidates=[0/1, -1/1, 1/2]. Give the Yes/No membership list for N.

View answer and explanation[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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r01-natural-number-0076/)

Natural number · qf-density11-natural-rational-membership-mask · candidate





## 77. Exact Rational Candidates for N

N includes 0. Ordered rational candidates=[2/2, 0/3, -1/2]. Give the Yes/No membership list for N.

View answer and explanation[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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r01-natural-number-0077/)

Natural number · qf-density11-natural-rational-membership-mask · candidate





## 78. Exact Rational Candidates for N

N includes 0. Ordered rational candidates=[2/1, -2/1, 3/5]. Give the Yes/No membership list for N.

View answer and explanation[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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r01-natural-number-0078/)

Natural number · qf-density11-natural-rational-membership-mask · candidate





## 79. Exact Rational Candidates for N

N includes 0. Ordered rational candidates=[3/1, -3/1, 4/7]. Give the Yes/No membership list for N.

View answer and explanation[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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r01-natural-number-0079/)

Natural number · qf-density11-natural-rational-membership-mask · candidate





## 80. Exact Rational Candidates for N

N includes 0. Ordered rational candidates=[4/1, -4/1, 5/9]. Give the Yes/No membership list for N.

View answer and explanation[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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r01-natural-number-0080/)

Natural number · qf-density11-natural-rational-membership-mask · candidate





## 81. Exact Rational Candidates for N

N includes 0. Ordered rational candidates=[5/1, -5/1, 6/11]. Give the Yes/No membership list for N.

View answer and explanation[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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r01-natural-number-0081/)

Natural number · qf-density11-natural-rational-membership-mask · candidate





## 82. Exact Rational Candidates for N

N includes 0. Ordered rational candidates=[6/1, -6/1, 7/13]. Give the Yes/No membership list for N.

View answer and explanation[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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r01-natural-number-0082/)

Natural number · qf-density11-natural-rational-membership-mask · candidate





## 83. Exact Rational Candidates for N

N includes 0. Ordered rational candidates=[7/1, -7/1, 8/15]. Give the Yes/No membership list for N.

View answer and explanation[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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r01-natural-number-0083/)

Natural number · qf-density11-natural-rational-membership-mask · candidate





## 84. Exact Rational Candidates for N

N includes 0. Ordered rational candidates=[8/1, -8/1, 9/17]. Give the Yes/No membership list for N.

View answer and explanation[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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r01-natural-number-0084/)

Natural number · qf-density11-natural-rational-membership-mask · candidate





## 85. Exact Rational Candidates for N

N includes 0. Ordered rational candidates=[9/1, -9/1, 10/19]. Give the Yes/No membership list for N.

View answer and explanation[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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r01-natural-number-0085/)

Natural number · qf-density11-natural-rational-membership-mask · candidate





## 86. Exact Rational Candidates for N

N includes 0. Ordered rational candidates=[10/1, -10/1, 11/21]. Give the Yes/No membership list for N.

View answer and explanation[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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r01-natural-number-0086/)

Natural number · qf-density11-natural-rational-membership-mask · candidate





## 87. Exact Rational Candidates for N

N includes 0. Ordered rational candidates=[11/1, -11/1, 12/23]. Give the Yes/No membership list for N.

View answer and explanation[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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r01-natural-number-0087/)

Natural number · qf-density11-natural-rational-membership-mask · candidate





## 88. Exact Rational Candidates for N

N includes 0. Ordered rational candidates=[12/1, -12/1, 13/25]. Give the Yes/No membership list for N.

View answer and explanation[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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r01-natural-number-0088/)

Natural number · qf-density11-natural-rational-membership-mask · candidate





## 89. Exact Rational Candidates for N

N includes 0. Ordered rational candidates=[13/1, -13/1, 14/27]. Give the Yes/No membership list for N.

View answer and explanation[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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r01-natural-number-0089/)

Natural number · qf-density11-natural-rational-membership-mask · candidate





## 90. Exact Rational Candidates for N

N includes 0. Ordered rational candidates=[14/1, -14/1, 15/29]. Give the Yes/No membership list for N.

View answer and explanation[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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r01-natural-number-0090/)

Natural number · qf-density11-natural-rational-membership-mask · candidate





## 91. Exact Rational Candidates for N

N includes 0. Ordered rational candidates=[15/1, -15/1, 16/31]. Give the Yes/No membership list for N.

View answer and explanation[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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r01-natural-number-0091/)

Natural number · qf-density11-natural-rational-membership-mask · candidate





## 92. Exact Rational Candidates for N

N includes 0. Ordered rational candidates=[16/1, -16/1, 17/33]. Give the Yes/No membership list for N.

View answer and explanation[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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r01-natural-number-0092/)

Natural number · qf-density11-natural-rational-membership-mask · candidate





## 93. Exact Rational Candidates for N

N includes 0. Ordered rational candidates=[17/1, -17/1, 18/35]. Give the Yes/No membership list for N.

View answer and explanation[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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r01-natural-number-0093/)

Natural number · qf-density11-natural-rational-membership-mask · candidate





## 94. Exact Rational Candidates for N

N includes 0. Ordered rational candidates=[18/1, -18/1, 19/37]. Give the Yes/No membership list for N.

View answer and explanation[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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r01-natural-number-0094/)

Natural number · qf-density11-natural-rational-membership-mask · candidate





## 95. Exact Rational Candidates for N

N includes 0. Ordered rational candidates=[19/1, -19/1, 20/39]. Give the Yes/No membership list for N.

View answer and explanation[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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r01-natural-number-0095/)

Natural number · qf-density11-natural-rational-membership-mask · candidate





## 96. Exact Rational Candidates for N

N includes 0. Ordered rational candidates=[20/1, -20/1, 21/41]. Give the Yes/No membership list for N.

View answer and explanation[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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r01-natural-number-0096/)

Natural number · qf-density11-natural-rational-membership-mask · candidate





## 97. Exact Rational Candidates for N

N includes 0. Ordered rational candidates=[21/1, -21/1, 22/43]. Give the Yes/No membership list for N.

View answer and explanation[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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r01-natural-number-0097/)

Natural number · qf-density11-natural-rational-membership-mask · candidate





## 98. Exact Rational Candidates for N

N includes 0. Ordered rational candidates=[22/1, -22/1, 23/45]. Give the Yes/No membership list for N.

View answer and explanation[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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r01-natural-number-0098/)

Natural number · qf-density11-natural-rational-membership-mask · candidate





## 99. Exact Rational Candidates for N

N includes 0. Ordered rational candidates=[23/1, -23/1, 24/47]. Give the Yes/No membership list for N.

View answer and explanation[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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r01-natural-number-0099/)

Natural number · qf-density11-natural-rational-membership-mask · candidate





## 100. Exact Rational Candidates for N

N includes 0. Ordered rational candidates=[24/1, -24/1, 25/49]. Give the Yes/No membership list for N.

View answer and explanation[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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r01-natural-number-0100/)

Natural number · qf-density11-natural-rational-membership-mask · candidate



[Canonical question JSONL](https://ocme.evemisslab.com/data/questions/batches/batch-20261008-advance-20261011-r01-natural-number-foundations/questions.jsonl) · [English JSONL](https://ocme.evemisslab.com/data/i18n/questions/batch-20261008-advance-20261011-r01-natural-number-foundations.en.jsonl) · [Validation report](https://ocme.evemisslab.com/data/questions/batches/batch-20261008-advance-20261011-r01-natural-number-foundations/validation-report.json)