Finite Predicate Partitions and Ordered Quantifier Counts
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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-20261010-r07-quantifier-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. Complete Finite Predicate Partition
The explicitly finite integer domain is U=[] and P(x) means x≥0. Give (elements satisfying P, elements not satisfying P), each sorted increasingly.
View answer and explanation
([], [])
Evaluate every domain element; the two disjoint parts cover U. Do not extend this finite enumeration to an infinite domain. The exact result, retaining the requested order, is ([], []).
Reviewed step-by-step derivation
Quantifiers and explicit domains · qf-density10-quantifier-truth-partition · candidate
2. Complete Finite Predicate Partition
The explicitly finite integer domain is U=[-2, 0, 2] and P(x) means 2 divides x (nonzero divisor). Give (elements satisfying P, elements not satisfying P), each sorted increasingly.
View answer and explanation
([-2, 0, 2], [])
Evaluate every domain element; the two disjoint parts cover U. Do not extend this finite enumeration to an infinite domain. The exact result, retaining the requested order, is ([-2, 0, 2], []).
Reviewed step-by-step derivation
Quantifiers and explicit domains · qf-density10-quantifier-truth-partition · candidate
3. Complete Finite Predicate Partition
The explicitly finite integer domain is U=[-1, 0, 1] and P(x) means x=0. Give (elements satisfying P, elements not satisfying P), each sorted increasingly.
View answer and explanation
([0], [-1, 1])
Evaluate every domain element; the two disjoint parts cover U. Do not extend this finite enumeration to an infinite domain. The exact result, retaining the requested order, is ([0], [-1, 1]).
Reviewed step-by-step derivation
Quantifiers and explicit domains · qf-density10-quantifier-truth-partition · candidate
4. Complete Finite Predicate Partition
The explicitly finite integer domain is U=[-3, -1, 0, 1, 3] and P(x) means x≥0. Give (elements satisfying P, elements not satisfying P), each sorted increasingly.
View answer and explanation
([0, 1, 3], [-3, -1])
Evaluate every domain element; the two disjoint parts cover U. Do not extend this finite enumeration to an infinite domain. The exact result, retaining the requested order, is ([0, 1, 3], [-3, -1]).
Reviewed step-by-step derivation
Quantifiers and explicit domains · qf-density10-quantifier-truth-partition · candidate
5. Complete Finite Predicate Partition
The explicitly finite integer domain is U=[-4, -1, 0, 1, 4] and P(x) means x=1. Give (elements satisfying P, elements not satisfying P), each sorted increasingly.
View answer and explanation
([1], [-4, -1, 0, 4])
Evaluate every domain element; the two disjoint parts cover U. Do not extend this finite enumeration to an infinite domain. The exact result, retaining the requested order, is ([1], [-4, -1, 0, 4]).
Reviewed step-by-step derivation
Quantifiers and explicit domains · qf-density10-quantifier-truth-partition · candidate
6. Complete Finite Predicate Partition
The explicitly finite integer domain is U=[-5, -1, 0, 1, 5] and P(x) means 2 divides x (nonzero divisor). Give (elements satisfying P, elements not satisfying P), each sorted increasingly.
View answer and explanation
([0], [-5, -1, 1, 5])
Evaluate every domain element; the two disjoint parts cover U. Do not extend this finite enumeration to an infinite domain. The exact result, retaining the requested order, is ([0], [-5, -1, 1, 5]).
Reviewed step-by-step derivation
Quantifiers and explicit domains · qf-density10-quantifier-truth-partition · candidate
7. Complete Finite Predicate Partition
The explicitly finite integer domain is U=[-6, -1, 0, 1, 6] and P(x) means x≥0. Give (elements satisfying P, elements not satisfying P), each sorted increasingly.
View answer and explanation
([0, 1, 6], [-6, -1])
Evaluate every domain element; the two disjoint parts cover U. Do not extend this finite enumeration to an infinite domain. The exact result, retaining the requested order, is ([0, 1, 6], [-6, -1]).
Reviewed step-by-step derivation
Quantifiers and explicit domains · qf-density10-quantifier-truth-partition · candidate
8. Complete Finite Predicate Partition
The explicitly finite integer domain is U=[-7, -1, 0, 1, 7] and P(x) means x=1. Give (elements satisfying P, elements not satisfying P), each sorted increasingly.
View answer and explanation
([1], [-7, -1, 0, 7])
Evaluate every domain element; the two disjoint parts cover U. Do not extend this finite enumeration to an infinite domain. The exact result, retaining the requested order, is ([1], [-7, -1, 0, 7]).
Reviewed step-by-step derivation
Quantifiers and explicit domains · qf-density10-quantifier-truth-partition · candidate
9. Complete Finite Predicate Partition
The explicitly finite integer domain is U=[-8, -1, 0, 1, 8] and P(x) means 2 divides x (nonzero divisor). Give (elements satisfying P, elements not satisfying P), each sorted increasingly.
View answer and explanation
([-8, 0, 8], [-1, 1])
Evaluate every domain element; the two disjoint parts cover U. Do not extend this finite enumeration to an infinite domain. The exact result, retaining the requested order, is ([-8, 0, 8], [-1, 1]).
Reviewed step-by-step derivation
Quantifiers and explicit domains · qf-density10-quantifier-truth-partition · candidate
10. Complete Finite Predicate Partition
The explicitly finite integer domain is U=[-9, -1, 0, 1, 9] and P(x) means x≥0. Give (elements satisfying P, elements not satisfying P), each sorted increasingly.
View answer and explanation
([0, 1, 9], [-9, -1])
Evaluate every domain element; the two disjoint parts cover U. Do not extend this finite enumeration to an infinite domain. The exact result, retaining the requested order, is ([0, 1, 9], [-9, -1]).
Reviewed step-by-step derivation
Quantifiers and explicit domains · qf-density10-quantifier-truth-partition · candidate
11. Complete Finite Predicate Partition
The explicitly finite integer domain is U=[-10, -1, 0, 1, 10] and P(x) means x=1. Give (elements satisfying P, elements not satisfying P), each sorted increasingly.
View answer and explanation
([1], [-10, -1, 0, 10])
Evaluate every domain element; the two disjoint parts cover U. Do not extend this finite enumeration to an infinite domain. The exact result, retaining the requested order, is ([1], [-10, -1, 0, 10]).
Reviewed step-by-step derivation
Quantifiers and explicit domains · qf-density10-quantifier-truth-partition · candidate
12. Complete Finite Predicate Partition
The explicitly finite integer domain is U=[-11, -1, 0, 1, 11] and P(x) means 2 divides x (nonzero divisor). Give (elements satisfying P, elements not satisfying P), each sorted increasingly.
View answer and explanation
([0], [-11, -1, 1, 11])
Evaluate every domain element; the two disjoint parts cover U. Do not extend this finite enumeration to an infinite domain. The exact result, retaining the requested order, is ([0], [-11, -1, 1, 11]).
Reviewed step-by-step derivation
Quantifiers and explicit domains · qf-density10-quantifier-truth-partition · candidate
13. Complete Finite Predicate Partition
The explicitly finite integer domain is U=[-12, -1, 0, 1, 12] and P(x) means x≥0. Give (elements satisfying P, elements not satisfying P), each sorted increasingly.
View answer and explanation
([0, 1, 12], [-12, -1])
Evaluate every domain element; the two disjoint parts cover U. Do not extend this finite enumeration to an infinite domain. The exact result, retaining the requested order, is ([0, 1, 12], [-12, -1]).
Reviewed step-by-step derivation
Quantifiers and explicit domains · qf-density10-quantifier-truth-partition · candidate
14. Complete Finite Predicate Partition
The explicitly finite integer domain is U=[-13, -1, 0, 1, 13] and P(x) means x=1. Give (elements satisfying P, elements not satisfying P), each sorted increasingly.
View answer and explanation
([1], [-13, -1, 0, 13])
Evaluate every domain element; the two disjoint parts cover U. Do not extend this finite enumeration to an infinite domain. The exact result, retaining the requested order, is ([1], [-13, -1, 0, 13]).
Reviewed step-by-step derivation
Quantifiers and explicit domains · qf-density10-quantifier-truth-partition · candidate
15. Complete Finite Predicate Partition
The explicitly finite integer domain is U=[-14, -1, 0, 1, 14] and P(x) means 2 divides x (nonzero divisor). Give (elements satisfying P, elements not satisfying P), each sorted increasingly.
View answer and explanation
([-14, 0, 14], [-1, 1])
Evaluate every domain element; the two disjoint parts cover U. Do not extend this finite enumeration to an infinite domain. The exact result, retaining the requested order, is ([-14, 0, 14], [-1, 1]).
Reviewed step-by-step derivation
Quantifiers and explicit domains · qf-density10-quantifier-truth-partition · candidate
16. Complete Finite Predicate Partition
The explicitly finite integer domain is U=[-15, -1, 0, 1, 15] and P(x) means x≥0. Give (elements satisfying P, elements not satisfying P), each sorted increasingly.
View answer and explanation
([0, 1, 15], [-15, -1])
Evaluate every domain element; the two disjoint parts cover U. Do not extend this finite enumeration to an infinite domain. The exact result, retaining the requested order, is ([0, 1, 15], [-15, -1]).
Reviewed step-by-step derivation
Quantifiers and explicit domains · qf-density10-quantifier-truth-partition · candidate
17. Complete Finite Predicate Partition
The explicitly finite integer domain is U=[-16, -1, 0, 1, 16] and P(x) means x=1. Give (elements satisfying P, elements not satisfying P), each sorted increasingly.
View answer and explanation
([1], [-16, -1, 0, 16])
Evaluate every domain element; the two disjoint parts cover U. Do not extend this finite enumeration to an infinite domain. The exact result, retaining the requested order, is ([1], [-16, -1, 0, 16]).
Reviewed step-by-step derivation
Quantifiers and explicit domains · qf-density10-quantifier-truth-partition · candidate
18. Complete Finite Predicate Partition
The explicitly finite integer domain is U=[-17, -1, 0, 1, 17] and P(x) means 2 divides x (nonzero divisor). Give (elements satisfying P, elements not satisfying P), each sorted increasingly.
View answer and explanation
([0], [-17, -1, 1, 17])
Evaluate every domain element; the two disjoint parts cover U. Do not extend this finite enumeration to an infinite domain. The exact result, retaining the requested order, is ([0], [-17, -1, 1, 17]).
Reviewed step-by-step derivation
Quantifiers and explicit domains · qf-density10-quantifier-truth-partition · candidate
19. Complete Finite Predicate Partition
The explicitly finite integer domain is U=[-18, -1, 0, 1, 18] and P(x) means x≥0. Give (elements satisfying P, elements not satisfying P), each sorted increasingly.
View answer and explanation
([0, 1, 18], [-18, -1])
Evaluate every domain element; the two disjoint parts cover U. Do not extend this finite enumeration to an infinite domain. The exact result, retaining the requested order, is ([0, 1, 18], [-18, -1]).
Reviewed step-by-step derivation
Quantifiers and explicit domains · qf-density10-quantifier-truth-partition · candidate
20. Complete Finite Predicate Partition
The explicitly finite integer domain is U=[-19, -1, 0, 1, 19] and P(x) means x=1. Give (elements satisfying P, elements not satisfying P), each sorted increasingly.
View answer and explanation
([1], [-19, -1, 0, 19])
Evaluate every domain element; the two disjoint parts cover U. Do not extend this finite enumeration to an infinite domain. The exact result, retaining the requested order, is ([1], [-19, -1, 0, 19]).
Reviewed step-by-step derivation
Quantifiers and explicit domains · qf-density10-quantifier-truth-partition · candidate
21. Complete Finite Predicate Partition
The explicitly finite integer domain is U=[-20, -1, 0, 1, 20] and P(x) means 2 divides x (nonzero divisor). Give (elements satisfying P, elements not satisfying P), each sorted increasingly.
View answer and explanation
([-20, 0, 20], [-1, 1])
Evaluate every domain element; the two disjoint parts cover U. Do not extend this finite enumeration to an infinite domain. The exact result, retaining the requested order, is ([-20, 0, 20], [-1, 1]).
Reviewed step-by-step derivation
Quantifiers and explicit domains · qf-density10-quantifier-truth-partition · candidate
22. Complete Finite Predicate Partition
The explicitly finite integer domain is U=[-21, -1, 0, 1, 21] and P(x) means x≥0. Give (elements satisfying P, elements not satisfying P), each sorted increasingly.
View answer and explanation
([0, 1, 21], [-21, -1])
Evaluate every domain element; the two disjoint parts cover U. Do not extend this finite enumeration to an infinite domain. The exact result, retaining the requested order, is ([0, 1, 21], [-21, -1]).
Reviewed step-by-step derivation
Quantifiers and explicit domains · qf-density10-quantifier-truth-partition · candidate
23. Complete Finite Predicate Partition
The explicitly finite integer domain is U=[-22, -1, 0, 1, 22] and P(x) means x=1. Give (elements satisfying P, elements not satisfying P), each sorted increasingly.
View answer and explanation
([1], [-22, -1, 0, 22])
Evaluate every domain element; the two disjoint parts cover U. Do not extend this finite enumeration to an infinite domain. The exact result, retaining the requested order, is ([1], [-22, -1, 0, 22]).
Reviewed step-by-step derivation
Quantifiers and explicit domains · qf-density10-quantifier-truth-partition · candidate
24. Complete Finite Predicate Partition
The explicitly finite integer domain is U=[-23, -1, 0, 1, 23] and P(x) means 2 divides x (nonzero divisor). Give (elements satisfying P, elements not satisfying P), each sorted increasingly.
View answer and explanation
([0], [-23, -1, 1, 23])
Evaluate every domain element; the two disjoint parts cover U. Do not extend this finite enumeration to an infinite domain. The exact result, retaining the requested order, is ([0], [-23, -1, 1, 23]).
Reviewed step-by-step derivation
Quantifiers and explicit domains · qf-density10-quantifier-truth-partition · candidate
25. Complete Finite Predicate Partition
The explicitly finite integer domain is U=[-24, -1, 0, 1, 24] and P(x) means x≥0. Give (elements satisfying P, elements not satisfying P), each sorted increasingly.
View answer and explanation
([0, 1, 24], [-24, -1])
Evaluate every domain element; the two disjoint parts cover U. Do not extend this finite enumeration to an infinite domain. The exact result, retaining the requested order, is ([0, 1, 24], [-24, -1]).
Reviewed step-by-step derivation
Quantifiers and explicit domains · qf-density10-quantifier-truth-partition · candidate
26. Negated Universal and Existential Pair
The explicitly finite integer domain is U=[] and P(x) means x≥0. Give (¬∀x∈U P(x), ¬∃x∈U P(x)) using True/False, retaining the empty-domain convention.
View answer and explanation
(False, True)
A false element negates the universal; absence of a true element negates the existential. The empty-domain pair is (False, True). Do not extend this finite enumeration to an infinite domain. The exact result, retaining the requested order, is (False, True).
Reviewed step-by-step derivation
Quantifiers and explicit domains · qf-density10-quantifier-negation-pair · candidate
27. Negated Universal and Existential Pair
The explicitly finite integer domain is U=[-1] and P(x) means x≥0. Give (¬∀x∈U P(x), ¬∃x∈U P(x)) using True/False, retaining the empty-domain convention.
View answer and explanation
(True, True)
A false element negates the universal; absence of a true element negates the existential. The empty-domain pair is (False, True). Do not extend this finite enumeration to an infinite domain. The exact result, retaining the requested order, is (True, True).
Reviewed step-by-step derivation
Quantifiers and explicit domains · qf-density10-quantifier-negation-pair · candidate
28. Negated Universal and Existential Pair
The explicitly finite integer domain is U=[1] and P(x) means x≥0. Give (¬∀x∈U P(x), ¬∃x∈U P(x)) using True/False, retaining the empty-domain convention.
View answer and explanation
(False, False)
A false element negates the universal; absence of a true element negates the existential. The empty-domain pair is (False, True). Do not extend this finite enumeration to an infinite domain. The exact result, retaining the requested order, is (False, False).
Reviewed step-by-step derivation
Quantifiers and explicit domains · qf-density10-quantifier-negation-pair · candidate
29. Negated Universal and Existential Pair
The explicitly finite integer domain is U=[-2, -1] and P(x) means x≥0. Give (¬∀x∈U P(x), ¬∃x∈U P(x)) using True/False, retaining the empty-domain convention.
View answer and explanation
(True, True)
A false element negates the universal; absence of a true element negates the existential. The empty-domain pair is (False, True). Do not extend this finite enumeration to an infinite domain. The exact result, retaining the requested order, is (True, True).
Reviewed step-by-step derivation
Quantifiers and explicit domains · qf-density10-quantifier-negation-pair · candidate
30. Negated Universal and Existential Pair
The explicitly finite integer domain is U=[1, -1] and P(x) means x≥0. Give (¬∀x∈U P(x), ¬∃x∈U P(x)) using True/False, retaining the empty-domain convention.
View answer and explanation
(True, False)
A false element negates the universal; absence of a true element negates the existential. The empty-domain pair is (False, True). Do not extend this finite enumeration to an infinite domain. The exact result, retaining the requested order, is (True, False).
Reviewed step-by-step derivation
Quantifiers and explicit domains · qf-density10-quantifier-negation-pair · candidate
31. Negated Universal and Existential Pair
The explicitly finite integer domain is U=[1, 2] and P(x) means x≥0. Give (¬∀x∈U P(x), ¬∃x∈U P(x)) using True/False, retaining the empty-domain convention.
View answer and explanation
(False, False)
A false element negates the universal; absence of a true element negates the existential. The empty-domain pair is (False, True). Do not extend this finite enumeration to an infinite domain. The exact result, retaining the requested order, is (False, False).
Reviewed step-by-step derivation
Quantifiers and explicit domains · qf-density10-quantifier-negation-pair · candidate
32. Negated Universal and Existential Pair
The explicitly finite integer domain is U=[-3, -2, -1] and P(x) means x≥0. Give (¬∀x∈U P(x), ¬∃x∈U P(x)) using True/False, retaining the empty-domain convention.
View answer and explanation
(True, True)
A false element negates the universal; absence of a true element negates the existential. The empty-domain pair is (False, True). Do not extend this finite enumeration to an infinite domain. The exact result, retaining the requested order, is (True, True).
Reviewed step-by-step derivation
Quantifiers and explicit domains · qf-density10-quantifier-negation-pair · candidate
33. Negated Universal and Existential Pair
The explicitly finite integer domain is U=[1, -2, -1] and P(x) means x≥0. Give (¬∀x∈U P(x), ¬∃x∈U P(x)) using True/False, retaining the empty-domain convention.
View answer and explanation
(True, False)
A false element negates the universal; absence of a true element negates the existential. The empty-domain pair is (False, True). Do not extend this finite enumeration to an infinite domain. The exact result, retaining the requested order, is (True, False).
Reviewed step-by-step derivation
Quantifiers and explicit domains · qf-density10-quantifier-negation-pair · candidate
34. Negated Universal and Existential Pair
The explicitly finite integer domain is U=[1, 2, -1] and P(x) means x≥0. Give (¬∀x∈U P(x), ¬∃x∈U P(x)) using True/False, retaining the empty-domain convention.
View answer and explanation
(True, False)
A false element negates the universal; absence of a true element negates the existential. The empty-domain pair is (False, True). Do not extend this finite enumeration to an infinite domain. The exact result, retaining the requested order, is (True, False).
Reviewed step-by-step derivation
Quantifiers and explicit domains · qf-density10-quantifier-negation-pair · candidate
35. Negated Universal and Existential Pair
The explicitly finite integer domain is U=[1, 2, 3] and P(x) means x≥0. Give (¬∀x∈U P(x), ¬∃x∈U P(x)) using True/False, retaining the empty-domain convention.
View answer and explanation
(False, False)
A false element negates the universal; absence of a true element negates the existential. The empty-domain pair is (False, True). Do not extend this finite enumeration to an infinite domain. The exact result, retaining the requested order, is (False, False).
Reviewed step-by-step derivation
Quantifiers and explicit domains · qf-density10-quantifier-negation-pair · candidate
36. Negated Universal and Existential Pair
The explicitly finite integer domain is U=[-4, -3, -2, -1] and P(x) means x≥0. Give (¬∀x∈U P(x), ¬∃x∈U P(x)) using True/False, retaining the empty-domain convention.
View answer and explanation
(True, True)
A false element negates the universal; absence of a true element negates the existential. The empty-domain pair is (False, True). Do not extend this finite enumeration to an infinite domain. The exact result, retaining the requested order, is (True, True).
Reviewed step-by-step derivation
Quantifiers and explicit domains · qf-density10-quantifier-negation-pair · candidate
37. Negated Universal and Existential Pair
The explicitly finite integer domain is U=[1, -3, -2, -1] and P(x) means x≥0. Give (¬∀x∈U P(x), ¬∃x∈U P(x)) using True/False, retaining the empty-domain convention.
View answer and explanation
(True, False)
A false element negates the universal; absence of a true element negates the existential. The empty-domain pair is (False, True). Do not extend this finite enumeration to an infinite domain. The exact result, retaining the requested order, is (True, False).
Reviewed step-by-step derivation
Quantifiers and explicit domains · qf-density10-quantifier-negation-pair · candidate
38. Negated Universal and Existential Pair
The explicitly finite integer domain is U=[1, 2, -2, -1] and P(x) means x≥0. Give (¬∀x∈U P(x), ¬∃x∈U P(x)) using True/False, retaining the empty-domain convention.
View answer and explanation
(True, False)
A false element negates the universal; absence of a true element negates the existential. The empty-domain pair is (False, True). Do not extend this finite enumeration to an infinite domain. The exact result, retaining the requested order, is (True, False).
Reviewed step-by-step derivation
Quantifiers and explicit domains · qf-density10-quantifier-negation-pair · candidate
39. Negated Universal and Existential Pair
The explicitly finite integer domain is U=[1, 2, 3, -1] and P(x) means x≥0. Give (¬∀x∈U P(x), ¬∃x∈U P(x)) using True/False, retaining the empty-domain convention.
View answer and explanation
(True, False)
A false element negates the universal; absence of a true element negates the existential. The empty-domain pair is (False, True). Do not extend this finite enumeration to an infinite domain. The exact result, retaining the requested order, is (True, False).
Reviewed step-by-step derivation
Quantifiers and explicit domains · qf-density10-quantifier-negation-pair · candidate
40. Negated Universal and Existential Pair
The explicitly finite integer domain is U=[1, 2, 3, 4] and P(x) means x≥0. Give (¬∀x∈U P(x), ¬∃x∈U P(x)) using True/False, retaining the empty-domain convention.
View answer and explanation
(False, False)
A false element negates the universal; absence of a true element negates the existential. The empty-domain pair is (False, True). Do not extend this finite enumeration to an infinite domain. The exact result, retaining the requested order, is (False, False).
Reviewed step-by-step derivation
Quantifiers and explicit domains · qf-density10-quantifier-negation-pair · candidate
41. Negated Universal and Existential Pair
The explicitly finite integer domain is U=[-5, -4, -3, -2, -1] and P(x) means x≥0. Give (¬∀x∈U P(x), ¬∃x∈U P(x)) using True/False, retaining the empty-domain convention.
View answer and explanation
(True, True)
A false element negates the universal; absence of a true element negates the existential. The empty-domain pair is (False, True). Do not extend this finite enumeration to an infinite domain. The exact result, retaining the requested order, is (True, True).
Reviewed step-by-step derivation
Quantifiers and explicit domains · qf-density10-quantifier-negation-pair · candidate
42. Negated Universal and Existential Pair
The explicitly finite integer domain is U=[1, -4, -3, -2, -1] and P(x) means x≥0. Give (¬∀x∈U P(x), ¬∃x∈U P(x)) using True/False, retaining the empty-domain convention.
View answer and explanation
(True, False)
A false element negates the universal; absence of a true element negates the existential. The empty-domain pair is (False, True). Do not extend this finite enumeration to an infinite domain. The exact result, retaining the requested order, is (True, False).
Reviewed step-by-step derivation
Quantifiers and explicit domains · qf-density10-quantifier-negation-pair · candidate
43. Negated Universal and Existential Pair
The explicitly finite integer domain is U=[1, 2, -3, -2, -1] and P(x) means x≥0. Give (¬∀x∈U P(x), ¬∃x∈U P(x)) using True/False, retaining the empty-domain convention.
View answer and explanation
(True, False)
A false element negates the universal; absence of a true element negates the existential. The empty-domain pair is (False, True). Do not extend this finite enumeration to an infinite domain. The exact result, retaining the requested order, is (True, False).
Reviewed step-by-step derivation
Quantifiers and explicit domains · qf-density10-quantifier-negation-pair · candidate
44. Negated Universal and Existential Pair
The explicitly finite integer domain is U=[1, 2, 3, -2, -1] and P(x) means x≥0. Give (¬∀x∈U P(x), ¬∃x∈U P(x)) using True/False, retaining the empty-domain convention.
View answer and explanation
(True, False)
A false element negates the universal; absence of a true element negates the existential. The empty-domain pair is (False, True). Do not extend this finite enumeration to an infinite domain. The exact result, retaining the requested order, is (True, False).
Reviewed step-by-step derivation
Quantifiers and explicit domains · qf-density10-quantifier-negation-pair · candidate
45. Negated Universal and Existential Pair
The explicitly finite integer domain is U=[1, 2, 3, 4, -1] and P(x) means x≥0. Give (¬∀x∈U P(x), ¬∃x∈U P(x)) using True/False, retaining the empty-domain convention.
View answer and explanation
(True, False)
A false element negates the universal; absence of a true element negates the existential. The empty-domain pair is (False, True). Do not extend this finite enumeration to an infinite domain. The exact result, retaining the requested order, is (True, False).
Reviewed step-by-step derivation
Quantifiers and explicit domains · qf-density10-quantifier-negation-pair · candidate
46. Negated Universal and Existential Pair
The explicitly finite integer domain is U=[1, 2, 3, 4, 5] and P(x) means x≥0. Give (¬∀x∈U P(x), ¬∃x∈U P(x)) using True/False, retaining the empty-domain convention.
View answer and explanation
(False, False)
A false element negates the universal; absence of a true element negates the existential. The empty-domain pair is (False, True). Do not extend this finite enumeration to an infinite domain. The exact result, retaining the requested order, is (False, False).
Reviewed step-by-step derivation
Quantifiers and explicit domains · qf-density10-quantifier-negation-pair · candidate
47. Negated Universal and Existential Pair
The explicitly finite integer domain is U=[-6, -5, -4, -3, -2, -1] and P(x) means x≥0. Give (¬∀x∈U P(x), ¬∃x∈U P(x)) using True/False, retaining the empty-domain convention.
View answer and explanation
(True, True)
A false element negates the universal; absence of a true element negates the existential. The empty-domain pair is (False, True). Do not extend this finite enumeration to an infinite domain. The exact result, retaining the requested order, is (True, True).
Reviewed step-by-step derivation
Quantifiers and explicit domains · qf-density10-quantifier-negation-pair · candidate
48. Negated Universal and Existential Pair
The explicitly finite integer domain is U=[1, -5, -4, -3, -2, -1] and P(x) means x≥0. Give (¬∀x∈U P(x), ¬∃x∈U P(x)) using True/False, retaining the empty-domain convention.
View answer and explanation
(True, False)
A false element negates the universal; absence of a true element negates the existential. The empty-domain pair is (False, True). Do not extend this finite enumeration to an infinite domain. The exact result, retaining the requested order, is (True, False).
Reviewed step-by-step derivation
Quantifiers and explicit domains · qf-density10-quantifier-negation-pair · candidate
49. Negated Universal and Existential Pair
The explicitly finite integer domain is U=[1, 2, -4, -3, -2, -1] and P(x) means x≥0. Give (¬∀x∈U P(x), ¬∃x∈U P(x)) using True/False, retaining the empty-domain convention.
View answer and explanation
(True, False)
A false element negates the universal; absence of a true element negates the existential. The empty-domain pair is (False, True). Do not extend this finite enumeration to an infinite domain. The exact result, retaining the requested order, is (True, False).
Reviewed step-by-step derivation
Quantifiers and explicit domains · qf-density10-quantifier-negation-pair · candidate
50. Negated Universal and Existential Pair
The explicitly finite integer domain is U=[1, 2, 3, -3, -2, -1] and P(x) means x≥0. Give (¬∀x∈U P(x), ¬∃x∈U P(x)) using True/False, retaining the empty-domain convention.
View answer and explanation
(True, False)
A false element negates the universal; absence of a true element negates the existential. The empty-domain pair is (False, True). Do not extend this finite enumeration to an infinite domain. The exact result, retaining the requested order, is (True, False).
Reviewed step-by-step derivation
Quantifiers and explicit domains · qf-density10-quantifier-negation-pair · candidate
51. In-Domain Counterexample Candidate Mask
The explicitly finite integer domain is U=[] and P(x) means x≥0. Ordered candidates=[100, 101, 102]. For each, answer Yes/No to membership in U together with failure of P; outside candidates are ineligible.
View answer and explanation
[No, No, No]
Check the domain before the negated predicate; outside candidates and satisfying domain elements are not counterexamples. Do not extend this finite enumeration to an infinite domain. The exact result, retaining the requested order, is [No, No, No].
Reviewed step-by-step derivation
Quantifiers and explicit domains · qf-density10-quantifier-candidate-mask · candidate
52. In-Domain Counterexample Candidate Mask
The explicitly finite integer domain is U=[-3, -2, -1, 1, 2, 3] and P(x) means x≥0. Ordered candidates=[-1, 101, 102]. For each, answer Yes/No to membership in U together with failure of P; outside candidates are ineligible.
View answer and explanation
[Yes, No, No]
Check the domain before the negated predicate; outside candidates and satisfying domain elements are not counterexamples. Do not extend this finite enumeration to an infinite domain. The exact result, retaining the requested order, is [Yes, No, No].
Reviewed step-by-step derivation
Quantifiers and explicit domains · qf-density10-quantifier-candidate-mask · candidate
53. In-Domain Counterexample Candidate Mask
The explicitly finite integer domain is U=[-3, -2, -1, 1, 2, 3] and P(x) means x≥0. Ordered candidates=[1, 2, 102]. For each, answer Yes/No to membership in U together with failure of P; outside candidates are ineligible.
View answer and explanation
[No, No, No]
Check the domain before the negated predicate; outside candidates and satisfying domain elements are not counterexamples. Do not extend this finite enumeration to an infinite domain. The exact result, retaining the requested order, is [No, No, No].
Reviewed step-by-step derivation
Quantifiers and explicit domains · qf-density10-quantifier-candidate-mask · candidate
54. In-Domain Counterexample Candidate Mask
The explicitly finite integer domain is U=[-3, -2, -1, 1, 2, 3] and P(x) means x≥0. Ordered candidates=[-1, 2, 102]. For each, answer Yes/No to membership in U together with failure of P; outside candidates are ineligible.
View answer and explanation
[Yes, No, No]
Check the domain before the negated predicate; outside candidates and satisfying domain elements are not counterexamples. Do not extend this finite enumeration to an infinite domain. The exact result, retaining the requested order, is [Yes, No, No].
Reviewed step-by-step derivation
Quantifiers and explicit domains · qf-density10-quantifier-candidate-mask · candidate
55. In-Domain Counterexample Candidate Mask
The explicitly finite integer domain is U=[-3, -2, -1, 1, 2, 3] and P(x) means x≥0. Ordered candidates=[100, -2, 102]. For each, answer Yes/No to membership in U together with failure of P; outside candidates are ineligible.
View answer and explanation
[No, Yes, No]
Check the domain before the negated predicate; outside candidates and satisfying domain elements are not counterexamples. Do not extend this finite enumeration to an infinite domain. The exact result, retaining the requested order, is [No, Yes, No].
Reviewed step-by-step derivation
Quantifiers and explicit domains · qf-density10-quantifier-candidate-mask · candidate
56. In-Domain Counterexample Candidate Mask
The explicitly finite integer domain is U=[-3, -2, -1, 1, 2, 3] and P(x) means x≥0. Ordered candidates=[1, -2, 102]. For each, answer Yes/No to membership in U together with failure of P; outside candidates are ineligible.
View answer and explanation
[No, Yes, No]
Check the domain before the negated predicate; outside candidates and satisfying domain elements are not counterexamples. Do not extend this finite enumeration to an infinite domain. The exact result, retaining the requested order, is [No, Yes, No].
Reviewed step-by-step derivation
Quantifiers and explicit domains · qf-density10-quantifier-candidate-mask · candidate
57. In-Domain Counterexample Candidate Mask
The explicitly finite integer domain is U=[-3, -2, -1, 1, 2, 3] and P(x) means x≥0. Ordered candidates=[-1, -2, 102]. For each, answer Yes/No to membership in U together with failure of P; outside candidates are ineligible.
View answer and explanation
[Yes, Yes, No]
Check the domain before the negated predicate; outside candidates and satisfying domain elements are not counterexamples. Do not extend this finite enumeration to an infinite domain. The exact result, retaining the requested order, is [Yes, Yes, No].
Reviewed step-by-step derivation
Quantifiers and explicit domains · qf-density10-quantifier-candidate-mask · candidate
58. In-Domain Counterexample Candidate Mask
The explicitly finite integer domain is U=[-3, -2, -1, 1, 2, 3] and P(x) means x≥0. Ordered candidates=[100, 101, 3]. For each, answer Yes/No to membership in U together with failure of P; outside candidates are ineligible.
View answer and explanation
[No, No, No]
Check the domain before the negated predicate; outside candidates and satisfying domain elements are not counterexamples. Do not extend this finite enumeration to an infinite domain. The exact result, retaining the requested order, is [No, No, No].
Reviewed step-by-step derivation
Quantifiers and explicit domains · qf-density10-quantifier-candidate-mask · candidate
59. In-Domain Counterexample Candidate Mask
The explicitly finite integer domain is U=[-3, -2, -1, 1, 2, 3] and P(x) means x≥0. Ordered candidates=[1, 101, 3]. For each, answer Yes/No to membership in U together with failure of P; outside candidates are ineligible.
View answer and explanation
[No, No, No]
Check the domain before the negated predicate; outside candidates and satisfying domain elements are not counterexamples. Do not extend this finite enumeration to an infinite domain. The exact result, retaining the requested order, is [No, No, No].
Reviewed step-by-step derivation
Quantifiers and explicit domains · qf-density10-quantifier-candidate-mask · candidate
60. In-Domain Counterexample Candidate Mask
The explicitly finite integer domain is U=[-3, -2, -1, 1, 2, 3] and P(x) means x≥0. Ordered candidates=[-1, 101, 3]. For each, answer Yes/No to membership in U together with failure of P; outside candidates are ineligible.
View answer and explanation
[Yes, No, No]
Check the domain before the negated predicate; outside candidates and satisfying domain elements are not counterexamples. Do not extend this finite enumeration to an infinite domain. The exact result, retaining the requested order, is [Yes, No, No].
Reviewed step-by-step derivation
Quantifiers and explicit domains · qf-density10-quantifier-candidate-mask · candidate
61. In-Domain Counterexample Candidate Mask
The explicitly finite integer domain is U=[-3, -2, -1, 1, 2, 3] and P(x) means x≥0. Ordered candidates=[100, 2, 3]. For each, answer Yes/No to membership in U together with failure of P; outside candidates are ineligible.
View answer and explanation
[No, No, No]
Check the domain before the negated predicate; outside candidates and satisfying domain elements are not counterexamples. Do not extend this finite enumeration to an infinite domain. The exact result, retaining the requested order, is [No, No, No].
Reviewed step-by-step derivation
Quantifiers and explicit domains · qf-density10-quantifier-candidate-mask · candidate
62. In-Domain Counterexample Candidate Mask
The explicitly finite integer domain is U=[-3, -2, -1, 1, 2, 3] and P(x) means x≥0. Ordered candidates=[1, 2, 3]. For each, answer Yes/No to membership in U together with failure of P; outside candidates are ineligible.
View answer and explanation
[No, No, No]
Check the domain before the negated predicate; outside candidates and satisfying domain elements are not counterexamples. Do not extend this finite enumeration to an infinite domain. The exact result, retaining the requested order, is [No, No, No].
Reviewed step-by-step derivation
Quantifiers and explicit domains · qf-density10-quantifier-candidate-mask · candidate
63. In-Domain Counterexample Candidate Mask
The explicitly finite integer domain is U=[-3, -2, -1, 1, 2, 3] and P(x) means x≥0. Ordered candidates=[-1, 2, 3]. For each, answer Yes/No to membership in U together with failure of P; outside candidates are ineligible.
View answer and explanation
[Yes, No, No]
Check the domain before the negated predicate; outside candidates and satisfying domain elements are not counterexamples. Do not extend this finite enumeration to an infinite domain. The exact result, retaining the requested order, is [Yes, No, No].
Reviewed step-by-step derivation
Quantifiers and explicit domains · qf-density10-quantifier-candidate-mask · candidate
64. In-Domain Counterexample Candidate Mask
The explicitly finite integer domain is U=[-3, -2, -1, 1, 2, 3] and P(x) means x≥0. Ordered candidates=[100, -2, 3]. For each, answer Yes/No to membership in U together with failure of P; outside candidates are ineligible.
View answer and explanation
[No, Yes, No]
Check the domain before the negated predicate; outside candidates and satisfying domain elements are not counterexamples. Do not extend this finite enumeration to an infinite domain. The exact result, retaining the requested order, is [No, Yes, No].
Reviewed step-by-step derivation
Quantifiers and explicit domains · qf-density10-quantifier-candidate-mask · candidate
65. In-Domain Counterexample Candidate Mask
The explicitly finite integer domain is U=[-3, -2, -1, 1, 2, 3] and P(x) means x≥0. Ordered candidates=[1, -2, 3]. For each, answer Yes/No to membership in U together with failure of P; outside candidates are ineligible.
View answer and explanation
[No, Yes, No]
Check the domain before the negated predicate; outside candidates and satisfying domain elements are not counterexamples. Do not extend this finite enumeration to an infinite domain. The exact result, retaining the requested order, is [No, Yes, No].
Reviewed step-by-step derivation
Quantifiers and explicit domains · qf-density10-quantifier-candidate-mask · candidate
66. In-Domain Counterexample Candidate Mask
The explicitly finite integer domain is U=[-3, -2, -1, 1, 2, 3] and P(x) means x≥0. Ordered candidates=[-1, -2, 3]. For each, answer Yes/No to membership in U together with failure of P; outside candidates are ineligible.
View answer and explanation
[Yes, Yes, No]
Check the domain before the negated predicate; outside candidates and satisfying domain elements are not counterexamples. Do not extend this finite enumeration to an infinite domain. The exact result, retaining the requested order, is [Yes, Yes, No].
Reviewed step-by-step derivation
Quantifiers and explicit domains · qf-density10-quantifier-candidate-mask · candidate
67. In-Domain Counterexample Candidate Mask
The explicitly finite integer domain is U=[-3, -2, -1, 1, 2, 3] and P(x) means x≥0. Ordered candidates=[100, 101, -3]. For each, answer Yes/No to membership in U together with failure of P; outside candidates are ineligible.
View answer and explanation
[No, No, Yes]
Check the domain before the negated predicate; outside candidates and satisfying domain elements are not counterexamples. Do not extend this finite enumeration to an infinite domain. The exact result, retaining the requested order, is [No, No, Yes].
Reviewed step-by-step derivation
Quantifiers and explicit domains · qf-density10-quantifier-candidate-mask · candidate
68. In-Domain Counterexample Candidate Mask
The explicitly finite integer domain is U=[-3, -2, -1, 1, 2, 3] and P(x) means x≥0. Ordered candidates=[1, 101, -3]. For each, answer Yes/No to membership in U together with failure of P; outside candidates are ineligible.
View answer and explanation
[No, No, Yes]
Check the domain before the negated predicate; outside candidates and satisfying domain elements are not counterexamples. Do not extend this finite enumeration to an infinite domain. The exact result, retaining the requested order, is [No, No, Yes].
Reviewed step-by-step derivation
Quantifiers and explicit domains · qf-density10-quantifier-candidate-mask · candidate
69. In-Domain Counterexample Candidate Mask
The explicitly finite integer domain is U=[-3, -2, -1, 1, 2, 3] and P(x) means x≥0. Ordered candidates=[-1, 101, -3]. For each, answer Yes/No to membership in U together with failure of P; outside candidates are ineligible.
View answer and explanation
[Yes, No, Yes]
Check the domain before the negated predicate; outside candidates and satisfying domain elements are not counterexamples. Do not extend this finite enumeration to an infinite domain. The exact result, retaining the requested order, is [Yes, No, Yes].
Reviewed step-by-step derivation
Quantifiers and explicit domains · qf-density10-quantifier-candidate-mask · candidate
70. In-Domain Counterexample Candidate Mask
The explicitly finite integer domain is U=[-3, -2, -1, 1, 2, 3] and P(x) means x≥0. Ordered candidates=[100, 2, -3]. For each, answer Yes/No to membership in U together with failure of P; outside candidates are ineligible.
View answer and explanation
[No, No, Yes]
Check the domain before the negated predicate; outside candidates and satisfying domain elements are not counterexamples. Do not extend this finite enumeration to an infinite domain. The exact result, retaining the requested order, is [No, No, Yes].
Reviewed step-by-step derivation
Quantifiers and explicit domains · qf-density10-quantifier-candidate-mask · candidate
71. In-Domain Counterexample Candidate Mask
The explicitly finite integer domain is U=[-3, -2, -1, 1, 2, 3] and P(x) means x≥0. Ordered candidates=[1, 2, -3]. For each, answer Yes/No to membership in U together with failure of P; outside candidates are ineligible.
View answer and explanation
[No, No, Yes]
Check the domain before the negated predicate; outside candidates and satisfying domain elements are not counterexamples. Do not extend this finite enumeration to an infinite domain. The exact result, retaining the requested order, is [No, No, Yes].
Reviewed step-by-step derivation
Quantifiers and explicit domains · qf-density10-quantifier-candidate-mask · candidate
72. In-Domain Counterexample Candidate Mask
The explicitly finite integer domain is U=[-3, -2, -1, 1, 2, 3] and P(x) means x≥0. Ordered candidates=[-1, 2, -3]. For each, answer Yes/No to membership in U together with failure of P; outside candidates are ineligible.
View answer and explanation
[Yes, No, Yes]
Check the domain before the negated predicate; outside candidates and satisfying domain elements are not counterexamples. Do not extend this finite enumeration to an infinite domain. The exact result, retaining the requested order, is [Yes, No, Yes].
Reviewed step-by-step derivation
Quantifiers and explicit domains · qf-density10-quantifier-candidate-mask · candidate
73. In-Domain Counterexample Candidate Mask
The explicitly finite integer domain is U=[-3, -2, -1, 1, 2, 3] and P(x) means x≥0. Ordered candidates=[100, -2, -3]. For each, answer Yes/No to membership in U together with failure of P; outside candidates are ineligible.
View answer and explanation
[No, Yes, Yes]
Check the domain before the negated predicate; outside candidates and satisfying domain elements are not counterexamples. Do not extend this finite enumeration to an infinite domain. The exact result, retaining the requested order, is [No, Yes, Yes].
Reviewed step-by-step derivation
Quantifiers and explicit domains · qf-density10-quantifier-candidate-mask · candidate
74. In-Domain Counterexample Candidate Mask
The explicitly finite integer domain is U=[-3, -2, -1, 1, 2, 3] and P(x) means x≥0. Ordered candidates=[1, -2, -3]. For each, answer Yes/No to membership in U together with failure of P; outside candidates are ineligible.
View answer and explanation
[No, Yes, Yes]
Check the domain before the negated predicate; outside candidates and satisfying domain elements are not counterexamples. Do not extend this finite enumeration to an infinite domain. The exact result, retaining the requested order, is [No, Yes, Yes].
Reviewed step-by-step derivation
Quantifiers and explicit domains · qf-density10-quantifier-candidate-mask · candidate
75. In-Domain Counterexample Candidate Mask
The explicitly finite integer domain is U=[-3, -2, -1, 1, 2, 3] and P(x) means x≥0. Ordered candidates=[-1, -2, -3]. For each, answer Yes/No to membership in U together with failure of P; outside candidates are ineligible.
View answer and explanation
[Yes, Yes, Yes]
Check the domain before the negated predicate; outside candidates and satisfying domain elements are not counterexamples. Do not extend this finite enumeration to an infinite domain. The exact result, retaining the requested order, is [Yes, Yes, Yes].
Reviewed step-by-step derivation
Quantifiers and explicit domains · qf-density10-quantifier-candidate-mask · candidate
76. Two Ordered Nested-Quantifier Counts
U=[], V=[] are explicitly finite integer domains. P(x,y) means x=y. Give (number of x∈U with some y∈V satisfying P, number of y∈V satisfying P for every x∈U).
View answer and explanation
(0, 0)
Enumerate the specified inner existential or universal without exchanging quantifiers; an empty-domain universal is true and existential is false. Do not extend this finite enumeration to an infinite domain. The exact result, retaining the requested order, is (0, 0).
Reviewed step-by-step derivation
Quantifiers and explicit domains · qf-density10-quantifier-nested-counts · candidate
77. Two Ordered Nested-Quantifier Counts
U=[], V=[-1, 1] are explicitly finite integer domains. P(x,y) means x≤y. Give (number of x∈U with some y∈V satisfying P, number of y∈V satisfying P for every x∈U).
View answer and explanation
(0, 2)
Enumerate the specified inner existential or universal without exchanging quantifiers; an empty-domain universal is true and existential is false. Do not extend this finite enumeration to an infinite domain. The exact result, retaining the requested order, is (0, 2).
Reviewed step-by-step derivation
Quantifiers and explicit domains · qf-density10-quantifier-nested-counts · candidate
78. Two Ordered Nested-Quantifier Counts
U=[1, 2], V=[] are explicitly finite integer domains. P(x,y) means x≤y. Give (number of x∈U with some y∈V satisfying P, number of y∈V satisfying P for every x∈U).
View answer and explanation
(0, 0)
Enumerate the specified inner existential or universal without exchanging quantifiers; an empty-domain universal is true and existential is false. Do not extend this finite enumeration to an infinite domain. The exact result, retaining the requested order, is (0, 0).
Reviewed step-by-step derivation
Quantifiers and explicit domains · qf-density10-quantifier-nested-counts · candidate
79. Two Ordered Nested-Quantifier Counts
U=[-1, 0, 1], V=[-2, 0, 2] are explicitly finite integer domains. P(x,y) means x≤y. Give (number of x∈U with some y∈V satisfying P, number of y∈V satisfying P for every x∈U).
View answer and explanation
(3, 1)
Enumerate the specified inner existential or universal without exchanging quantifiers; an empty-domain universal is true and existential is false. Do not extend this finite enumeration to an infinite domain. The exact result, retaining the requested order, is (3, 1).
Reviewed step-by-step derivation
Quantifiers and explicit domains · qf-density10-quantifier-nested-counts · candidate
80. Two Ordered Nested-Quantifier Counts
U=[-2, -1, 0, 1, 2], V=[-1, 0, 1] are explicitly finite integer domains. P(x,y) means x≤y. Give (number of x∈U with some y∈V satisfying P, number of y∈V satisfying P for every x∈U).
View answer and explanation
(4, 0)
Enumerate the specified inner existential or universal without exchanging quantifiers; an empty-domain universal is true and existential is false. Do not extend this finite enumeration to an infinite domain. The exact result, retaining the requested order, is (4, 0).
Reviewed step-by-step derivation
Quantifiers and explicit domains · qf-density10-quantifier-nested-counts · candidate
81. Two Ordered Nested-Quantifier Counts
U=[-1], V=[100] are explicitly finite integer domains. P(x,y) means x=y. Give (number of x∈U with some y∈V satisfying P, number of y∈V satisfying P for every x∈U).
View answer and explanation
(0, 0)
Enumerate the specified inner existential or universal without exchanging quantifiers; an empty-domain universal is true and existential is false. Do not extend this finite enumeration to an infinite domain. The exact result, retaining the requested order, is (0, 0).
Reviewed step-by-step derivation
Quantifiers and explicit domains · qf-density10-quantifier-nested-counts · candidate
82. Two Ordered Nested-Quantifier Counts
U=[0], V=[0] are explicitly finite integer domains. P(x,y) means x=y. Give (number of x∈U with some y∈V satisfying P, number of y∈V satisfying P for every x∈U).
View answer and explanation
(1, 1)
Enumerate the specified inner existential or universal without exchanging quantifiers; an empty-domain universal is true and existential is false. Do not extend this finite enumeration to an infinite domain. The exact result, retaining the requested order, is (1, 1).
Reviewed step-by-step derivation
Quantifiers and explicit domains · qf-density10-quantifier-nested-counts · candidate
83. Two Ordered Nested-Quantifier Counts
U=[-1], V=[100, 101] are explicitly finite integer domains. P(x,y) means x=y. Give (number of x∈U with some y∈V satisfying P, number of y∈V satisfying P for every x∈U).
View answer and explanation
(0, 0)
Enumerate the specified inner existential or universal without exchanging quantifiers; an empty-domain universal is true and existential is false. Do not extend this finite enumeration to an infinite domain. The exact result, retaining the requested order, is (0, 0).
Reviewed step-by-step derivation
Quantifiers and explicit domains · qf-density10-quantifier-nested-counts · candidate
84. Two Ordered Nested-Quantifier Counts
U=[0], V=[0, 100] are explicitly finite integer domains. P(x,y) means x=y. Give (number of x∈U with some y∈V satisfying P, number of y∈V satisfying P for every x∈U).
View answer and explanation
(1, 1)
Enumerate the specified inner existential or universal without exchanging quantifiers; an empty-domain universal is true and existential is false. Do not extend this finite enumeration to an infinite domain. The exact result, retaining the requested order, is (1, 1).
Reviewed step-by-step derivation
Quantifiers and explicit domains · qf-density10-quantifier-nested-counts · candidate
85. Two Ordered Nested-Quantifier Counts
U=[-1], V=[100, 101, 102] are explicitly finite integer domains. P(x,y) means x=y. Give (number of x∈U with some y∈V satisfying P, number of y∈V satisfying P for every x∈U).
View answer and explanation
(0, 0)
Enumerate the specified inner existential or universal without exchanging quantifiers; an empty-domain universal is true and existential is false. Do not extend this finite enumeration to an infinite domain. The exact result, retaining the requested order, is (0, 0).
Reviewed step-by-step derivation
Quantifiers and explicit domains · qf-density10-quantifier-nested-counts · candidate
86. Two Ordered Nested-Quantifier Counts
U=[0], V=[0, 100, 101] are explicitly finite integer domains. P(x,y) means x=y. Give (number of x∈U with some y∈V satisfying P, number of y∈V satisfying P for every x∈U).
View answer and explanation
(1, 1)
Enumerate the specified inner existential or universal without exchanging quantifiers; an empty-domain universal is true and existential is false. Do not extend this finite enumeration to an infinite domain. The exact result, retaining the requested order, is (1, 1).
Reviewed step-by-step derivation
Quantifiers and explicit domains · qf-density10-quantifier-nested-counts · candidate
87. Two Ordered Nested-Quantifier Counts
U=[-1], V=[100, 101, 102, 103] are explicitly finite integer domains. P(x,y) means x=y. Give (number of x∈U with some y∈V satisfying P, number of y∈V satisfying P for every x∈U).
View answer and explanation
(0, 0)
Enumerate the specified inner existential or universal without exchanging quantifiers; an empty-domain universal is true and existential is false. Do not extend this finite enumeration to an infinite domain. The exact result, retaining the requested order, is (0, 0).
Reviewed step-by-step derivation
Quantifiers and explicit domains · qf-density10-quantifier-nested-counts · candidate
88. Two Ordered Nested-Quantifier Counts
U=[0], V=[0, 100, 101, 102] are explicitly finite integer domains. P(x,y) means x=y. Give (number of x∈U with some y∈V satisfying P, number of y∈V satisfying P for every x∈U).
View answer and explanation
(1, 1)
Enumerate the specified inner existential or universal without exchanging quantifiers; an empty-domain universal is true and existential is false. Do not extend this finite enumeration to an infinite domain. The exact result, retaining the requested order, is (1, 1).
Reviewed step-by-step derivation
Quantifiers and explicit domains · qf-density10-quantifier-nested-counts · candidate
89. Two Ordered Nested-Quantifier Counts
U=[-2, -1], V=[100] are explicitly finite integer domains. P(x,y) means x=y. Give (number of x∈U with some y∈V satisfying P, number of y∈V satisfying P for every x∈U).
View answer and explanation
(0, 0)
Enumerate the specified inner existential or universal without exchanging quantifiers; an empty-domain universal is true and existential is false. Do not extend this finite enumeration to an infinite domain. The exact result, retaining the requested order, is (0, 0).
Reviewed step-by-step derivation
Quantifiers and explicit domains · qf-density10-quantifier-nested-counts · candidate
90. Two Ordered Nested-Quantifier Counts
U=[0, -1], V=[0] are explicitly finite integer domains. P(x,y) means x=y. Give (number of x∈U with some y∈V satisfying P, number of y∈V satisfying P for every x∈U).
View answer and explanation
(1, 0)
Enumerate the specified inner existential or universal without exchanging quantifiers; an empty-domain universal is true and existential is false. Do not extend this finite enumeration to an infinite domain. The exact result, retaining the requested order, is (1, 0).
Reviewed step-by-step derivation
Quantifiers and explicit domains · qf-density10-quantifier-nested-counts · candidate
91. Two Ordered Nested-Quantifier Counts
U=[-2, -1], V=[100, 101] are explicitly finite integer domains. P(x,y) means x=y. Give (number of x∈U with some y∈V satisfying P, number of y∈V satisfying P for every x∈U).
View answer and explanation
(0, 0)
Enumerate the specified inner existential or universal without exchanging quantifiers; an empty-domain universal is true and existential is false. Do not extend this finite enumeration to an infinite domain. The exact result, retaining the requested order, is (0, 0).
Reviewed step-by-step derivation
Quantifiers and explicit domains · qf-density10-quantifier-nested-counts · candidate
92. Two Ordered Nested-Quantifier Counts
U=[0, -1], V=[0, 100] are explicitly finite integer domains. P(x,y) means x=y. Give (number of x∈U with some y∈V satisfying P, number of y∈V satisfying P for every x∈U).
View answer and explanation
(1, 0)
Enumerate the specified inner existential or universal without exchanging quantifiers; an empty-domain universal is true and existential is false. Do not extend this finite enumeration to an infinite domain. The exact result, retaining the requested order, is (1, 0).
Reviewed step-by-step derivation
Quantifiers and explicit domains · qf-density10-quantifier-nested-counts · candidate
93. Two Ordered Nested-Quantifier Counts
U=[0, 1], V=[0, 1] are explicitly finite integer domains. P(x,y) means x=y. Give (number of x∈U with some y∈V satisfying P, number of y∈V satisfying P for every x∈U).
View answer and explanation
(2, 0)
Enumerate the specified inner existential or universal without exchanging quantifiers; an empty-domain universal is true and existential is false. Do not extend this finite enumeration to an infinite domain. The exact result, retaining the requested order, is (2, 0).
Reviewed step-by-step derivation
Quantifiers and explicit domains · qf-density10-quantifier-nested-counts · candidate
94. Two Ordered Nested-Quantifier Counts
U=[-2, -1], V=[100, 101, 102] are explicitly finite integer domains. P(x,y) means x=y. Give (number of x∈U with some y∈V satisfying P, number of y∈V satisfying P for every x∈U).
View answer and explanation
(0, 0)
Enumerate the specified inner existential or universal without exchanging quantifiers; an empty-domain universal is true and existential is false. Do not extend this finite enumeration to an infinite domain. The exact result, retaining the requested order, is (0, 0).
Reviewed step-by-step derivation
Quantifiers and explicit domains · qf-density10-quantifier-nested-counts · candidate
95. Two Ordered Nested-Quantifier Counts
U=[0, -1], V=[0, 100, 101] are explicitly finite integer domains. P(x,y) means x=y. Give (number of x∈U with some y∈V satisfying P, number of y∈V satisfying P for every x∈U).
View answer and explanation
(1, 0)
Enumerate the specified inner existential or universal without exchanging quantifiers; an empty-domain universal is true and existential is false. Do not extend this finite enumeration to an infinite domain. The exact result, retaining the requested order, is (1, 0).
Reviewed step-by-step derivation
Quantifiers and explicit domains · qf-density10-quantifier-nested-counts · candidate
96. Two Ordered Nested-Quantifier Counts
U=[0, 1], V=[0, 1, 100] are explicitly finite integer domains. P(x,y) means x=y. Give (number of x∈U with some y∈V satisfying P, number of y∈V satisfying P for every x∈U).
View answer and explanation
(2, 0)
Enumerate the specified inner existential or universal without exchanging quantifiers; an empty-domain universal is true and existential is false. Do not extend this finite enumeration to an infinite domain. The exact result, retaining the requested order, is (2, 0).
Reviewed step-by-step derivation
Quantifiers and explicit domains · qf-density10-quantifier-nested-counts · candidate
97. Two Ordered Nested-Quantifier Counts
U=[-2, -1], V=[100, 101, 102, 103] are explicitly finite integer domains. P(x,y) means x=y. Give (number of x∈U with some y∈V satisfying P, number of y∈V satisfying P for every x∈U).
View answer and explanation
(0, 0)
Enumerate the specified inner existential or universal without exchanging quantifiers; an empty-domain universal is true and existential is false. Do not extend this finite enumeration to an infinite domain. The exact result, retaining the requested order, is (0, 0).
Reviewed step-by-step derivation
Quantifiers and explicit domains · qf-density10-quantifier-nested-counts · candidate
98. Two Ordered Nested-Quantifier Counts
U=[0, -1], V=[0, 100, 101, 102] are explicitly finite integer domains. P(x,y) means x=y. Give (number of x∈U with some y∈V satisfying P, number of y∈V satisfying P for every x∈U).
View answer and explanation
(1, 0)
Enumerate the specified inner existential or universal without exchanging quantifiers; an empty-domain universal is true and existential is false. Do not extend this finite enumeration to an infinite domain. The exact result, retaining the requested order, is (1, 0).
Reviewed step-by-step derivation
Quantifiers and explicit domains · qf-density10-quantifier-nested-counts · candidate
99. Two Ordered Nested-Quantifier Counts
U=[0, 1], V=[0, 1, 100, 101] are explicitly finite integer domains. P(x,y) means x=y. Give (number of x∈U with some y∈V satisfying P, number of y∈V satisfying P for every x∈U).
View answer and explanation
(2, 0)
Enumerate the specified inner existential or universal without exchanging quantifiers; an empty-domain universal is true and existential is false. Do not extend this finite enumeration to an infinite domain. The exact result, retaining the requested order, is (2, 0).
Reviewed step-by-step derivation
Quantifiers and explicit domains · qf-density10-quantifier-nested-counts · candidate
100. Two Ordered Nested-Quantifier Counts
U=[-3, -2, -1], V=[100] are explicitly finite integer domains. P(x,y) means x=y. Give (number of x∈U with some y∈V satisfying P, number of y∈V satisfying P for every x∈U).
View answer and explanation
(0, 0)
Enumerate the specified inner existential or universal without exchanging quantifiers; an empty-domain universal is true and existential is false. Do not extend this finite enumeration to an infinite domain. The exact result, retaining the requested order, is (0, 0).
Reviewed step-by-step derivation
Quantifiers and explicit domains · qf-density10-quantifier-nested-counts · candidate
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