Typed Membership and Finite Predicate Extensions
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This question bank has corresponding bilingual step-by-step derivations. Source candidate and English ai_candidate states are retained. AI review, finite computation and Lean formal proof are labelled separately.
batch-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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
Set membership · qf-density12-membership-predicate-extension · candidate
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