# Typed Membership and Finite Predicate Extensions

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-20261012-r04-set-membership-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. Typed Membership Queries and Truth Values

The finite type is Fin(0)=∅. Complete characteristic table bits=[] uses 0/1 membership from index 0; ordered queries=[-1, 0, 1]. Give (type-eligibility mask, membership truth values or ill-typed).

View answer and explanation([No, No, No], [ill-typed, ill-typed, ill-typed])

Check typing before reading the complete characteristic table; an outside-type query is not a False membership value. Finite membership does not constitute all of set theory. The exact result, retaining the requested order, is ([No, No, No], [ill-typed, ill-typed, ill-typed]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r04-set-membership-0001/)

Set membership · qf-density12-membership-indexed-predicate-values · candidate





## 2. Typed Membership Queries and Truth Values

The finite type is Fin(6)={0,…,5}. Complete characteristic table bits=[0, 0, 0, 1, 1, 1] uses 0/1 membership from index 0; ordered queries=[3, 101, 102]. Give (type-eligibility mask, membership truth values or ill-typed).

View answer and explanation([Yes, No, No], [True, ill-typed, ill-typed])

Check typing before reading the complete characteristic table; an outside-type query is not a False membership value. Finite membership does not constitute all of set theory. The exact result, retaining the requested order, is ([Yes, No, No], [True, ill-typed, ill-typed]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r04-set-membership-0002/)

Set membership · qf-density12-membership-indexed-predicate-values · candidate





## 3. Typed Membership Queries and Truth Values

The finite type is Fin(6)={0,…,5}. Complete characteristic table bits=[0, 0, 0, 1, 1, 1] uses 0/1 membership from index 0; ordered queries=[0, 1, 102]. Give (type-eligibility mask, membership truth values or ill-typed).

View answer and explanation([Yes, Yes, No], [False, False, ill-typed])

Check typing before reading the complete characteristic table; an outside-type query is not a False membership value. Finite membership does not constitute all of set theory. The exact result, retaining the requested order, is ([Yes, Yes, No], [False, False, ill-typed]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r04-set-membership-0003/)

Set membership · qf-density12-membership-indexed-predicate-values · candidate





## 4. Typed Membership Queries and Truth Values

The finite type is Fin(6)={0,…,5}. Complete characteristic table bits=[0, 0, 0, 1, 1, 1] uses 0/1 membership from index 0; ordered queries=[3, 1, 102]. Give (type-eligibility mask, membership truth values or ill-typed).

View answer and explanation([Yes, Yes, No], [True, False, ill-typed])

Check typing before reading the complete characteristic table; an outside-type query is not a False membership value. Finite membership does not constitute all of set theory. The exact result, retaining the requested order, is ([Yes, Yes, No], [True, False, ill-typed]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r04-set-membership-0004/)

Set membership · qf-density12-membership-indexed-predicate-values · candidate





## 5. Typed Membership Queries and Truth Values

The finite type is Fin(6)={0,…,5}. Complete characteristic table bits=[0, 0, 0, 1, 1, 1] uses 0/1 membership from index 0; ordered queries=[100, 4, 102]. Give (type-eligibility mask, membership truth values or ill-typed).

View answer and explanation([No, Yes, No], [ill-typed, True, ill-typed])

Check typing before reading the complete characteristic table; an outside-type query is not a False membership value. Finite membership does not constitute all of set theory. The exact result, retaining the requested order, is ([No, Yes, No], [ill-typed, True, ill-typed]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r04-set-membership-0005/)

Set membership · qf-density12-membership-indexed-predicate-values · candidate





## 6. Typed Membership Queries and Truth Values

The finite type is Fin(6)={0,…,5}. Complete characteristic table bits=[0, 0, 0, 1, 1, 1] uses 0/1 membership from index 0; ordered queries=[0, 4, 102]. Give (type-eligibility mask, membership truth values or ill-typed).

View answer and explanation([Yes, Yes, No], [False, True, ill-typed])

Check typing before reading the complete characteristic table; an outside-type query is not a False membership value. Finite membership does not constitute all of set theory. The exact result, retaining the requested order, is ([Yes, Yes, No], [False, True, ill-typed]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r04-set-membership-0006/)

Set membership · qf-density12-membership-indexed-predicate-values · candidate





## 7. Typed Membership Queries and Truth Values

The finite type is Fin(6)={0,…,5}. Complete characteristic table bits=[0, 0, 0, 1, 1, 1] uses 0/1 membership from index 0; ordered queries=[3, 4, 102]. Give (type-eligibility mask, membership truth values or ill-typed).

View answer and explanation([Yes, Yes, No], [True, True, ill-typed])

Check typing before reading the complete characteristic table; an outside-type query is not a False membership value. Finite membership does not constitute all of set theory. The exact result, retaining the requested order, is ([Yes, Yes, No], [True, True, ill-typed]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r04-set-membership-0007/)

Set membership · qf-density12-membership-indexed-predicate-values · candidate





## 8. Typed Membership Queries and Truth Values

The finite type is Fin(6)={0,…,5}. Complete characteristic table bits=[0, 0, 0, 1, 1, 1] uses 0/1 membership from index 0; ordered queries=[100, 101, 2]. Give (type-eligibility mask, membership truth values or ill-typed).

View answer and explanation([No, No, Yes], [ill-typed, ill-typed, False])

Check typing before reading the complete characteristic table; an outside-type query is not a False membership value. Finite membership does not constitute all of set theory. The exact result, retaining the requested order, is ([No, No, Yes], [ill-typed, ill-typed, False]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r04-set-membership-0008/)

Set membership · qf-density12-membership-indexed-predicate-values · candidate





## 9. Typed Membership Queries and Truth Values

The finite type is Fin(6)={0,…,5}. Complete characteristic table bits=[0, 0, 0, 1, 1, 1] uses 0/1 membership from index 0; ordered queries=[0, 101, 2]. Give (type-eligibility mask, membership truth values or ill-typed).

View answer and explanation([Yes, No, Yes], [False, ill-typed, False])

Check typing before reading the complete characteristic table; an outside-type query is not a False membership value. Finite membership does not constitute all of set theory. The exact result, retaining the requested order, is ([Yes, No, Yes], [False, ill-typed, False]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r04-set-membership-0009/)

Set membership · qf-density12-membership-indexed-predicate-values · candidate





## 10. Typed Membership Queries and Truth Values

The finite type is Fin(6)={0,…,5}. Complete characteristic table bits=[0, 0, 0, 1, 1, 1] uses 0/1 membership from index 0; ordered queries=[3, 101, 2]. Give (type-eligibility mask, membership truth values or ill-typed).

View answer and explanation([Yes, No, Yes], [True, ill-typed, False])

Check typing before reading the complete characteristic table; an outside-type query is not a False membership value. Finite membership does not constitute all of set theory. The exact result, retaining the requested order, is ([Yes, No, Yes], [True, ill-typed, False]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r04-set-membership-0010/)

Set membership · qf-density12-membership-indexed-predicate-values · candidate





## 11. Typed Membership Queries and Truth Values

The finite type is Fin(6)={0,…,5}. Complete characteristic table bits=[0, 0, 0, 1, 1, 1] uses 0/1 membership from index 0; ordered queries=[100, 1, 2]. Give (type-eligibility mask, membership truth values or ill-typed).

View answer and explanation([No, Yes, Yes], [ill-typed, False, False])

Check typing before reading the complete characteristic table; an outside-type query is not a False membership value. Finite membership does not constitute all of set theory. The exact result, retaining the requested order, is ([No, Yes, Yes], [ill-typed, False, False]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r04-set-membership-0011/)

Set membership · qf-density12-membership-indexed-predicate-values · candidate





## 12. Typed Membership Queries and Truth Values

The finite type is Fin(6)={0,…,5}. Complete characteristic table bits=[0, 0, 0, 1, 1, 1] uses 0/1 membership from index 0; ordered queries=[0, 1, 2]. Give (type-eligibility mask, membership truth values or ill-typed).

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

Check typing before reading the complete characteristic table; an outside-type query is not a False membership value. Finite membership does not constitute all of set theory. The exact result, retaining the requested order, is ([Yes, Yes, Yes], [False, False, False]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r04-set-membership-0012/)

Set membership · qf-density12-membership-indexed-predicate-values · candidate





## 13. Typed Membership Queries and Truth Values

The finite type is Fin(6)={0,…,5}. Complete characteristic table bits=[0, 0, 0, 1, 1, 1] uses 0/1 membership from index 0; ordered queries=[3, 1, 2]. Give (type-eligibility mask, membership truth values or ill-typed).

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

Check typing before reading the complete characteristic table; an outside-type query is not a False membership value. Finite membership does not constitute all of set theory. The exact result, retaining the requested order, is ([Yes, Yes, Yes], [True, False, False]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r04-set-membership-0013/)

Set membership · qf-density12-membership-indexed-predicate-values · candidate





## 14. Typed Membership Queries and Truth Values

The finite type is Fin(6)={0,…,5}. Complete characteristic table bits=[0, 0, 0, 1, 1, 1] uses 0/1 membership from index 0; ordered queries=[100, 4, 2]. Give (type-eligibility mask, membership truth values or ill-typed).

View answer and explanation([No, Yes, Yes], [ill-typed, True, False])

Check typing before reading the complete characteristic table; an outside-type query is not a False membership value. Finite membership does not constitute all of set theory. The exact result, retaining the requested order, is ([No, Yes, Yes], [ill-typed, True, False]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r04-set-membership-0014/)

Set membership · qf-density12-membership-indexed-predicate-values · candidate





## 15. Typed Membership Queries and Truth Values

The finite type is Fin(6)={0,…,5}. Complete characteristic table bits=[0, 0, 0, 1, 1, 1] uses 0/1 membership from index 0; ordered queries=[0, 4, 2]. Give (type-eligibility mask, membership truth values or ill-typed).

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

Check typing before reading the complete characteristic table; an outside-type query is not a False membership value. Finite membership does not constitute all of set theory. The exact result, retaining the requested order, is ([Yes, Yes, Yes], [False, True, False]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r04-set-membership-0015/)

Set membership · qf-density12-membership-indexed-predicate-values · candidate





## 16. Typed Membership Queries and Truth Values

The finite type is Fin(6)={0,…,5}. Complete characteristic table bits=[0, 0, 0, 1, 1, 1] uses 0/1 membership from index 0; ordered queries=[3, 4, 2]. Give (type-eligibility mask, membership truth values or ill-typed).

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

Check typing before reading the complete characteristic table; an outside-type query is not a False membership value. Finite membership does not constitute all of set theory. The exact result, retaining the requested order, is ([Yes, Yes, Yes], [True, True, False]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r04-set-membership-0016/)

Set membership · qf-density12-membership-indexed-predicate-values · candidate





## 17. Typed Membership Queries and Truth Values

The finite type is Fin(6)={0,…,5}. Complete characteristic table bits=[0, 0, 0, 1, 1, 1] uses 0/1 membership from index 0; ordered queries=[100, 101, 5]. Give (type-eligibility mask, membership truth values or ill-typed).

View answer and explanation([No, No, Yes], [ill-typed, ill-typed, True])

Check typing before reading the complete characteristic table; an outside-type query is not a False membership value. Finite membership does not constitute all of set theory. The exact result, retaining the requested order, is ([No, No, Yes], [ill-typed, ill-typed, True]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r04-set-membership-0017/)

Set membership · qf-density12-membership-indexed-predicate-values · candidate





## 18. Typed Membership Queries and Truth Values

The finite type is Fin(6)={0,…,5}. Complete characteristic table bits=[0, 0, 0, 1, 1, 1] uses 0/1 membership from index 0; ordered queries=[0, 101, 5]. Give (type-eligibility mask, membership truth values or ill-typed).

View answer and explanation([Yes, No, Yes], [False, ill-typed, True])

Check typing before reading the complete characteristic table; an outside-type query is not a False membership value. Finite membership does not constitute all of set theory. The exact result, retaining the requested order, is ([Yes, No, Yes], [False, ill-typed, True]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r04-set-membership-0018/)

Set membership · qf-density12-membership-indexed-predicate-values · candidate





## 19. Typed Membership Queries and Truth Values

The finite type is Fin(6)={0,…,5}. Complete characteristic table bits=[0, 0, 0, 1, 1, 1] uses 0/1 membership from index 0; ordered queries=[3, 101, 5]. Give (type-eligibility mask, membership truth values or ill-typed).

View answer and explanation([Yes, No, Yes], [True, ill-typed, True])

Check typing before reading the complete characteristic table; an outside-type query is not a False membership value. Finite membership does not constitute all of set theory. The exact result, retaining the requested order, is ([Yes, No, Yes], [True, ill-typed, True]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r04-set-membership-0019/)

Set membership · qf-density12-membership-indexed-predicate-values · candidate





## 20. Typed Membership Queries and Truth Values

The finite type is Fin(6)={0,…,5}. Complete characteristic table bits=[0, 0, 0, 1, 1, 1] uses 0/1 membership from index 0; ordered queries=[100, 1, 5]. Give (type-eligibility mask, membership truth values or ill-typed).

View answer and explanation([No, Yes, Yes], [ill-typed, False, True])

Check typing before reading the complete characteristic table; an outside-type query is not a False membership value. Finite membership does not constitute all of set theory. The exact result, retaining the requested order, is ([No, Yes, Yes], [ill-typed, False, True]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r04-set-membership-0020/)

Set membership · qf-density12-membership-indexed-predicate-values · candidate





## 21. Typed Membership Queries and Truth Values

The finite type is Fin(6)={0,…,5}. Complete characteristic table bits=[0, 0, 0, 1, 1, 1] uses 0/1 membership from index 0; ordered queries=[0, 1, 5]. Give (type-eligibility mask, membership truth values or ill-typed).

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

Check typing before reading the complete characteristic table; an outside-type query is not a False membership value. Finite membership does not constitute all of set theory. The exact result, retaining the requested order, is ([Yes, Yes, Yes], [False, False, True]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r04-set-membership-0021/)

Set membership · qf-density12-membership-indexed-predicate-values · candidate





## 22. Typed Membership Queries and Truth Values

The finite type is Fin(6)={0,…,5}. Complete characteristic table bits=[0, 0, 0, 1, 1, 1] uses 0/1 membership from index 0; ordered queries=[3, 1, 5]. Give (type-eligibility mask, membership truth values or ill-typed).

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

Check typing before reading the complete characteristic table; an outside-type query is not a False membership value. Finite membership does not constitute all of set theory. The exact result, retaining the requested order, is ([Yes, Yes, Yes], [True, False, True]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r04-set-membership-0022/)

Set membership · qf-density12-membership-indexed-predicate-values · candidate





## 23. Typed Membership Queries and Truth Values

The finite type is Fin(6)={0,…,5}. Complete characteristic table bits=[0, 0, 0, 1, 1, 1] uses 0/1 membership from index 0; ordered queries=[100, 4, 5]. Give (type-eligibility mask, membership truth values or ill-typed).

View answer and explanation([No, Yes, Yes], [ill-typed, True, True])

Check typing before reading the complete characteristic table; an outside-type query is not a False membership value. Finite membership does not constitute all of set theory. The exact result, retaining the requested order, is ([No, Yes, Yes], [ill-typed, True, True]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r04-set-membership-0023/)

Set membership · qf-density12-membership-indexed-predicate-values · candidate





## 24. Typed Membership Queries and Truth Values

The finite type is Fin(6)={0,…,5}. Complete characteristic table bits=[0, 0, 0, 1, 1, 1] uses 0/1 membership from index 0; ordered queries=[0, 4, 5]. Give (type-eligibility mask, membership truth values or ill-typed).

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

Check typing before reading the complete characteristic table; an outside-type query is not a False membership value. Finite membership does not constitute all of set theory. The exact result, retaining the requested order, is ([Yes, Yes, Yes], [False, True, True]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r04-set-membership-0024/)

Set membership · qf-density12-membership-indexed-predicate-values · candidate





## 25. Typed Membership Queries and Truth Values

The finite type is Fin(1)={0,…,0}. Complete characteristic table bits=[1] uses 0/1 membership from index 0; ordered queries=[0, 0, 0]. Give (type-eligibility mask, membership truth values or ill-typed).

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

Check typing before reading the complete characteristic table; an outside-type query is not a False membership value. Finite membership does not constitute all of set theory. The exact result, retaining the requested order, is ([Yes, Yes, Yes], [True, True, True]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r04-set-membership-0025/)

Set membership · qf-density12-membership-indexed-predicate-values · candidate





## 26. Original Membership and Set Insertion

Finite integer set A=[], x=0. Give (original truth of x∈A, sorted deduplicated A∪{x}).

View answer and explanation(False, [0])

Check original membership before inserting once; an existing element is not duplicated, and the finite operation is not a new general set-theory proof. The exact result, retaining the requested order, is (False, [0]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r04-set-membership-0026/)

Set membership · qf-density12-membership-set-insertion-delta · candidate





## 27. Original Membership and Set Insertion

Finite integer set A=[0], x=0. Give (original truth of x∈A, sorted deduplicated A∪{x}).

View answer and explanation(True, [0])

Check original membership before inserting once; an existing element is not duplicated, and the finite operation is not a new general set-theory proof. The exact result, retaining the requested order, is (True, [0]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r04-set-membership-0027/)

Set membership · qf-density12-membership-set-insertion-delta · candidate





## 28. Original Membership and Set Insertion

Finite integer set A=[-2, 0], x=2. Give (original truth of x∈A, sorted deduplicated A∪{x}).

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

Check original membership before inserting once; an existing element is not duplicated, and the finite operation is not a new general set-theory proof. The exact result, retaining the requested order, is (False, [-2, 0, 2]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r04-set-membership-0028/)

Set membership · qf-density12-membership-set-insertion-delta · candidate





## 29. Original Membership and Set Insertion

Finite integer set A=[-3, 0], x=-3. Give (original truth of x∈A, sorted deduplicated A∪{x}).

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

Check original membership before inserting once; an existing element is not duplicated, and the finite operation is not a new general set-theory proof. The exact result, retaining the requested order, is (True, [-3, 0]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r04-set-membership-0029/)

Set membership · qf-density12-membership-set-insertion-delta · candidate





## 30. Original Membership and Set Insertion

Finite integer set A=[-4, 0], x=4. Give (original truth of x∈A, sorted deduplicated A∪{x}).

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

Check original membership before inserting once; an existing element is not duplicated, and the finite operation is not a new general set-theory proof. The exact result, retaining the requested order, is (False, [-4, 0, 4]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r04-set-membership-0030/)

Set membership · qf-density12-membership-set-insertion-delta · candidate





## 31. Original Membership and Set Insertion

Finite integer set A=[-5, 0], x=-5. Give (original truth of x∈A, sorted deduplicated A∪{x}).

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

Check original membership before inserting once; an existing element is not duplicated, and the finite operation is not a new general set-theory proof. The exact result, retaining the requested order, is (True, [-5, 0]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r04-set-membership-0031/)

Set membership · qf-density12-membership-set-insertion-delta · candidate





## 32. Original Membership and Set Insertion

Finite integer set A=[-6, 0], x=6. Give (original truth of x∈A, sorted deduplicated A∪{x}).

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

Check original membership before inserting once; an existing element is not duplicated, and the finite operation is not a new general set-theory proof. The exact result, retaining the requested order, is (False, [-6, 0, 6]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r04-set-membership-0032/)

Set membership · qf-density12-membership-set-insertion-delta · candidate





## 33. Original Membership and Set Insertion

Finite integer set A=[-7, 0], x=-7. Give (original truth of x∈A, sorted deduplicated A∪{x}).

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

Check original membership before inserting once; an existing element is not duplicated, and the finite operation is not a new general set-theory proof. The exact result, retaining the requested order, is (True, [-7, 0]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r04-set-membership-0033/)

Set membership · qf-density12-membership-set-insertion-delta · candidate





## 34. Original Membership and Set Insertion

Finite integer set A=[-8, 0], x=8. Give (original truth of x∈A, sorted deduplicated A∪{x}).

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

Check original membership before inserting once; an existing element is not duplicated, and the finite operation is not a new general set-theory proof. The exact result, retaining the requested order, is (False, [-8, 0, 8]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r04-set-membership-0034/)

Set membership · qf-density12-membership-set-insertion-delta · candidate





## 35. Original Membership and Set Insertion

Finite integer set A=[-9, 0], x=-9. Give (original truth of x∈A, sorted deduplicated A∪{x}).

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

Check original membership before inserting once; an existing element is not duplicated, and the finite operation is not a new general set-theory proof. The exact result, retaining the requested order, is (True, [-9, 0]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r04-set-membership-0035/)

Set membership · qf-density12-membership-set-insertion-delta · candidate





## 36. Original Membership and Set Insertion

Finite integer set A=[-10, 0], x=10. Give (original truth of x∈A, sorted deduplicated A∪{x}).

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

Check original membership before inserting once; an existing element is not duplicated, and the finite operation is not a new general set-theory proof. The exact result, retaining the requested order, is (False, [-10, 0, 10]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r04-set-membership-0036/)

Set membership · qf-density12-membership-set-insertion-delta · candidate





## 37. Original Membership and Set Insertion

Finite integer set A=[-11, 0], x=-11. Give (original truth of x∈A, sorted deduplicated A∪{x}).

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

Check original membership before inserting once; an existing element is not duplicated, and the finite operation is not a new general set-theory proof. The exact result, retaining the requested order, is (True, [-11, 0]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r04-set-membership-0037/)

Set membership · qf-density12-membership-set-insertion-delta · candidate





## 38. Original Membership and Set Insertion

Finite integer set A=[-12, 0], x=12. Give (original truth of x∈A, sorted deduplicated A∪{x}).

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

Check original membership before inserting once; an existing element is not duplicated, and the finite operation is not a new general set-theory proof. The exact result, retaining the requested order, is (False, [-12, 0, 12]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r04-set-membership-0038/)

Set membership · qf-density12-membership-set-insertion-delta · candidate





## 39. Original Membership and Set Insertion

Finite integer set A=[-13, 0], x=-13. Give (original truth of x∈A, sorted deduplicated A∪{x}).

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

Check original membership before inserting once; an existing element is not duplicated, and the finite operation is not a new general set-theory proof. The exact result, retaining the requested order, is (True, [-13, 0]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r04-set-membership-0039/)

Set membership · qf-density12-membership-set-insertion-delta · candidate





## 40. Original Membership and Set Insertion

Finite integer set A=[-14, 0], x=14. Give (original truth of x∈A, sorted deduplicated A∪{x}).

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

Check original membership before inserting once; an existing element is not duplicated, and the finite operation is not a new general set-theory proof. The exact result, retaining the requested order, is (False, [-14, 0, 14]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r04-set-membership-0040/)

Set membership · qf-density12-membership-set-insertion-delta · candidate





## 41. Original Membership and Set Insertion

Finite integer set A=[-15, 0], x=-15. Give (original truth of x∈A, sorted deduplicated A∪{x}).

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

Check original membership before inserting once; an existing element is not duplicated, and the finite operation is not a new general set-theory proof. The exact result, retaining the requested order, is (True, [-15, 0]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r04-set-membership-0041/)

Set membership · qf-density12-membership-set-insertion-delta · candidate





## 42. Original Membership and Set Insertion

Finite integer set A=[-16, 0], x=16. Give (original truth of x∈A, sorted deduplicated A∪{x}).

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

Check original membership before inserting once; an existing element is not duplicated, and the finite operation is not a new general set-theory proof. The exact result, retaining the requested order, is (False, [-16, 0, 16]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r04-set-membership-0042/)

Set membership · qf-density12-membership-set-insertion-delta · candidate





## 43. Original Membership and Set Insertion

Finite integer set A=[-17, 0], x=-17. Give (original truth of x∈A, sorted deduplicated A∪{x}).

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

Check original membership before inserting once; an existing element is not duplicated, and the finite operation is not a new general set-theory proof. The exact result, retaining the requested order, is (True, [-17, 0]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r04-set-membership-0043/)

Set membership · qf-density12-membership-set-insertion-delta · candidate





## 44. Original Membership and Set Insertion

Finite integer set A=[-18, 0], x=18. Give (original truth of x∈A, sorted deduplicated A∪{x}).

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

Check original membership before inserting once; an existing element is not duplicated, and the finite operation is not a new general set-theory proof. The exact result, retaining the requested order, is (False, [-18, 0, 18]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r04-set-membership-0044/)

Set membership · qf-density12-membership-set-insertion-delta · candidate





## 45. Original Membership and Set Insertion

Finite integer set A=[-19, 0], x=-19. Give (original truth of x∈A, sorted deduplicated A∪{x}).

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

Check original membership before inserting once; an existing element is not duplicated, and the finite operation is not a new general set-theory proof. The exact result, retaining the requested order, is (True, [-19, 0]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r04-set-membership-0045/)

Set membership · qf-density12-membership-set-insertion-delta · candidate





## 46. Original Membership and Set Insertion

Finite integer set A=[-20, 0], x=20. Give (original truth of x∈A, sorted deduplicated A∪{x}).

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

Check original membership before inserting once; an existing element is not duplicated, and the finite operation is not a new general set-theory proof. The exact result, retaining the requested order, is (False, [-20, 0, 20]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r04-set-membership-0046/)

Set membership · qf-density12-membership-set-insertion-delta · candidate





## 47. Original Membership and Set Insertion

Finite integer set A=[-21, 0], x=-21. Give (original truth of x∈A, sorted deduplicated A∪{x}).

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

Check original membership before inserting once; an existing element is not duplicated, and the finite operation is not a new general set-theory proof. The exact result, retaining the requested order, is (True, [-21, 0]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r04-set-membership-0047/)

Set membership · qf-density12-membership-set-insertion-delta · candidate





## 48. Original Membership and Set Insertion

Finite integer set A=[-22, 0], x=22. Give (original truth of x∈A, sorted deduplicated A∪{x}).

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

Check original membership before inserting once; an existing element is not duplicated, and the finite operation is not a new general set-theory proof. The exact result, retaining the requested order, is (False, [-22, 0, 22]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r04-set-membership-0048/)

Set membership · qf-density12-membership-set-insertion-delta · candidate





## 49. Original Membership and Set Insertion

Finite integer set A=[-23, 0], x=-23. Give (original truth of x∈A, sorted deduplicated A∪{x}).

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

Check original membership before inserting once; an existing element is not duplicated, and the finite operation is not a new general set-theory proof. The exact result, retaining the requested order, is (True, [-23, 0]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r04-set-membership-0049/)

Set membership · qf-density12-membership-set-insertion-delta · candidate





## 50. Original Membership and Set Insertion

Finite integer set A=[-24, 0], x=24. Give (original truth of x∈A, sorted deduplicated A∪{x}).

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

Check original membership before inserting once; an existing element is not duplicated, and the finite operation is not a new general set-theory proof. The exact result, retaining the requested order, is (False, [-24, 0, 24]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r04-set-membership-0050/)

Set membership · qf-density12-membership-set-insertion-delta · candidate





## 51. Original Membership and Set Deletion

Finite integer set A=[0], x=0. Give (original truth of x∈A, increasing A\{x}), retaining [].

View answer and explanation(True, [])

Judge membership in the original set, then remove elements equal to x; a candidate outside A leaves the set unchanged. The exact result, retaining the requested order, is (True, []).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r04-set-membership-0051/)

Set membership · qf-density12-membership-deletion-residual · candidate





## 52. Original Membership and Set Deletion

Finite integer set A=[], x=99. Give (original truth of x∈A, increasing A\{x}), retaining [].

View answer and explanation(False, [])

Judge membership in the original set, then remove elements equal to x; a candidate outside A leaves the set unchanged. The exact result, retaining the requested order, is (False, []).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r04-set-membership-0052/)

Set membership · qf-density12-membership-deletion-residual · candidate





## 53. Original Membership and Set Deletion

Finite integer set A=[-2, 0, 2], x=3. Give (original truth of x∈A, increasing A\{x}), retaining [].

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

Judge membership in the original set, then remove elements equal to x; a candidate outside A leaves the set unchanged. The exact result, retaining the requested order, is (False, [-2, 0, 2]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r04-set-membership-0053/)

Set membership · qf-density12-membership-deletion-residual · candidate





## 54. Original Membership and Set Deletion

Finite integer set A=[-3, 0, 3], x=-3. Give (original truth of x∈A, increasing A\{x}), retaining [].

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

Judge membership in the original set, then remove elements equal to x; a candidate outside A leaves the set unchanged. The exact result, retaining the requested order, is (True, [0, 3]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r04-set-membership-0054/)

Set membership · qf-density12-membership-deletion-residual · candidate





## 55. Original Membership and Set Deletion

Finite integer set A=[-4, 0, 4], x=5. Give (original truth of x∈A, increasing A\{x}), retaining [].

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

Judge membership in the original set, then remove elements equal to x; a candidate outside A leaves the set unchanged. The exact result, retaining the requested order, is (False, [-4, 0, 4]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r04-set-membership-0055/)

Set membership · qf-density12-membership-deletion-residual · candidate





## 56. Original Membership and Set Deletion

Finite integer set A=[-5, 0, 5], x=-5. Give (original truth of x∈A, increasing A\{x}), retaining [].

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

Judge membership in the original set, then remove elements equal to x; a candidate outside A leaves the set unchanged. The exact result, retaining the requested order, is (True, [0, 5]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r04-set-membership-0056/)

Set membership · qf-density12-membership-deletion-residual · candidate





## 57. Original Membership and Set Deletion

Finite integer set A=[-6, 0, 6], x=7. Give (original truth of x∈A, increasing A\{x}), retaining [].

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

Judge membership in the original set, then remove elements equal to x; a candidate outside A leaves the set unchanged. The exact result, retaining the requested order, is (False, [-6, 0, 6]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r04-set-membership-0057/)

Set membership · qf-density12-membership-deletion-residual · candidate





## 58. Original Membership and Set Deletion

Finite integer set A=[-7, 0, 7], x=-7. Give (original truth of x∈A, increasing A\{x}), retaining [].

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

Judge membership in the original set, then remove elements equal to x; a candidate outside A leaves the set unchanged. The exact result, retaining the requested order, is (True, [0, 7]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r04-set-membership-0058/)

Set membership · qf-density12-membership-deletion-residual · candidate





## 59. Original Membership and Set Deletion

Finite integer set A=[-8, 0, 8], x=9. Give (original truth of x∈A, increasing A\{x}), retaining [].

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

Judge membership in the original set, then remove elements equal to x; a candidate outside A leaves the set unchanged. The exact result, retaining the requested order, is (False, [-8, 0, 8]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r04-set-membership-0059/)

Set membership · qf-density12-membership-deletion-residual · candidate





## 60. Original Membership and Set Deletion

Finite integer set A=[-9, 0, 9], x=-9. Give (original truth of x∈A, increasing A\{x}), retaining [].

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

Judge membership in the original set, then remove elements equal to x; a candidate outside A leaves the set unchanged. The exact result, retaining the requested order, is (True, [0, 9]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r04-set-membership-0060/)

Set membership · qf-density12-membership-deletion-residual · candidate





## 61. Original Membership and Set Deletion

Finite integer set A=[-10, 0, 10], x=11. Give (original truth of x∈A, increasing A\{x}), retaining [].

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

Judge membership in the original set, then remove elements equal to x; a candidate outside A leaves the set unchanged. The exact result, retaining the requested order, is (False, [-10, 0, 10]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r04-set-membership-0061/)

Set membership · qf-density12-membership-deletion-residual · candidate





## 62. Original Membership and Set Deletion

Finite integer set A=[-11, 0, 11], x=-11. Give (original truth of x∈A, increasing A\{x}), retaining [].

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

Judge membership in the original set, then remove elements equal to x; a candidate outside A leaves the set unchanged. The exact result, retaining the requested order, is (True, [0, 11]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r04-set-membership-0062/)

Set membership · qf-density12-membership-deletion-residual · candidate





## 63. Original Membership and Set Deletion

Finite integer set A=[-12, 0, 12], x=13. Give (original truth of x∈A, increasing A\{x}), retaining [].

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

Judge membership in the original set, then remove elements equal to x; a candidate outside A leaves the set unchanged. The exact result, retaining the requested order, is (False, [-12, 0, 12]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r04-set-membership-0063/)

Set membership · qf-density12-membership-deletion-residual · candidate





## 64. Original Membership and Set Deletion

Finite integer set A=[-13, 0, 13], x=-13. Give (original truth of x∈A, increasing A\{x}), retaining [].

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

Judge membership in the original set, then remove elements equal to x; a candidate outside A leaves the set unchanged. The exact result, retaining the requested order, is (True, [0, 13]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r04-set-membership-0064/)

Set membership · qf-density12-membership-deletion-residual · candidate





## 65. Original Membership and Set Deletion

Finite integer set A=[-14, 0, 14], x=15. Give (original truth of x∈A, increasing A\{x}), retaining [].

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

Judge membership in the original set, then remove elements equal to x; a candidate outside A leaves the set unchanged. The exact result, retaining the requested order, is (False, [-14, 0, 14]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r04-set-membership-0065/)

Set membership · qf-density12-membership-deletion-residual · candidate





## 66. Original Membership and Set Deletion

Finite integer set A=[-15, 0, 15], x=-15. Give (original truth of x∈A, increasing A\{x}), retaining [].

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

Judge membership in the original set, then remove elements equal to x; a candidate outside A leaves the set unchanged. The exact result, retaining the requested order, is (True, [0, 15]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r04-set-membership-0066/)

Set membership · qf-density12-membership-deletion-residual · candidate





## 67. Original Membership and Set Deletion

Finite integer set A=[-16, 0, 16], x=17. Give (original truth of x∈A, increasing A\{x}), retaining [].

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

Judge membership in the original set, then remove elements equal to x; a candidate outside A leaves the set unchanged. The exact result, retaining the requested order, is (False, [-16, 0, 16]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r04-set-membership-0067/)

Set membership · qf-density12-membership-deletion-residual · candidate





## 68. Original Membership and Set Deletion

Finite integer set A=[-17, 0, 17], x=-17. Give (original truth of x∈A, increasing A\{x}), retaining [].

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

Judge membership in the original set, then remove elements equal to x; a candidate outside A leaves the set unchanged. The exact result, retaining the requested order, is (True, [0, 17]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r04-set-membership-0068/)

Set membership · qf-density12-membership-deletion-residual · candidate





## 69. Original Membership and Set Deletion

Finite integer set A=[-18, 0, 18], x=19. Give (original truth of x∈A, increasing A\{x}), retaining [].

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

Judge membership in the original set, then remove elements equal to x; a candidate outside A leaves the set unchanged. The exact result, retaining the requested order, is (False, [-18, 0, 18]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r04-set-membership-0069/)

Set membership · qf-density12-membership-deletion-residual · candidate





## 70. Original Membership and Set Deletion

Finite integer set A=[-19, 0, 19], x=-19. Give (original truth of x∈A, increasing A\{x}), retaining [].

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

Judge membership in the original set, then remove elements equal to x; a candidate outside A leaves the set unchanged. The exact result, retaining the requested order, is (True, [0, 19]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r04-set-membership-0070/)

Set membership · qf-density12-membership-deletion-residual · candidate





## 71. Original Membership and Set Deletion

Finite integer set A=[-20, 0, 20], x=21. Give (original truth of x∈A, increasing A\{x}), retaining [].

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

Judge membership in the original set, then remove elements equal to x; a candidate outside A leaves the set unchanged. The exact result, retaining the requested order, is (False, [-20, 0, 20]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r04-set-membership-0071/)

Set membership · qf-density12-membership-deletion-residual · candidate





## 72. Original Membership and Set Deletion

Finite integer set A=[-21, 0, 21], x=-21. Give (original truth of x∈A, increasing A\{x}), retaining [].

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

Judge membership in the original set, then remove elements equal to x; a candidate outside A leaves the set unchanged. The exact result, retaining the requested order, is (True, [0, 21]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r04-set-membership-0072/)

Set membership · qf-density12-membership-deletion-residual · candidate





## 73. Original Membership and Set Deletion

Finite integer set A=[-22, 0, 22], x=23. Give (original truth of x∈A, increasing A\{x}), retaining [].

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

Judge membership in the original set, then remove elements equal to x; a candidate outside A leaves the set unchanged. The exact result, retaining the requested order, is (False, [-22, 0, 22]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r04-set-membership-0073/)

Set membership · qf-density12-membership-deletion-residual · candidate





## 74. Original Membership and Set Deletion

Finite integer set A=[-23, 0, 23], x=-23. Give (original truth of x∈A, increasing A\{x}), retaining [].

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

Judge membership in the original set, then remove elements equal to x; a candidate outside A leaves the set unchanged. The exact result, retaining the requested order, is (True, [0, 23]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r04-set-membership-0074/)

Set membership · qf-density12-membership-deletion-residual · candidate





## 75. Original Membership and Set Deletion

Finite integer set A=[-24, 0, 24], x=25. Give (original truth of x∈A, increasing A\{x}), retaining [].

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

Judge membership in the original set, then remove elements equal to x; a candidate outside A leaves the set unchanged. The exact result, retaining the requested order, is (False, [-24, 0, 24]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r04-set-membership-0075/)

Set membership · qf-density12-membership-deletion-residual · candidate





## 76. Finite Predicate Extension and Aligned Truths

Finite integer set U=[]; P(x) means x≥0. Normalize U increasingly, then give (complete satisfying extension A, truth list aligned to sorted U).

View answer and explanation([], [])

Evaluate the declared decidable predicate for every domain element; divisibility requires a nonzero divisor. This is a finite extension, 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-20261012-r04-set-membership-0076/)

Set membership · qf-density12-membership-predicate-extension · candidate





## 77. Finite Predicate Extension and Aligned Truths

Finite integer set U=[-2, 0, 2]; P(x) means 2 divides x (nonzero divisor). Normalize U increasingly, then give (complete satisfying extension A, truth list aligned to sorted U).

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

Evaluate the declared decidable predicate for every domain element; divisibility requires a nonzero divisor. This is a finite extension, not unrestricted comprehension. The exact result, retaining the requested order, is ([-2, 0, 2], [True, True, True]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r04-set-membership-0077/)

Set membership · qf-density12-membership-predicate-extension · candidate





## 78. Finite Predicate Extension and Aligned Truths

Finite integer set U=[-1, 0, 1]; P(x) means x=0. Normalize U increasingly, then give (complete satisfying extension A, truth list aligned to sorted U).

View answer and explanation([0], [False, True, False])

Evaluate the declared decidable predicate for every domain element; divisibility requires a nonzero divisor. This is a finite extension, not unrestricted comprehension. The exact result, retaining the requested order, is ([0], [False, True, False]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r04-set-membership-0078/)

Set membership · qf-density12-membership-predicate-extension · candidate





## 79. Finite Predicate Extension and Aligned Truths

Finite integer set U=[-3, -1, 0, 1, 3]; P(x) means x≥0. Normalize U increasingly, then give (complete satisfying extension A, truth list aligned to sorted U).

View answer and explanation([0, 1, 3], [False, False, True, True, True])

Evaluate the declared decidable predicate for every domain element; divisibility requires a nonzero divisor. This is a finite extension, not unrestricted comprehension. The exact result, retaining the requested order, is ([0, 1, 3], [False, False, True, True, True]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r04-set-membership-0079/)

Set membership · qf-density12-membership-predicate-extension · candidate





## 80. Finite Predicate Extension and Aligned Truths

Finite integer set U=[-4, -1, 0, 1, 4]; P(x) means x=4. Normalize U increasingly, then give (complete satisfying extension A, truth list aligned to sorted U).

View answer and explanation([4], [False, False, False, False, True])

Evaluate the declared decidable predicate for every domain element; divisibility requires a nonzero divisor. This is a finite extension, not unrestricted comprehension. The exact result, retaining the requested order, is ([4], [False, False, False, False, True]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r04-set-membership-0080/)

Set membership · qf-density12-membership-predicate-extension · candidate





## 81. Finite Predicate Extension and Aligned Truths

Finite integer set U=[-5, -1, 0, 1, 5]; P(x) means 5 divides x (nonzero divisor). Normalize U increasingly, then give (complete satisfying extension A, truth list aligned to sorted U).

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

Evaluate the declared decidable predicate for every domain element; divisibility requires a nonzero divisor. This is a finite extension, not unrestricted comprehension. The exact result, retaining the requested order, is ([-5, 0, 5], [True, False, True, False, True]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r04-set-membership-0081/)

Set membership · qf-density12-membership-predicate-extension · candidate





## 82. Finite Predicate Extension and Aligned Truths

Finite integer set U=[-6, -1, 0, 1, 6]; P(x) means x≥0. Normalize U increasingly, then give (complete satisfying extension A, truth list aligned to sorted U).

View answer and explanation([0, 1, 6], [False, False, True, True, True])

Evaluate the declared decidable predicate for every domain element; divisibility requires a nonzero divisor. This is a finite extension, not unrestricted comprehension. The exact result, retaining the requested order, is ([0, 1, 6], [False, False, True, True, True]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r04-set-membership-0082/)

Set membership · qf-density12-membership-predicate-extension · candidate





## 83. Finite Predicate Extension and Aligned Truths

Finite integer set U=[-7, -1, 0, 1, 7]; P(x) means x=7. Normalize U increasingly, then give (complete satisfying extension A, truth list aligned to sorted U).

View answer and explanation([7], [False, False, False, False, True])

Evaluate the declared decidable predicate for every domain element; divisibility requires a nonzero divisor. This is a finite extension, not unrestricted comprehension. The exact result, retaining the requested order, is ([7], [False, False, False, False, True]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r04-set-membership-0083/)

Set membership · qf-density12-membership-predicate-extension · candidate





## 84. Finite Predicate Extension and Aligned Truths

Finite integer set U=[-8, -1, 0, 1, 8]; P(x) means 8 divides x (nonzero divisor). Normalize U increasingly, then give (complete satisfying extension A, truth list aligned to sorted U).

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

Evaluate the declared decidable predicate for every domain element; divisibility requires a nonzero divisor. This is a finite extension, not unrestricted comprehension. The exact result, retaining the requested order, is ([-8, 0, 8], [True, False, True, False, True]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r04-set-membership-0084/)

Set membership · qf-density12-membership-predicate-extension · candidate





## 85. Finite Predicate Extension and Aligned Truths

Finite integer set U=[-9, -1, 0, 1, 9]; P(x) means x≥0. Normalize U increasingly, then give (complete satisfying extension A, truth list aligned to sorted U).

View answer and explanation([0, 1, 9], [False, False, True, True, True])

Evaluate the declared decidable predicate for every domain element; divisibility requires a nonzero divisor. This is a finite extension, not unrestricted comprehension. The exact result, retaining the requested order, is ([0, 1, 9], [False, False, True, True, True]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r04-set-membership-0085/)

Set membership · qf-density12-membership-predicate-extension · candidate





## 86. Finite Predicate Extension and Aligned Truths

Finite integer set U=[-10, -1, 0, 1, 10]; P(x) means x=10. Normalize U increasingly, then give (complete satisfying extension A, truth list aligned to sorted U).

View answer and explanation([10], [False, False, False, False, True])

Evaluate the declared decidable predicate for every domain element; divisibility requires a nonzero divisor. This is a finite extension, not unrestricted comprehension. The exact result, retaining the requested order, is ([10], [False, False, False, False, True]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r04-set-membership-0086/)

Set membership · qf-density12-membership-predicate-extension · candidate





## 87. Finite Predicate Extension and Aligned Truths

Finite integer set U=[-11, -1, 0, 1, 11]; P(x) means 11 divides x (nonzero divisor). Normalize U increasingly, then give (complete satisfying extension A, truth list aligned to sorted U).

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

Evaluate the declared decidable predicate for every domain element; divisibility requires a nonzero divisor. This is a finite extension, not unrestricted comprehension. The exact result, retaining the requested order, is ([-11, 0, 11], [True, False, True, False, True]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r04-set-membership-0087/)

Set membership · qf-density12-membership-predicate-extension · candidate





## 88. Finite Predicate Extension and Aligned Truths

Finite integer set U=[-12, -1, 0, 1, 12]; P(x) means x≥0. Normalize U increasingly, then give (complete satisfying extension A, truth list aligned to sorted U).

View answer and explanation([0, 1, 12], [False, False, True, True, True])

Evaluate the declared decidable predicate for every domain element; divisibility requires a nonzero divisor. This is a finite extension, not unrestricted comprehension. The exact result, retaining the requested order, is ([0, 1, 12], [False, False, True, True, True]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r04-set-membership-0088/)

Set membership · qf-density12-membership-predicate-extension · candidate





## 89. Finite Predicate Extension and Aligned Truths

Finite integer set U=[-13, -1, 0, 1, 13]; P(x) means x=13. Normalize U increasingly, then give (complete satisfying extension A, truth list aligned to sorted U).

View answer and explanation([13], [False, False, False, False, True])

Evaluate the declared decidable predicate for every domain element; divisibility requires a nonzero divisor. This is a finite extension, not unrestricted comprehension. The exact result, retaining the requested order, is ([13], [False, False, False, False, True]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r04-set-membership-0089/)

Set membership · qf-density12-membership-predicate-extension · candidate





## 90. Finite Predicate Extension and Aligned Truths

Finite integer set U=[-14, -1, 0, 1, 14]; P(x) means 14 divides x (nonzero divisor). Normalize U increasingly, then give (complete satisfying extension A, truth list aligned to sorted U).

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

Evaluate the declared decidable predicate for every domain element; divisibility requires a nonzero divisor. This is a finite extension, not unrestricted comprehension. The exact result, retaining the requested order, is ([-14, 0, 14], [True, False, True, False, True]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r04-set-membership-0090/)

Set membership · qf-density12-membership-predicate-extension · candidate





## 91. Finite Predicate Extension and Aligned Truths

Finite integer set U=[-15, -1, 0, 1, 15]; P(x) means x≥0. Normalize U increasingly, then give (complete satisfying extension A, truth list aligned to sorted U).

View answer and explanation([0, 1, 15], [False, False, True, True, True])

Evaluate the declared decidable predicate for every domain element; divisibility requires a nonzero divisor. This is a finite extension, not unrestricted comprehension. The exact result, retaining the requested order, is ([0, 1, 15], [False, False, True, True, True]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r04-set-membership-0091/)

Set membership · qf-density12-membership-predicate-extension · candidate





## 92. Finite Predicate Extension and Aligned Truths

Finite integer set U=[-16, -1, 0, 1, 16]; P(x) means x=16. Normalize U increasingly, then give (complete satisfying extension A, truth list aligned to sorted U).

View answer and explanation([16], [False, False, False, False, True])

Evaluate the declared decidable predicate for every domain element; divisibility requires a nonzero divisor. This is a finite extension, not unrestricted comprehension. The exact result, retaining the requested order, is ([16], [False, False, False, False, True]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r04-set-membership-0092/)

Set membership · qf-density12-membership-predicate-extension · candidate





## 93. Finite Predicate Extension and Aligned Truths

Finite integer set U=[-17, -1, 0, 1, 17]; P(x) means 17 divides x (nonzero divisor). Normalize U increasingly, then give (complete satisfying extension A, truth list aligned to sorted U).

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

Evaluate the declared decidable predicate for every domain element; divisibility requires a nonzero divisor. This is a finite extension, not unrestricted comprehension. The exact result, retaining the requested order, is ([-17, 0, 17], [True, False, True, False, True]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r04-set-membership-0093/)

Set membership · qf-density12-membership-predicate-extension · candidate





## 94. Finite Predicate Extension and Aligned Truths

Finite integer set U=[-18, -1, 0, 1, 18]; P(x) means x≥0. Normalize U increasingly, then give (complete satisfying extension A, truth list aligned to sorted U).

View answer and explanation([0, 1, 18], [False, False, True, True, True])

Evaluate the declared decidable predicate for every domain element; divisibility requires a nonzero divisor. This is a finite extension, not unrestricted comprehension. The exact result, retaining the requested order, is ([0, 1, 18], [False, False, True, True, True]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r04-set-membership-0094/)

Set membership · qf-density12-membership-predicate-extension · candidate





## 95. Finite Predicate Extension and Aligned Truths

Finite integer set U=[-19, -1, 0, 1, 19]; P(x) means x=19. Normalize U increasingly, then give (complete satisfying extension A, truth list aligned to sorted U).

View answer and explanation([19], [False, False, False, False, True])

Evaluate the declared decidable predicate for every domain element; divisibility requires a nonzero divisor. This is a finite extension, not unrestricted comprehension. The exact result, retaining the requested order, is ([19], [False, False, False, False, True]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r04-set-membership-0095/)

Set membership · qf-density12-membership-predicate-extension · candidate





## 96. Finite Predicate Extension and Aligned Truths

Finite integer set U=[-20, -1, 0, 1, 20]; P(x) means 20 divides x (nonzero divisor). Normalize U increasingly, then give (complete satisfying extension A, truth list aligned to sorted U).

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

Evaluate the declared decidable predicate for every domain element; divisibility requires a nonzero divisor. This is a finite extension, not unrestricted comprehension. The exact result, retaining the requested order, is ([-20, 0, 20], [True, False, True, False, True]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r04-set-membership-0096/)

Set membership · qf-density12-membership-predicate-extension · candidate





## 97. Finite Predicate Extension and Aligned Truths

Finite integer set U=[-21, -1, 0, 1, 21]; P(x) means x≥0. Normalize U increasingly, then give (complete satisfying extension A, truth list aligned to sorted U).

View answer and explanation([0, 1, 21], [False, False, True, True, True])

Evaluate the declared decidable predicate for every domain element; divisibility requires a nonzero divisor. This is a finite extension, not unrestricted comprehension. The exact result, retaining the requested order, is ([0, 1, 21], [False, False, True, True, True]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r04-set-membership-0097/)

Set membership · qf-density12-membership-predicate-extension · candidate





## 98. Finite Predicate Extension and Aligned Truths

Finite integer set U=[-22, -1, 0, 1, 22]; P(x) means x=22. Normalize U increasingly, then give (complete satisfying extension A, truth list aligned to sorted U).

View answer and explanation([22], [False, False, False, False, True])

Evaluate the declared decidable predicate for every domain element; divisibility requires a nonzero divisor. This is a finite extension, not unrestricted comprehension. The exact result, retaining the requested order, is ([22], [False, False, False, False, True]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r04-set-membership-0098/)

Set membership · qf-density12-membership-predicate-extension · candidate





## 99. Finite Predicate Extension and Aligned Truths

Finite integer set U=[-23, -1, 0, 1, 23]; P(x) means 23 divides x (nonzero divisor). Normalize U increasingly, then give (complete satisfying extension A, truth list aligned to sorted U).

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

Evaluate the declared decidable predicate for every domain element; divisibility requires a nonzero divisor. This is a finite extension, not unrestricted comprehension. The exact result, retaining the requested order, is ([-23, 0, 23], [True, False, True, False, True]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r04-set-membership-0099/)

Set membership · qf-density12-membership-predicate-extension · candidate





## 100. Finite Predicate Extension and Aligned Truths

Finite integer set U=[-24, -1, 0, 1, 24]; P(x) means x≥0. Normalize U increasingly, then give (complete satisfying extension A, truth list aligned to sorted U).

View answer and explanation([0, 1, 24], [False, False, True, True, True])

Evaluate the declared decidable predicate for every domain element; divisibility requires a nonzero divisor. This is a finite extension, not unrestricted comprehension. The exact result, retaining the requested order, is ([0, 1, 24], [False, False, True, True, True]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261012-r04-set-membership-0100/)

Set membership · qf-density12-membership-predicate-extension · candidate



[Canonical question JSONL](https://ocme.evemisslab.com/data/questions/batches/batch-20261008-advance-20261012-r04-set-membership-foundations/questions.jsonl) · [English JSONL](https://ocme.evemisslab.com/data/i18n/questions/batch-20261008-advance-20261012-r04-set-membership-foundations.en.jsonl) · [Validation report](https://ocme.evemisslab.com/data/questions/batches/batch-20261008-advance-20261012-r04-set-membership-foundations/validation-report.json)