# Finite Set Extension, Equality, and Powersets

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-r04-set-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. Normalized Finite Set Enumeration

The raw integer list is entries=[]. Remove duplicates and enumerate every set element increasingly, using [] for the empty set.

View answer and explanation[]

Set extension depends on elements, not input order or repetition; list each element once. The exact result, retaining the requested order, is [].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r04-set-0001/)

Set · qf-density11-set-normalized-enumeration · candidate





## 2. Normalized Finite Set Enumeration

The raw integer list is entries=[0, 0]. Remove duplicates and enumerate every set element increasingly, using [] for the empty set.

View answer and explanation[0]

Set extension depends on elements, not input order or repetition; list each element once. The exact result, retaining the requested order, is [0].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r04-set-0002/)

Set · qf-density11-set-normalized-enumeration · candidate





## 3. Normalized Finite Set Enumeration

The raw integer list is entries=[-2, 1, -2]. Remove duplicates and enumerate every set element increasingly, using [] for the empty set.

View answer and explanation[-2, 1]

Set extension depends on elements, not input order or repetition; list each element once. The exact result, retaining the requested order, is [-2, 1].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r04-set-0003/)

Set · qf-density11-set-normalized-enumeration · candidate





## 4. Normalized Finite Set Enumeration

The raw integer list is entries=[3, -3, 0, 3]. Remove duplicates and enumerate every set element increasingly, using [] for the empty set.

View answer and explanation[-3, 0, 3]

Set extension depends on elements, not input order or repetition; list each element once. The exact result, retaining the requested order, is [-3, 0, 3].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r04-set-0004/)

Set · qf-density11-set-normalized-enumeration · candidate





## 5. Normalized Finite Set Enumeration

The raw integer list is entries=[4, -4, 0, 4]. Remove duplicates and enumerate every set element increasingly, using [] for the empty set.

View answer and explanation[-4, 0, 4]

Set extension depends on elements, not input order or repetition; list each element once. The exact result, retaining the requested order, is [-4, 0, 4].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r04-set-0005/)

Set · qf-density11-set-normalized-enumeration · candidate





## 6. Normalized Finite Set Enumeration

The raw integer list is entries=[5, -5, 0, 5]. Remove duplicates and enumerate every set element increasingly, using [] for the empty set.

View answer and explanation[-5, 0, 5]

Set extension depends on elements, not input order or repetition; list each element once. The exact result, retaining the requested order, is [-5, 0, 5].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r04-set-0006/)

Set · qf-density11-set-normalized-enumeration · candidate





## 7. Normalized Finite Set Enumeration

The raw integer list is entries=[6, -6, 0, 6]. Remove duplicates and enumerate every set element increasingly, using [] for the empty set.

View answer and explanation[-6, 0, 6]

Set extension depends on elements, not input order or repetition; list each element once. The exact result, retaining the requested order, is [-6, 0, 6].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r04-set-0007/)

Set · qf-density11-set-normalized-enumeration · candidate





## 8. Normalized Finite Set Enumeration

The raw integer list is entries=[7, -7, 0, 7]. Remove duplicates and enumerate every set element increasingly, using [] for the empty set.

View answer and explanation[-7, 0, 7]

Set extension depends on elements, not input order or repetition; list each element once. The exact result, retaining the requested order, is [-7, 0, 7].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r04-set-0008/)

Set · qf-density11-set-normalized-enumeration · candidate





## 9. Normalized Finite Set Enumeration

The raw integer list is entries=[8, -8, 0, 8]. Remove duplicates and enumerate every set element increasingly, using [] for the empty set.

View answer and explanation[-8, 0, 8]

Set extension depends on elements, not input order or repetition; list each element once. The exact result, retaining the requested order, is [-8, 0, 8].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r04-set-0009/)

Set · qf-density11-set-normalized-enumeration · candidate





## 10. Normalized Finite Set Enumeration

The raw integer list is entries=[9, -9, 0, 9]. Remove duplicates and enumerate every set element increasingly, using [] for the empty set.

View answer and explanation[-9, 0, 9]

Set extension depends on elements, not input order or repetition; list each element once. The exact result, retaining the requested order, is [-9, 0, 9].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r04-set-0010/)

Set · qf-density11-set-normalized-enumeration · candidate





## 11. Normalized Finite Set Enumeration

The raw integer list is entries=[10, -10, 0, 10]. Remove duplicates and enumerate every set element increasingly, using [] for the empty set.

View answer and explanation[-10, 0, 10]

Set extension depends on elements, not input order or repetition; list each element once. The exact result, retaining the requested order, is [-10, 0, 10].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r04-set-0011/)

Set · qf-density11-set-normalized-enumeration · candidate





## 12. Normalized Finite Set Enumeration

The raw integer list is entries=[11, -11, 0, 11]. Remove duplicates and enumerate every set element increasingly, using [] for the empty set.

View answer and explanation[-11, 0, 11]

Set extension depends on elements, not input order or repetition; list each element once. The exact result, retaining the requested order, is [-11, 0, 11].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r04-set-0012/)

Set · qf-density11-set-normalized-enumeration · candidate





## 13. Normalized Finite Set Enumeration

The raw integer list is entries=[12, -12, 0, 12]. Remove duplicates and enumerate every set element increasingly, using [] for the empty set.

View answer and explanation[-12, 0, 12]

Set extension depends on elements, not input order or repetition; list each element once. The exact result, retaining the requested order, is [-12, 0, 12].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r04-set-0013/)

Set · qf-density11-set-normalized-enumeration · candidate





## 14. Normalized Finite Set Enumeration

The raw integer list is entries=[13, -13, 0, 13]. Remove duplicates and enumerate every set element increasingly, using [] for the empty set.

View answer and explanation[-13, 0, 13]

Set extension depends on elements, not input order or repetition; list each element once. The exact result, retaining the requested order, is [-13, 0, 13].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r04-set-0014/)

Set · qf-density11-set-normalized-enumeration · candidate





## 15. Normalized Finite Set Enumeration

The raw integer list is entries=[14, -14, 0, 14]. Remove duplicates and enumerate every set element increasingly, using [] for the empty set.

View answer and explanation[-14, 0, 14]

Set extension depends on elements, not input order or repetition; list each element once. The exact result, retaining the requested order, is [-14, 0, 14].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r04-set-0015/)

Set · qf-density11-set-normalized-enumeration · candidate





## 16. Normalized Finite Set Enumeration

The raw integer list is entries=[15, -15, 0, 15]. Remove duplicates and enumerate every set element increasingly, using [] for the empty set.

View answer and explanation[-15, 0, 15]

Set extension depends on elements, not input order or repetition; list each element once. The exact result, retaining the requested order, is [-15, 0, 15].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r04-set-0016/)

Set · qf-density11-set-normalized-enumeration · candidate





## 17. Normalized Finite Set Enumeration

The raw integer list is entries=[16, -16, 0, 16]. Remove duplicates and enumerate every set element increasingly, using [] for the empty set.

View answer and explanation[-16, 0, 16]

Set extension depends on elements, not input order or repetition; list each element once. The exact result, retaining the requested order, is [-16, 0, 16].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r04-set-0017/)

Set · qf-density11-set-normalized-enumeration · candidate





## 18. Normalized Finite Set Enumeration

The raw integer list is entries=[17, -17, 0, 17]. Remove duplicates and enumerate every set element increasingly, using [] for the empty set.

View answer and explanation[-17, 0, 17]

Set extension depends on elements, not input order or repetition; list each element once. The exact result, retaining the requested order, is [-17, 0, 17].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r04-set-0018/)

Set · qf-density11-set-normalized-enumeration · candidate





## 19. Normalized Finite Set Enumeration

The raw integer list is entries=[18, -18, 0, 18]. Remove duplicates and enumerate every set element increasingly, using [] for the empty set.

View answer and explanation[-18, 0, 18]

Set extension depends on elements, not input order or repetition; list each element once. The exact result, retaining the requested order, is [-18, 0, 18].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r04-set-0019/)

Set · qf-density11-set-normalized-enumeration · candidate





## 20. Normalized Finite Set Enumeration

The raw integer list is entries=[19, -19, 0, 19]. Remove duplicates and enumerate every set element increasingly, using [] for the empty set.

View answer and explanation[-19, 0, 19]

Set extension depends on elements, not input order or repetition; list each element once. The exact result, retaining the requested order, is [-19, 0, 19].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r04-set-0020/)

Set · qf-density11-set-normalized-enumeration · candidate





## 21. Normalized Finite Set Enumeration

The raw integer list is entries=[20, -20, 0, 20]. Remove duplicates and enumerate every set element increasingly, using [] for the empty set.

View answer and explanation[-20, 0, 20]

Set extension depends on elements, not input order or repetition; list each element once. The exact result, retaining the requested order, is [-20, 0, 20].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r04-set-0021/)

Set · qf-density11-set-normalized-enumeration · candidate





## 22. Normalized Finite Set Enumeration

The raw integer list is entries=[21, -21, 0, 21]. Remove duplicates and enumerate every set element increasingly, using [] for the empty set.

View answer and explanation[-21, 0, 21]

Set extension depends on elements, not input order or repetition; list each element once. The exact result, retaining the requested order, is [-21, 0, 21].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r04-set-0022/)

Set · qf-density11-set-normalized-enumeration · candidate





## 23. Normalized Finite Set Enumeration

The raw integer list is entries=[22, -22, 0, 22]. Remove duplicates and enumerate every set element increasingly, using [] for the empty set.

View answer and explanation[-22, 0, 22]

Set extension depends on elements, not input order or repetition; list each element once. The exact result, retaining the requested order, is [-22, 0, 22].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r04-set-0023/)

Set · qf-density11-set-normalized-enumeration · candidate





## 24. Normalized Finite Set Enumeration

The raw integer list is entries=[23, -23, 0, 23]. Remove duplicates and enumerate every set element increasingly, using [] for the empty set.

View answer and explanation[-23, 0, 23]

Set extension depends on elements, not input order or repetition; list each element once. The exact result, retaining the requested order, is [-23, 0, 23].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r04-set-0024/)

Set · qf-density11-set-normalized-enumeration · candidate





## 25. Normalized Finite Set Enumeration

The raw integer list is entries=[24, -24, 0, 24]. Remove duplicates and enumerate every set element increasingly, using [] for the empty set.

View answer and explanation[-24, 0, 24]

Set extension depends on elements, not input order or repetition; list each element once. The exact result, retaining the requested order, is [-24, 0, 24].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r04-set-0025/)

Set · qf-density11-set-normalized-enumeration · candidate





## 26. Set Equality with Both Difference Witnesses

Finite integer sets A=[0, 1] and B=[1, 0, 1]; repeated elements count once. Give (are A and B equal?, A\B, B\A), with increasing difference lists.

View answer and explanation(Yes, [], [])

Equality requires both differences to be empty; equal cardinalities alone are insufficient. The exact result, retaining the requested order, is (Yes, [], []).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r04-set-0026/)

Set · qf-density11-set-equality-defect · candidate





## 27. Set Equality with Both Difference Witnesses

Finite integer sets A=[] and B=[-1]; repeated elements count once. Give (are A and B equal?, A\B, B\A), with increasing difference lists.

View answer and explanation(No, [], [-1])

Equality requires both differences to be empty; equal cardinalities alone are insufficient. The exact result, retaining the requested order, is (No, [], [-1]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r04-set-0027/)

Set · qf-density11-set-equality-defect · candidate





## 28. Set Equality with Both Difference Witnesses

Finite integer sets A=[-2] and B=[]; repeated elements count once. Give (are A and B equal?, A\B, B\A), with increasing difference lists.

View answer and explanation(No, [-2], [])

Equality requires both differences to be empty; equal cardinalities alone are insufficient. The exact result, retaining the requested order, is (No, [-2], []).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r04-set-0028/)

Set · qf-density11-set-equality-defect · candidate





## 29. Set Equality with Both Difference Witnesses

Finite integer sets A=[-3, 0, 3] and B=[0, 3, 4]; repeated elements count once. Give (are A and B equal?, A\B, B\A), with increasing difference lists.

View answer and explanation(No, [-3], [4])

Equality requires both differences to be empty; equal cardinalities alone are insufficient. The exact result, retaining the requested order, is (No, [-3], [4]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r04-set-0029/)

Set · qf-density11-set-equality-defect · candidate





## 30. Set Equality with Both Difference Witnesses

Finite integer sets A=[-4, 0, 4] and B=[0, 4, 5]; repeated elements count once. Give (are A and B equal?, A\B, B\A), with increasing difference lists.

View answer and explanation(No, [-4], [5])

Equality requires both differences to be empty; equal cardinalities alone are insufficient. The exact result, retaining the requested order, is (No, [-4], [5]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r04-set-0030/)

Set · qf-density11-set-equality-defect · candidate





## 31. Set Equality with Both Difference Witnesses

Finite integer sets A=[-5, 0, 5] and B=[0, 5, 6]; repeated elements count once. Give (are A and B equal?, A\B, B\A), with increasing difference lists.

View answer and explanation(No, [-5], [6])

Equality requires both differences to be empty; equal cardinalities alone are insufficient. The exact result, retaining the requested order, is (No, [-5], [6]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r04-set-0031/)

Set · qf-density11-set-equality-defect · candidate





## 32. Set Equality with Both Difference Witnesses

Finite integer sets A=[-6, 0, 6] and B=[0, 6, 7]; repeated elements count once. Give (are A and B equal?, A\B, B\A), with increasing difference lists.

View answer and explanation(No, [-6], [7])

Equality requires both differences to be empty; equal cardinalities alone are insufficient. The exact result, retaining the requested order, is (No, [-6], [7]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r04-set-0032/)

Set · qf-density11-set-equality-defect · candidate





## 33. Set Equality with Both Difference Witnesses

Finite integer sets A=[-7, 0, 7] and B=[0, 7, 8]; repeated elements count once. Give (are A and B equal?, A\B, B\A), with increasing difference lists.

View answer and explanation(No, [-7], [8])

Equality requires both differences to be empty; equal cardinalities alone are insufficient. The exact result, retaining the requested order, is (No, [-7], [8]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r04-set-0033/)

Set · qf-density11-set-equality-defect · candidate





## 34. Set Equality with Both Difference Witnesses

Finite integer sets A=[-8, 0, 8] and B=[0, 8, 9]; repeated elements count once. Give (are A and B equal?, A\B, B\A), with increasing difference lists.

View answer and explanation(No, [-8], [9])

Equality requires both differences to be empty; equal cardinalities alone are insufficient. The exact result, retaining the requested order, is (No, [-8], [9]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r04-set-0034/)

Set · qf-density11-set-equality-defect · candidate





## 35. Set Equality with Both Difference Witnesses

Finite integer sets A=[-9, 0, 9] and B=[0, 9, 10]; repeated elements count once. Give (are A and B equal?, A\B, B\A), with increasing difference lists.

View answer and explanation(No, [-9], [10])

Equality requires both differences to be empty; equal cardinalities alone are insufficient. The exact result, retaining the requested order, is (No, [-9], [10]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r04-set-0035/)

Set · qf-density11-set-equality-defect · candidate





## 36. Set Equality with Both Difference Witnesses

Finite integer sets A=[-10, 0, 10] and B=[0, 10, 11]; repeated elements count once. Give (are A and B equal?, A\B, B\A), with increasing difference lists.

View answer and explanation(No, [-10], [11])

Equality requires both differences to be empty; equal cardinalities alone are insufficient. The exact result, retaining the requested order, is (No, [-10], [11]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r04-set-0036/)

Set · qf-density11-set-equality-defect · candidate





## 37. Set Equality with Both Difference Witnesses

Finite integer sets A=[-11, 0, 11] and B=[0, 11, 12]; repeated elements count once. Give (are A and B equal?, A\B, B\A), with increasing difference lists.

View answer and explanation(No, [-11], [12])

Equality requires both differences to be empty; equal cardinalities alone are insufficient. The exact result, retaining the requested order, is (No, [-11], [12]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r04-set-0037/)

Set · qf-density11-set-equality-defect · candidate





## 38. Set Equality with Both Difference Witnesses

Finite integer sets A=[-12, 0, 12] and B=[0, 12, 13]; repeated elements count once. Give (are A and B equal?, A\B, B\A), with increasing difference lists.

View answer and explanation(No, [-12], [13])

Equality requires both differences to be empty; equal cardinalities alone are insufficient. The exact result, retaining the requested order, is (No, [-12], [13]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r04-set-0038/)

Set · qf-density11-set-equality-defect · candidate





## 39. Set Equality with Both Difference Witnesses

Finite integer sets A=[-13, 0, 13] and B=[0, 13, 14]; repeated elements count once. Give (are A and B equal?, A\B, B\A), with increasing difference lists.

View answer and explanation(No, [-13], [14])

Equality requires both differences to be empty; equal cardinalities alone are insufficient. The exact result, retaining the requested order, is (No, [-13], [14]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r04-set-0039/)

Set · qf-density11-set-equality-defect · candidate





## 40. Set Equality with Both Difference Witnesses

Finite integer sets A=[-14, 0, 14] and B=[0, 14, 15]; repeated elements count once. Give (are A and B equal?, A\B, B\A), with increasing difference lists.

View answer and explanation(No, [-14], [15])

Equality requires both differences to be empty; equal cardinalities alone are insufficient. The exact result, retaining the requested order, is (No, [-14], [15]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r04-set-0040/)

Set · qf-density11-set-equality-defect · candidate





## 41. Set Equality with Both Difference Witnesses

Finite integer sets A=[-15, 0, 15] and B=[0, 15, 16]; repeated elements count once. Give (are A and B equal?, A\B, B\A), with increasing difference lists.

View answer and explanation(No, [-15], [16])

Equality requires both differences to be empty; equal cardinalities alone are insufficient. The exact result, retaining the requested order, is (No, [-15], [16]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r04-set-0041/)

Set · qf-density11-set-equality-defect · candidate





## 42. Set Equality with Both Difference Witnesses

Finite integer sets A=[-16, 0, 16] and B=[0, 16, 17]; repeated elements count once. Give (are A and B equal?, A\B, B\A), with increasing difference lists.

View answer and explanation(No, [-16], [17])

Equality requires both differences to be empty; equal cardinalities alone are insufficient. The exact result, retaining the requested order, is (No, [-16], [17]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r04-set-0042/)

Set · qf-density11-set-equality-defect · candidate





## 43. Set Equality with Both Difference Witnesses

Finite integer sets A=[-17, 0, 17] and B=[0, 17, 18]; repeated elements count once. Give (are A and B equal?, A\B, B\A), with increasing difference lists.

View answer and explanation(No, [-17], [18])

Equality requires both differences to be empty; equal cardinalities alone are insufficient. The exact result, retaining the requested order, is (No, [-17], [18]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r04-set-0043/)

Set · qf-density11-set-equality-defect · candidate





## 44. Set Equality with Both Difference Witnesses

Finite integer sets A=[-18, 0, 18] and B=[0, 18, 19]; repeated elements count once. Give (are A and B equal?, A\B, B\A), with increasing difference lists.

View answer and explanation(No, [-18], [19])

Equality requires both differences to be empty; equal cardinalities alone are insufficient. The exact result, retaining the requested order, is (No, [-18], [19]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r04-set-0044/)

Set · qf-density11-set-equality-defect · candidate





## 45. Set Equality with Both Difference Witnesses

Finite integer sets A=[-19, 0, 19] and B=[0, 19, 20]; repeated elements count once. Give (are A and B equal?, A\B, B\A), with increasing difference lists.

View answer and explanation(No, [-19], [20])

Equality requires both differences to be empty; equal cardinalities alone are insufficient. The exact result, retaining the requested order, is (No, [-19], [20]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r04-set-0045/)

Set · qf-density11-set-equality-defect · candidate





## 46. Set Equality with Both Difference Witnesses

Finite integer sets A=[-20, 0, 20] and B=[0, 20, 21]; repeated elements count once. Give (are A and B equal?, A\B, B\A), with increasing difference lists.

View answer and explanation(No, [-20], [21])

Equality requires both differences to be empty; equal cardinalities alone are insufficient. The exact result, retaining the requested order, is (No, [-20], [21]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r04-set-0046/)

Set · qf-density11-set-equality-defect · candidate





## 47. Set Equality with Both Difference Witnesses

Finite integer sets A=[-21, 0, 21] and B=[0, 21, 22]; repeated elements count once. Give (are A and B equal?, A\B, B\A), with increasing difference lists.

View answer and explanation(No, [-21], [22])

Equality requires both differences to be empty; equal cardinalities alone are insufficient. The exact result, retaining the requested order, is (No, [-21], [22]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r04-set-0047/)

Set · qf-density11-set-equality-defect · candidate





## 48. Set Equality with Both Difference Witnesses

Finite integer sets A=[-22, 0, 22] and B=[0, 22, 23]; repeated elements count once. Give (are A and B equal?, A\B, B\A), with increasing difference lists.

View answer and explanation(No, [-22], [23])

Equality requires both differences to be empty; equal cardinalities alone are insufficient. The exact result, retaining the requested order, is (No, [-22], [23]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r04-set-0048/)

Set · qf-density11-set-equality-defect · candidate





## 49. Set Equality with Both Difference Witnesses

Finite integer sets A=[-23, 0, 23] and B=[0, 23, 24]; repeated elements count once. Give (are A and B equal?, A\B, B\A), with increasing difference lists.

View answer and explanation(No, [-23], [24])

Equality requires both differences to be empty; equal cardinalities alone are insufficient. The exact result, retaining the requested order, is (No, [-23], [24]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r04-set-0049/)

Set · qf-density11-set-equality-defect · candidate





## 50. Set Equality with Both Difference Witnesses

Finite integer sets A=[-24, 0, 24] and B=[0, 24, 25]; repeated elements count once. Give (are A and B equal?, A\B, B\A), with increasing difference lists.

View answer and explanation(No, [-24], [25])

Equality requires both differences to be empty; equal cardinalities alone are insufficient. The exact result, retaining the requested order, is (No, [-24], [25]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r04-set-0050/)

Set · qf-density11-set-equality-defect · candidate





## 51. Partition of an Explicit Candidate Set

Finite integer set A=[]; raw candidates=[] are treated as a set. Give (candidates belonging to A, candidates outside A), each sorted increasingly.

View answer and explanation([], [])

Normalize candidate repetition before membership tests; unlisted extra elements of A do not enter either output part. The exact result, retaining the requested order, is ([], []).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r04-set-0051/)

Set · qf-density11-set-candidate-partition · candidate





## 52. Partition of an Explicit Candidate Set

Finite integer set A=[0]; raw candidates=[0, 1, 1] are treated as a set. Give (candidates belonging to A, candidates outside A), each sorted increasingly.

View answer and explanation([0], [1])

Normalize candidate repetition before membership tests; unlisted extra elements of A do not enter either output part. The exact result, retaining the requested order, is ([0], [1]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r04-set-0052/)

Set · qf-density11-set-candidate-partition · candidate





## 53. Partition of an Explicit Candidate Set

Finite integer set A=[-2, 0]; raw candidates=[-2, 2, 0, 2] are treated as a set. Give (candidates belonging to A, candidates outside A), each sorted increasingly.

View answer and explanation([-2, 0], [2])

Normalize candidate repetition before membership tests; unlisted extra elements of A do not enter either output part. The exact result, retaining the requested order, is ([-2, 0], [2]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r04-set-0053/)

Set · qf-density11-set-candidate-partition · candidate





## 54. Partition of an Explicit Candidate Set

Finite integer set A=[-3, 0]; raw candidates=[-3, 3, 0, 3] are treated as a set. Give (candidates belonging to A, candidates outside A), each sorted increasingly.

View answer and explanation([-3, 0], [3])

Normalize candidate repetition before membership tests; unlisted extra elements of A do not enter either output part. The exact result, retaining the requested order, is ([-3, 0], [3]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r04-set-0054/)

Set · qf-density11-set-candidate-partition · candidate





## 55. Partition of an Explicit Candidate Set

Finite integer set A=[-4, 0]; raw candidates=[-4, 4, 0, 4] are treated as a set. Give (candidates belonging to A, candidates outside A), each sorted increasingly.

View answer and explanation([-4, 0], [4])

Normalize candidate repetition before membership tests; unlisted extra elements of A do not enter either output part. The exact result, retaining the requested order, is ([-4, 0], [4]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r04-set-0055/)

Set · qf-density11-set-candidate-partition · candidate





## 56. Partition of an Explicit Candidate Set

Finite integer set A=[-5, 0]; raw candidates=[-5, 5, 0, 5] are treated as a set. Give (candidates belonging to A, candidates outside A), each sorted increasingly.

View answer and explanation([-5, 0], [5])

Normalize candidate repetition before membership tests; unlisted extra elements of A do not enter either output part. The exact result, retaining the requested order, is ([-5, 0], [5]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r04-set-0056/)

Set · qf-density11-set-candidate-partition · candidate





## 57. Partition of an Explicit Candidate Set

Finite integer set A=[-6, 0]; raw candidates=[-6, 6, 0, 6] are treated as a set. Give (candidates belonging to A, candidates outside A), each sorted increasingly.

View answer and explanation([-6, 0], [6])

Normalize candidate repetition before membership tests; unlisted extra elements of A do not enter either output part. The exact result, retaining the requested order, is ([-6, 0], [6]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r04-set-0057/)

Set · qf-density11-set-candidate-partition · candidate





## 58. Partition of an Explicit Candidate Set

Finite integer set A=[-7, 0]; raw candidates=[-7, 7, 0, 7] are treated as a set. Give (candidates belonging to A, candidates outside A), each sorted increasingly.

View answer and explanation([-7, 0], [7])

Normalize candidate repetition before membership tests; unlisted extra elements of A do not enter either output part. The exact result, retaining the requested order, is ([-7, 0], [7]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r04-set-0058/)

Set · qf-density11-set-candidate-partition · candidate





## 59. Partition of an Explicit Candidate Set

Finite integer set A=[-8, 0]; raw candidates=[-8, 8, 0, 8] are treated as a set. Give (candidates belonging to A, candidates outside A), each sorted increasingly.

View answer and explanation([-8, 0], [8])

Normalize candidate repetition before membership tests; unlisted extra elements of A do not enter either output part. The exact result, retaining the requested order, is ([-8, 0], [8]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r04-set-0059/)

Set · qf-density11-set-candidate-partition · candidate





## 60. Partition of an Explicit Candidate Set

Finite integer set A=[-9, 0]; raw candidates=[-9, 9, 0, 9] are treated as a set. Give (candidates belonging to A, candidates outside A), each sorted increasingly.

View answer and explanation([-9, 0], [9])

Normalize candidate repetition before membership tests; unlisted extra elements of A do not enter either output part. The exact result, retaining the requested order, is ([-9, 0], [9]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r04-set-0060/)

Set · qf-density11-set-candidate-partition · candidate





## 61. Partition of an Explicit Candidate Set

Finite integer set A=[-10, 0]; raw candidates=[-10, 10, 0, 10] are treated as a set. Give (candidates belonging to A, candidates outside A), each sorted increasingly.

View answer and explanation([-10, 0], [10])

Normalize candidate repetition before membership tests; unlisted extra elements of A do not enter either output part. The exact result, retaining the requested order, is ([-10, 0], [10]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r04-set-0061/)

Set · qf-density11-set-candidate-partition · candidate





## 62. Partition of an Explicit Candidate Set

Finite integer set A=[-11, 0]; raw candidates=[-11, 11, 0, 11] are treated as a set. Give (candidates belonging to A, candidates outside A), each sorted increasingly.

View answer and explanation([-11, 0], [11])

Normalize candidate repetition before membership tests; unlisted extra elements of A do not enter either output part. The exact result, retaining the requested order, is ([-11, 0], [11]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r04-set-0062/)

Set · qf-density11-set-candidate-partition · candidate





## 63. Partition of an Explicit Candidate Set

Finite integer set A=[-12, 0]; raw candidates=[-12, 12, 0, 12] are treated as a set. Give (candidates belonging to A, candidates outside A), each sorted increasingly.

View answer and explanation([-12, 0], [12])

Normalize candidate repetition before membership tests; unlisted extra elements of A do not enter either output part. The exact result, retaining the requested order, is ([-12, 0], [12]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r04-set-0063/)

Set · qf-density11-set-candidate-partition · candidate





## 64. Partition of an Explicit Candidate Set

Finite integer set A=[-13, 0]; raw candidates=[-13, 13, 0, 13] are treated as a set. Give (candidates belonging to A, candidates outside A), each sorted increasingly.

View answer and explanation([-13, 0], [13])

Normalize candidate repetition before membership tests; unlisted extra elements of A do not enter either output part. The exact result, retaining the requested order, is ([-13, 0], [13]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r04-set-0064/)

Set · qf-density11-set-candidate-partition · candidate





## 65. Partition of an Explicit Candidate Set

Finite integer set A=[-14, 0]; raw candidates=[-14, 14, 0, 14] are treated as a set. Give (candidates belonging to A, candidates outside A), each sorted increasingly.

View answer and explanation([-14, 0], [14])

Normalize candidate repetition before membership tests; unlisted extra elements of A do not enter either output part. The exact result, retaining the requested order, is ([-14, 0], [14]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r04-set-0065/)

Set · qf-density11-set-candidate-partition · candidate





## 66. Partition of an Explicit Candidate Set

Finite integer set A=[-15, 0]; raw candidates=[-15, 15, 0, 15] are treated as a set. Give (candidates belonging to A, candidates outside A), each sorted increasingly.

View answer and explanation([-15, 0], [15])

Normalize candidate repetition before membership tests; unlisted extra elements of A do not enter either output part. The exact result, retaining the requested order, is ([-15, 0], [15]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r04-set-0066/)

Set · qf-density11-set-candidate-partition · candidate





## 67. Partition of an Explicit Candidate Set

Finite integer set A=[-16, 0]; raw candidates=[-16, 16, 0, 16] are treated as a set. Give (candidates belonging to A, candidates outside A), each sorted increasingly.

View answer and explanation([-16, 0], [16])

Normalize candidate repetition before membership tests; unlisted extra elements of A do not enter either output part. The exact result, retaining the requested order, is ([-16, 0], [16]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r04-set-0067/)

Set · qf-density11-set-candidate-partition · candidate





## 68. Partition of an Explicit Candidate Set

Finite integer set A=[-17, 0]; raw candidates=[-17, 17, 0, 17] are treated as a set. Give (candidates belonging to A, candidates outside A), each sorted increasingly.

View answer and explanation([-17, 0], [17])

Normalize candidate repetition before membership tests; unlisted extra elements of A do not enter either output part. The exact result, retaining the requested order, is ([-17, 0], [17]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r04-set-0068/)

Set · qf-density11-set-candidate-partition · candidate





## 69. Partition of an Explicit Candidate Set

Finite integer set A=[-18, 0]; raw candidates=[-18, 18, 0, 18] are treated as a set. Give (candidates belonging to A, candidates outside A), each sorted increasingly.

View answer and explanation([-18, 0], [18])

Normalize candidate repetition before membership tests; unlisted extra elements of A do not enter either output part. The exact result, retaining the requested order, is ([-18, 0], [18]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r04-set-0069/)

Set · qf-density11-set-candidate-partition · candidate





## 70. Partition of an Explicit Candidate Set

Finite integer set A=[-19, 0]; raw candidates=[-19, 19, 0, 19] are treated as a set. Give (candidates belonging to A, candidates outside A), each sorted increasingly.

View answer and explanation([-19, 0], [19])

Normalize candidate repetition before membership tests; unlisted extra elements of A do not enter either output part. The exact result, retaining the requested order, is ([-19, 0], [19]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r04-set-0070/)

Set · qf-density11-set-candidate-partition · candidate





## 71. Partition of an Explicit Candidate Set

Finite integer set A=[-20, 0]; raw candidates=[-20, 20, 0, 20] are treated as a set. Give (candidates belonging to A, candidates outside A), each sorted increasingly.

View answer and explanation([-20, 0], [20])

Normalize candidate repetition before membership tests; unlisted extra elements of A do not enter either output part. The exact result, retaining the requested order, is ([-20, 0], [20]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r04-set-0071/)

Set · qf-density11-set-candidate-partition · candidate





## 72. Partition of an Explicit Candidate Set

Finite integer set A=[-21, 0]; raw candidates=[-21, 21, 0, 21] are treated as a set. Give (candidates belonging to A, candidates outside A), each sorted increasingly.

View answer and explanation([-21, 0], [21])

Normalize candidate repetition before membership tests; unlisted extra elements of A do not enter either output part. The exact result, retaining the requested order, is ([-21, 0], [21]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r04-set-0072/)

Set · qf-density11-set-candidate-partition · candidate





## 73. Partition of an Explicit Candidate Set

Finite integer set A=[-22, 0]; raw candidates=[-22, 22, 0, 22] are treated as a set. Give (candidates belonging to A, candidates outside A), each sorted increasingly.

View answer and explanation([-22, 0], [22])

Normalize candidate repetition before membership tests; unlisted extra elements of A do not enter either output part. The exact result, retaining the requested order, is ([-22, 0], [22]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r04-set-0073/)

Set · qf-density11-set-candidate-partition · candidate





## 74. Partition of an Explicit Candidate Set

Finite integer set A=[-23, 0]; raw candidates=[-23, 23, 0, 23] are treated as a set. Give (candidates belonging to A, candidates outside A), each sorted increasingly.

View answer and explanation([-23, 0], [23])

Normalize candidate repetition before membership tests; unlisted extra elements of A do not enter either output part. The exact result, retaining the requested order, is ([-23, 0], [23]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r04-set-0074/)

Set · qf-density11-set-candidate-partition · candidate





## 75. Partition of an Explicit Candidate Set

Finite integer set A=[-24, 0]; raw candidates=[-24, 24, 0, 24] are treated as a set. Give (candidates belonging to A, candidates outside A), each sorted increasingly.

View answer and explanation([-24, 0], [24])

Normalize candidate repetition before membership tests; unlisted extra elements of A do not enter either output part. The exact result, retaining the requested order, is ([-24, 0], [24]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r04-set-0075/)

Set · qf-density11-set-candidate-partition · candidate





## 76. Complete Finite Powerset

Finite integer set A=[]. List all subsets, each increasingly ordered; order the outer list by size and then numeric lexicographic order. The empty set’s powerset contains the empty subset.

View answer and explanation[[]]

Branch on including or excluding each distinct element, then apply the declared order; this finite construction is not unrestricted comprehension. The exact result, retaining the requested order, is [[]].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r04-set-0076/)

Set · qf-density11-set-powerset-enumeration · candidate





## 77. Complete Finite Powerset

Finite integer set A=[0]. List all subsets, each increasingly ordered; order the outer list by size and then numeric lexicographic order. The empty set’s powerset contains the empty subset.

View answer and explanation[[], [0]]

Branch on including or excluding each distinct element, then apply the declared order; this finite construction is not unrestricted comprehension. The exact result, retaining the requested order, is [[], [0]].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r04-set-0077/)

Set · qf-density11-set-powerset-enumeration · candidate





## 78. Complete Finite Powerset

Finite integer set A=[-2, 1]. List all subsets, each increasingly ordered; order the outer list by size and then numeric lexicographic order. The empty set’s powerset contains the empty subset.

View answer and explanation[[], [-2], [1], [-2, 1]]

Branch on including or excluding each distinct element, then apply the declared order; this finite construction is not unrestricted comprehension. The exact result, retaining the requested order, is [[], [-2], [1], [-2, 1]].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r04-set-0078/)

Set · qf-density11-set-powerset-enumeration · candidate





## 79. Complete Finite Powerset

Finite integer set A=[-3, 0, 1, 3]. List all subsets, each increasingly ordered; order the outer list by size and then numeric lexicographic order. The empty set’s powerset contains the empty subset.

View answer and explanation[[], [-3], [0], [1], [3], [-3, 0], [-3, 1], [-3, 3], [0, 1], [0, 3], [1, 3], [-3, 0, 1], [-3, 0, 3], [-3, 1, 3], [0, 1, 3], [-3, 0, 1, 3]]

Branch on including or excluding each distinct element, then apply the declared order; this finite construction is not unrestricted comprehension. The exact result, retaining the requested order, is [[], [-3], [0], [1], [3], [-3, 0], [-3, 1], [-3, 3], [0, 1], [0, 3], [1, 3], [-3, 0, 1], [-3, 0, 3], [-3, 1, 3], [0, 1, 3], [-3, 0, 1, 3]].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r04-set-0079/)

Set · qf-density11-set-powerset-enumeration · candidate





## 80. Complete Finite Powerset

Finite integer set A=[-4, 0, 1, 4]. List all subsets, each increasingly ordered; order the outer list by size and then numeric lexicographic order. The empty set’s powerset contains the empty subset.

View answer and explanation[[], [-4], [0], [1], [4], [-4, 0], [-4, 1], [-4, 4], [0, 1], [0, 4], [1, 4], [-4, 0, 1], [-4, 0, 4], [-4, 1, 4], [0, 1, 4], [-4, 0, 1, 4]]

Branch on including or excluding each distinct element, then apply the declared order; this finite construction is not unrestricted comprehension. The exact result, retaining the requested order, is [[], [-4], [0], [1], [4], [-4, 0], [-4, 1], [-4, 4], [0, 1], [0, 4], [1, 4], [-4, 0, 1], [-4, 0, 4], [-4, 1, 4], [0, 1, 4], [-4, 0, 1, 4]].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r04-set-0080/)

Set · qf-density11-set-powerset-enumeration · candidate





## 81. Complete Finite Powerset

Finite integer set A=[-5, 0, 1, 5]. List all subsets, each increasingly ordered; order the outer list by size and then numeric lexicographic order. The empty set’s powerset contains the empty subset.

View answer and explanation[[], [-5], [0], [1], [5], [-5, 0], [-5, 1], [-5, 5], [0, 1], [0, 5], [1, 5], [-5, 0, 1], [-5, 0, 5], [-5, 1, 5], [0, 1, 5], [-5, 0, 1, 5]]

Branch on including or excluding each distinct element, then apply the declared order; this finite construction is not unrestricted comprehension. The exact result, retaining the requested order, is [[], [-5], [0], [1], [5], [-5, 0], [-5, 1], [-5, 5], [0, 1], [0, 5], [1, 5], [-5, 0, 1], [-5, 0, 5], [-5, 1, 5], [0, 1, 5], [-5, 0, 1, 5]].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r04-set-0081/)

Set · qf-density11-set-powerset-enumeration · candidate





## 82. Complete Finite Powerset

Finite integer set A=[-6, 0, 1, 6]. List all subsets, each increasingly ordered; order the outer list by size and then numeric lexicographic order. The empty set’s powerset contains the empty subset.

View answer and explanation[[], [-6], [0], [1], [6], [-6, 0], [-6, 1], [-6, 6], [0, 1], [0, 6], [1, 6], [-6, 0, 1], [-6, 0, 6], [-6, 1, 6], [0, 1, 6], [-6, 0, 1, 6]]

Branch on including or excluding each distinct element, then apply the declared order; this finite construction is not unrestricted comprehension. The exact result, retaining the requested order, is [[], [-6], [0], [1], [6], [-6, 0], [-6, 1], [-6, 6], [0, 1], [0, 6], [1, 6], [-6, 0, 1], [-6, 0, 6], [-6, 1, 6], [0, 1, 6], [-6, 0, 1, 6]].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r04-set-0082/)

Set · qf-density11-set-powerset-enumeration · candidate





## 83. Complete Finite Powerset

Finite integer set A=[-7, 0, 1, 7]. List all subsets, each increasingly ordered; order the outer list by size and then numeric lexicographic order. The empty set’s powerset contains the empty subset.

View answer and explanation[[], [-7], [0], [1], [7], [-7, 0], [-7, 1], [-7, 7], [0, 1], [0, 7], [1, 7], [-7, 0, 1], [-7, 0, 7], [-7, 1, 7], [0, 1, 7], [-7, 0, 1, 7]]

Branch on including or excluding each distinct element, then apply the declared order; this finite construction is not unrestricted comprehension. The exact result, retaining the requested order, is [[], [-7], [0], [1], [7], [-7, 0], [-7, 1], [-7, 7], [0, 1], [0, 7], [1, 7], [-7, 0, 1], [-7, 0, 7], [-7, 1, 7], [0, 1, 7], [-7, 0, 1, 7]].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r04-set-0083/)

Set · qf-density11-set-powerset-enumeration · candidate





## 84. Complete Finite Powerset

Finite integer set A=[-8, 0, 1, 8]. List all subsets, each increasingly ordered; order the outer list by size and then numeric lexicographic order. The empty set’s powerset contains the empty subset.

View answer and explanation[[], [-8], [0], [1], [8], [-8, 0], [-8, 1], [-8, 8], [0, 1], [0, 8], [1, 8], [-8, 0, 1], [-8, 0, 8], [-8, 1, 8], [0, 1, 8], [-8, 0, 1, 8]]

Branch on including or excluding each distinct element, then apply the declared order; this finite construction is not unrestricted comprehension. The exact result, retaining the requested order, is [[], [-8], [0], [1], [8], [-8, 0], [-8, 1], [-8, 8], [0, 1], [0, 8], [1, 8], [-8, 0, 1], [-8, 0, 8], [-8, 1, 8], [0, 1, 8], [-8, 0, 1, 8]].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r04-set-0084/)

Set · qf-density11-set-powerset-enumeration · candidate





## 85. Complete Finite Powerset

Finite integer set A=[-9, 0, 1, 9]. List all subsets, each increasingly ordered; order the outer list by size and then numeric lexicographic order. The empty set’s powerset contains the empty subset.

View answer and explanation[[], [-9], [0], [1], [9], [-9, 0], [-9, 1], [-9, 9], [0, 1], [0, 9], [1, 9], [-9, 0, 1], [-9, 0, 9], [-9, 1, 9], [0, 1, 9], [-9, 0, 1, 9]]

Branch on including or excluding each distinct element, then apply the declared order; this finite construction is not unrestricted comprehension. The exact result, retaining the requested order, is [[], [-9], [0], [1], [9], [-9, 0], [-9, 1], [-9, 9], [0, 1], [0, 9], [1, 9], [-9, 0, 1], [-9, 0, 9], [-9, 1, 9], [0, 1, 9], [-9, 0, 1, 9]].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r04-set-0085/)

Set · qf-density11-set-powerset-enumeration · candidate





## 86. Complete Finite Powerset

Finite integer set A=[-10, 0, 1, 10]. List all subsets, each increasingly ordered; order the outer list by size and then numeric lexicographic order. The empty set’s powerset contains the empty subset.

View answer and explanation[[], [-10], [0], [1], [10], [-10, 0], [-10, 1], [-10, 10], [0, 1], [0, 10], [1, 10], [-10, 0, 1], [-10, 0, 10], [-10, 1, 10], [0, 1, 10], [-10, 0, 1, 10]]

Branch on including or excluding each distinct element, then apply the declared order; this finite construction is not unrestricted comprehension. The exact result, retaining the requested order, is [[], [-10], [0], [1], [10], [-10, 0], [-10, 1], [-10, 10], [0, 1], [0, 10], [1, 10], [-10, 0, 1], [-10, 0, 10], [-10, 1, 10], [0, 1, 10], [-10, 0, 1, 10]].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r04-set-0086/)

Set · qf-density11-set-powerset-enumeration · candidate





## 87. Complete Finite Powerset

Finite integer set A=[-11, 0, 1, 11]. List all subsets, each increasingly ordered; order the outer list by size and then numeric lexicographic order. The empty set’s powerset contains the empty subset.

View answer and explanation[[], [-11], [0], [1], [11], [-11, 0], [-11, 1], [-11, 11], [0, 1], [0, 11], [1, 11], [-11, 0, 1], [-11, 0, 11], [-11, 1, 11], [0, 1, 11], [-11, 0, 1, 11]]

Branch on including or excluding each distinct element, then apply the declared order; this finite construction is not unrestricted comprehension. The exact result, retaining the requested order, is [[], [-11], [0], [1], [11], [-11, 0], [-11, 1], [-11, 11], [0, 1], [0, 11], [1, 11], [-11, 0, 1], [-11, 0, 11], [-11, 1, 11], [0, 1, 11], [-11, 0, 1, 11]].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r04-set-0087/)

Set · qf-density11-set-powerset-enumeration · candidate





## 88. Complete Finite Powerset

Finite integer set A=[-12, 0, 1, 12]. List all subsets, each increasingly ordered; order the outer list by size and then numeric lexicographic order. The empty set’s powerset contains the empty subset.

View answer and explanation[[], [-12], [0], [1], [12], [-12, 0], [-12, 1], [-12, 12], [0, 1], [0, 12], [1, 12], [-12, 0, 1], [-12, 0, 12], [-12, 1, 12], [0, 1, 12], [-12, 0, 1, 12]]

Branch on including or excluding each distinct element, then apply the declared order; this finite construction is not unrestricted comprehension. The exact result, retaining the requested order, is [[], [-12], [0], [1], [12], [-12, 0], [-12, 1], [-12, 12], [0, 1], [0, 12], [1, 12], [-12, 0, 1], [-12, 0, 12], [-12, 1, 12], [0, 1, 12], [-12, 0, 1, 12]].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r04-set-0088/)

Set · qf-density11-set-powerset-enumeration · candidate





## 89. Complete Finite Powerset

Finite integer set A=[-13, 0, 1, 13]. List all subsets, each increasingly ordered; order the outer list by size and then numeric lexicographic order. The empty set’s powerset contains the empty subset.

View answer and explanation[[], [-13], [0], [1], [13], [-13, 0], [-13, 1], [-13, 13], [0, 1], [0, 13], [1, 13], [-13, 0, 1], [-13, 0, 13], [-13, 1, 13], [0, 1, 13], [-13, 0, 1, 13]]

Branch on including or excluding each distinct element, then apply the declared order; this finite construction is not unrestricted comprehension. The exact result, retaining the requested order, is [[], [-13], [0], [1], [13], [-13, 0], [-13, 1], [-13, 13], [0, 1], [0, 13], [1, 13], [-13, 0, 1], [-13, 0, 13], [-13, 1, 13], [0, 1, 13], [-13, 0, 1, 13]].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r04-set-0089/)

Set · qf-density11-set-powerset-enumeration · candidate





## 90. Complete Finite Powerset

Finite integer set A=[-14, 0, 1, 14]. List all subsets, each increasingly ordered; order the outer list by size and then numeric lexicographic order. The empty set’s powerset contains the empty subset.

View answer and explanation[[], [-14], [0], [1], [14], [-14, 0], [-14, 1], [-14, 14], [0, 1], [0, 14], [1, 14], [-14, 0, 1], [-14, 0, 14], [-14, 1, 14], [0, 1, 14], [-14, 0, 1, 14]]

Branch on including or excluding each distinct element, then apply the declared order; this finite construction is not unrestricted comprehension. The exact result, retaining the requested order, is [[], [-14], [0], [1], [14], [-14, 0], [-14, 1], [-14, 14], [0, 1], [0, 14], [1, 14], [-14, 0, 1], [-14, 0, 14], [-14, 1, 14], [0, 1, 14], [-14, 0, 1, 14]].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r04-set-0090/)

Set · qf-density11-set-powerset-enumeration · candidate





## 91. Complete Finite Powerset

Finite integer set A=[-15, 0, 1, 15]. List all subsets, each increasingly ordered; order the outer list by size and then numeric lexicographic order. The empty set’s powerset contains the empty subset.

View answer and explanation[[], [-15], [0], [1], [15], [-15, 0], [-15, 1], [-15, 15], [0, 1], [0, 15], [1, 15], [-15, 0, 1], [-15, 0, 15], [-15, 1, 15], [0, 1, 15], [-15, 0, 1, 15]]

Branch on including or excluding each distinct element, then apply the declared order; this finite construction is not unrestricted comprehension. The exact result, retaining the requested order, is [[], [-15], [0], [1], [15], [-15, 0], [-15, 1], [-15, 15], [0, 1], [0, 15], [1, 15], [-15, 0, 1], [-15, 0, 15], [-15, 1, 15], [0, 1, 15], [-15, 0, 1, 15]].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r04-set-0091/)

Set · qf-density11-set-powerset-enumeration · candidate





## 92. Complete Finite Powerset

Finite integer set A=[-16, 0, 1, 16]. List all subsets, each increasingly ordered; order the outer list by size and then numeric lexicographic order. The empty set’s powerset contains the empty subset.

View answer and explanation[[], [-16], [0], [1], [16], [-16, 0], [-16, 1], [-16, 16], [0, 1], [0, 16], [1, 16], [-16, 0, 1], [-16, 0, 16], [-16, 1, 16], [0, 1, 16], [-16, 0, 1, 16]]

Branch on including or excluding each distinct element, then apply the declared order; this finite construction is not unrestricted comprehension. The exact result, retaining the requested order, is [[], [-16], [0], [1], [16], [-16, 0], [-16, 1], [-16, 16], [0, 1], [0, 16], [1, 16], [-16, 0, 1], [-16, 0, 16], [-16, 1, 16], [0, 1, 16], [-16, 0, 1, 16]].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r04-set-0092/)

Set · qf-density11-set-powerset-enumeration · candidate





## 93. Complete Finite Powerset

Finite integer set A=[-17, 0, 1, 17]. List all subsets, each increasingly ordered; order the outer list by size and then numeric lexicographic order. The empty set’s powerset contains the empty subset.

View answer and explanation[[], [-17], [0], [1], [17], [-17, 0], [-17, 1], [-17, 17], [0, 1], [0, 17], [1, 17], [-17, 0, 1], [-17, 0, 17], [-17, 1, 17], [0, 1, 17], [-17, 0, 1, 17]]

Branch on including or excluding each distinct element, then apply the declared order; this finite construction is not unrestricted comprehension. The exact result, retaining the requested order, is [[], [-17], [0], [1], [17], [-17, 0], [-17, 1], [-17, 17], [0, 1], [0, 17], [1, 17], [-17, 0, 1], [-17, 0, 17], [-17, 1, 17], [0, 1, 17], [-17, 0, 1, 17]].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r04-set-0093/)

Set · qf-density11-set-powerset-enumeration · candidate





## 94. Complete Finite Powerset

Finite integer set A=[-18, 0, 1, 18]. List all subsets, each increasingly ordered; order the outer list by size and then numeric lexicographic order. The empty set’s powerset contains the empty subset.

View answer and explanation[[], [-18], [0], [1], [18], [-18, 0], [-18, 1], [-18, 18], [0, 1], [0, 18], [1, 18], [-18, 0, 1], [-18, 0, 18], [-18, 1, 18], [0, 1, 18], [-18, 0, 1, 18]]

Branch on including or excluding each distinct element, then apply the declared order; this finite construction is not unrestricted comprehension. The exact result, retaining the requested order, is [[], [-18], [0], [1], [18], [-18, 0], [-18, 1], [-18, 18], [0, 1], [0, 18], [1, 18], [-18, 0, 1], [-18, 0, 18], [-18, 1, 18], [0, 1, 18], [-18, 0, 1, 18]].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r04-set-0094/)

Set · qf-density11-set-powerset-enumeration · candidate





## 95. Complete Finite Powerset

Finite integer set A=[-19, 0, 1, 19]. List all subsets, each increasingly ordered; order the outer list by size and then numeric lexicographic order. The empty set’s powerset contains the empty subset.

View answer and explanation[[], [-19], [0], [1], [19], [-19, 0], [-19, 1], [-19, 19], [0, 1], [0, 19], [1, 19], [-19, 0, 1], [-19, 0, 19], [-19, 1, 19], [0, 1, 19], [-19, 0, 1, 19]]

Branch on including or excluding each distinct element, then apply the declared order; this finite construction is not unrestricted comprehension. The exact result, retaining the requested order, is [[], [-19], [0], [1], [19], [-19, 0], [-19, 1], [-19, 19], [0, 1], [0, 19], [1, 19], [-19, 0, 1], [-19, 0, 19], [-19, 1, 19], [0, 1, 19], [-19, 0, 1, 19]].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r04-set-0095/)

Set · qf-density11-set-powerset-enumeration · candidate





## 96. Complete Finite Powerset

Finite integer set A=[-20, 0, 1, 20]. List all subsets, each increasingly ordered; order the outer list by size and then numeric lexicographic order. The empty set’s powerset contains the empty subset.

View answer and explanation[[], [-20], [0], [1], [20], [-20, 0], [-20, 1], [-20, 20], [0, 1], [0, 20], [1, 20], [-20, 0, 1], [-20, 0, 20], [-20, 1, 20], [0, 1, 20], [-20, 0, 1, 20]]

Branch on including or excluding each distinct element, then apply the declared order; this finite construction is not unrestricted comprehension. The exact result, retaining the requested order, is [[], [-20], [0], [1], [20], [-20, 0], [-20, 1], [-20, 20], [0, 1], [0, 20], [1, 20], [-20, 0, 1], [-20, 0, 20], [-20, 1, 20], [0, 1, 20], [-20, 0, 1, 20]].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r04-set-0096/)

Set · qf-density11-set-powerset-enumeration · candidate





## 97. Complete Finite Powerset

Finite integer set A=[-21, 0, 1, 21]. List all subsets, each increasingly ordered; order the outer list by size and then numeric lexicographic order. The empty set’s powerset contains the empty subset.

View answer and explanation[[], [-21], [0], [1], [21], [-21, 0], [-21, 1], [-21, 21], [0, 1], [0, 21], [1, 21], [-21, 0, 1], [-21, 0, 21], [-21, 1, 21], [0, 1, 21], [-21, 0, 1, 21]]

Branch on including or excluding each distinct element, then apply the declared order; this finite construction is not unrestricted comprehension. The exact result, retaining the requested order, is [[], [-21], [0], [1], [21], [-21, 0], [-21, 1], [-21, 21], [0, 1], [0, 21], [1, 21], [-21, 0, 1], [-21, 0, 21], [-21, 1, 21], [0, 1, 21], [-21, 0, 1, 21]].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r04-set-0097/)

Set · qf-density11-set-powerset-enumeration · candidate





## 98. Complete Finite Powerset

Finite integer set A=[-22, 0, 1, 22]. List all subsets, each increasingly ordered; order the outer list by size and then numeric lexicographic order. The empty set’s powerset contains the empty subset.

View answer and explanation[[], [-22], [0], [1], [22], [-22, 0], [-22, 1], [-22, 22], [0, 1], [0, 22], [1, 22], [-22, 0, 1], [-22, 0, 22], [-22, 1, 22], [0, 1, 22], [-22, 0, 1, 22]]

Branch on including or excluding each distinct element, then apply the declared order; this finite construction is not unrestricted comprehension. The exact result, retaining the requested order, is [[], [-22], [0], [1], [22], [-22, 0], [-22, 1], [-22, 22], [0, 1], [0, 22], [1, 22], [-22, 0, 1], [-22, 0, 22], [-22, 1, 22], [0, 1, 22], [-22, 0, 1, 22]].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r04-set-0098/)

Set · qf-density11-set-powerset-enumeration · candidate





## 99. Complete Finite Powerset

Finite integer set A=[-23, 0, 1, 23]. List all subsets, each increasingly ordered; order the outer list by size and then numeric lexicographic order. The empty set’s powerset contains the empty subset.

View answer and explanation[[], [-23], [0], [1], [23], [-23, 0], [-23, 1], [-23, 23], [0, 1], [0, 23], [1, 23], [-23, 0, 1], [-23, 0, 23], [-23, 1, 23], [0, 1, 23], [-23, 0, 1, 23]]

Branch on including or excluding each distinct element, then apply the declared order; this finite construction is not unrestricted comprehension. The exact result, retaining the requested order, is [[], [-23], [0], [1], [23], [-23, 0], [-23, 1], [-23, 23], [0, 1], [0, 23], [1, 23], [-23, 0, 1], [-23, 0, 23], [-23, 1, 23], [0, 1, 23], [-23, 0, 1, 23]].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r04-set-0099/)

Set · qf-density11-set-powerset-enumeration · candidate





## 100. Complete Finite Powerset

Finite integer set A=[-24, 0, 1, 24]. List all subsets, each increasingly ordered; order the outer list by size and then numeric lexicographic order. The empty set’s powerset contains the empty subset.

View answer and explanation[[], [-24], [0], [1], [24], [-24, 0], [-24, 1], [-24, 24], [0, 1], [0, 24], [1, 24], [-24, 0, 1], [-24, 0, 24], [-24, 1, 24], [0, 1, 24], [-24, 0, 1, 24]]

Branch on including or excluding each distinct element, then apply the declared order; this finite construction is not unrestricted comprehension. The exact result, retaining the requested order, is [[], [-24], [0], [1], [24], [-24, 0], [-24, 1], [-24, 24], [0, 1], [0, 24], [1, 24], [-24, 0, 1], [-24, 0, 24], [-24, 1, 24], [0, 1, 24], [-24, 0, 1, 24]].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261011-r04-set-0100/)

Set · qf-density11-set-powerset-enumeration · candidate



[Canonical question JSONL](https://ocme.evemisslab.com/data/questions/batches/batch-20261008-advance-20261011-r04-set-foundations/questions.jsonl) · [English JSONL](https://ocme.evemisslab.com/data/i18n/questions/batch-20261008-advance-20261011-r04-set-foundations.en.jsonl) · [Validation report](https://ocme.evemisslab.com/data/questions/batches/batch-20261008-advance-20261011-r04-set-foundations/validation-report.json)