# Finite Predicate Partitions and Ordered Quantifier Counts

This batch has passed Schema, deduplication, source-snapshot, and independent-answer recomputation checks; it is still not marked as human-reviewed.

This question bank has corresponding bilingual step-by-step derivations. Source candidate and English ai_candidate states are retained. AI review, finite computation and Lean formal proof are labelled separately.

batch-20261008-advance-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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r07-quantifier-0001/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r07-quantifier-0002/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r07-quantifier-0003/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r07-quantifier-0004/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r07-quantifier-0005/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r07-quantifier-0006/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r07-quantifier-0007/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r07-quantifier-0008/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r07-quantifier-0009/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r07-quantifier-0010/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r07-quantifier-0011/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r07-quantifier-0012/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r07-quantifier-0013/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r07-quantifier-0014/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r07-quantifier-0015/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r07-quantifier-0016/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r07-quantifier-0017/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r07-quantifier-0018/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r07-quantifier-0019/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r07-quantifier-0020/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r07-quantifier-0021/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r07-quantifier-0022/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r07-quantifier-0023/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r07-quantifier-0024/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r07-quantifier-0025/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r07-quantifier-0026/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r07-quantifier-0027/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r07-quantifier-0028/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r07-quantifier-0029/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r07-quantifier-0030/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r07-quantifier-0031/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r07-quantifier-0032/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r07-quantifier-0033/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r07-quantifier-0034/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r07-quantifier-0035/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r07-quantifier-0036/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r07-quantifier-0037/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r07-quantifier-0038/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r07-quantifier-0039/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r07-quantifier-0040/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r07-quantifier-0041/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r07-quantifier-0042/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r07-quantifier-0043/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r07-quantifier-0044/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r07-quantifier-0045/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r07-quantifier-0046/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r07-quantifier-0047/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r07-quantifier-0048/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r07-quantifier-0049/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r07-quantifier-0050/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r07-quantifier-0051/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r07-quantifier-0052/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r07-quantifier-0053/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r07-quantifier-0054/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r07-quantifier-0055/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r07-quantifier-0056/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r07-quantifier-0057/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r07-quantifier-0058/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r07-quantifier-0059/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r07-quantifier-0060/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r07-quantifier-0061/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r07-quantifier-0062/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r07-quantifier-0063/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r07-quantifier-0064/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r07-quantifier-0065/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r07-quantifier-0066/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r07-quantifier-0067/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r07-quantifier-0068/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r07-quantifier-0069/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r07-quantifier-0070/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r07-quantifier-0071/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r07-quantifier-0072/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r07-quantifier-0073/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r07-quantifier-0074/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r07-quantifier-0075/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r07-quantifier-0076/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r07-quantifier-0077/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r07-quantifier-0078/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r07-quantifier-0079/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r07-quantifier-0080/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r07-quantifier-0081/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r07-quantifier-0082/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r07-quantifier-0083/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r07-quantifier-0084/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r07-quantifier-0085/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r07-quantifier-0086/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r07-quantifier-0087/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r07-quantifier-0088/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r07-quantifier-0089/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r07-quantifier-0090/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r07-quantifier-0091/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r07-quantifier-0092/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r07-quantifier-0093/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r07-quantifier-0094/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r07-quantifier-0095/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r07-quantifier-0096/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r07-quantifier-0097/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r07-quantifier-0098/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r07-quantifier-0099/)

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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r07-quantifier-0100/)

Quantifiers and explicit domains · qf-density10-quantifier-nested-counts · candidate



[Canonical question JSONL](https://ocme.evemisslab.com/data/questions/batches/batch-20261008-advance-20261010-r07-quantifier-foundations/questions.jsonl) · [English JSONL](https://ocme.evemisslab.com/data/i18n/questions/batch-20261008-advance-20261010-r07-quantifier-foundations.en.jsonl) · [Validation report](https://ocme.evemisslab.com/data/questions/batches/batch-20261008-advance-20261010-r07-quantifier-foundations/validation-report.json)