# Counterexamples and Domain Eligibility

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-20261003-advance-20261004-r02-counterexample-foundations · 2026-10-03 · 100 questions

This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.



## 1. Domain Eligibility and Predicate Failure

Using classical truth semantics, the explicit finite integer domain is U=∅ and P(x) means x≥0. For the claim ∀x∈U P(x), is c=0 a valid counterexample?

View answer and explanationNo

c does not belong to U and its integer predicate value is true, so the answer is No. An outside value cannot refute this restricted claim even when its integer predicate fails. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r02-counterexample-0001/)

Counterexamples and domain eligibility · qf-counterexample-witness-validity · candidate





## 2. Domain Eligibility and Predicate Failure

Using classical truth semantics, the explicit finite integer domain is U={-2, -1, 0} and P(x) means x≥0. For the claim ∀x∈U P(x), is c=-1 a valid counterexample?

View answer and explanationYes

c belongs to U and its integer predicate value is false, so the answer is Yes. An outside value cannot refute this restricted claim even when its integer predicate fails. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r02-counterexample-0002/)

Counterexamples and domain eligibility · qf-counterexample-witness-validity · candidate





## 3. Domain Eligibility and Predicate Failure

Using classical truth semantics, the explicit finite integer domain is U={-2, -1, 0} and P(x) means x≥0. For the claim ∀x∈U P(x), is c=0 a valid counterexample?

View answer and explanationNo

c belongs to U and its integer predicate value is true, so the answer is No. An outside value cannot refute this restricted claim even when its integer predicate fails. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r02-counterexample-0003/)

Counterexamples and domain eligibility · qf-counterexample-witness-validity · candidate





## 4. Domain Eligibility and Predicate Failure

Using classical truth semantics, the explicit finite integer domain is U={-2, -1, 0} and P(x) means x≥0. For the claim ∀x∈U P(x), is c=-3 a valid counterexample?

View answer and explanationNo

c does not belong to U and its integer predicate value is false, so the answer is No. An outside value cannot refute this restricted claim even when its integer predicate fails. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r02-counterexample-0004/)

Counterexamples and domain eligibility · qf-counterexample-witness-validity · candidate





## 5. Domain Eligibility and Predicate Failure

Using classical truth semantics, the explicit finite integer domain is U={-8, -7, -6} and P(x) means x≥-7. For the claim ∀x∈U P(x), is c=-8 a valid counterexample?

View answer and explanationYes

c belongs to U and its integer predicate value is false, so the answer is Yes. An outside value cannot refute this restricted claim even when its integer predicate fails. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r02-counterexample-0005/)

Counterexamples and domain eligibility · qf-counterexample-witness-validity · candidate





## 6. Domain Eligibility and Predicate Failure

Using classical truth semantics, the explicit finite integer domain is U={-7, -6, -5} and P(x) means x≤-6. For the claim ∀x∈U P(x), is c=-6 a valid counterexample?

View answer and explanationNo

c belongs to U and its integer predicate value is true, so the answer is No. An outside value cannot refute this restricted claim even when its integer predicate fails. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r02-counterexample-0006/)

Counterexamples and domain eligibility · qf-counterexample-witness-validity · candidate





## 7. Domain Eligibility and Predicate Failure

Using classical truth semantics, the explicit finite integer domain is U={-6, -5, -4} and P(x) means x=-5. For the claim ∀x∈U P(x), is c=-2 a valid counterexample?

View answer and explanationNo

c does not belong to U and its integer predicate value is false, so the answer is No. An outside value cannot refute this restricted claim even when its integer predicate fails. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r02-counterexample-0007/)

Counterexamples and domain eligibility · qf-counterexample-witness-validity · candidate





## 8. Domain Eligibility and Predicate Failure

Using classical truth semantics, the explicit finite integer domain is U={-5, -4, -3} and P(x) means x is an integer multiple of (3). For the claim ∀x∈U P(x), is c=-5 a valid counterexample?

View answer and explanationYes

c belongs to U and its integer predicate value is false, so the answer is Yes. An outside value cannot refute this restricted claim even when its integer predicate fails. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r02-counterexample-0008/)

Counterexamples and domain eligibility · qf-counterexample-witness-validity · candidate





## 9. Domain Eligibility and Predicate Failure

Using classical truth semantics, the explicit finite integer domain is U={-4, -3, -2} and P(x) means x≥-3. For the claim ∀x∈U P(x), is c=-3 a valid counterexample?

View answer and explanationNo

c belongs to U and its integer predicate value is true, so the answer is No. An outside value cannot refute this restricted claim even when its integer predicate fails. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r02-counterexample-0009/)

Counterexamples and domain eligibility · qf-counterexample-witness-validity · candidate





## 10. Domain Eligibility and Predicate Failure

Using classical truth semantics, the explicit finite integer domain is U={-3, -2, -1} and P(x) means x≤-2. For the claim ∀x∈U P(x), is c=1 a valid counterexample?

View answer and explanationNo

c does not belong to U and its integer predicate value is false, so the answer is No. An outside value cannot refute this restricted claim even when its integer predicate fails. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r02-counterexample-0010/)

Counterexamples and domain eligibility · qf-counterexample-witness-validity · candidate





## 11. Domain Eligibility and Predicate Failure

Using classical truth semantics, the explicit finite integer domain is U={-2, -1, 0} and P(x) means x=-1. For the claim ∀x∈U P(x), is c=-2 a valid counterexample?

View answer and explanationYes

c belongs to U and its integer predicate value is false, so the answer is Yes. An outside value cannot refute this restricted claim even when its integer predicate fails. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r02-counterexample-0011/)

Counterexamples and domain eligibility · qf-counterexample-witness-validity · candidate





## 12. Domain Eligibility and Predicate Failure

Using classical truth semantics, the explicit finite integer domain is U={-1, 0, 1} and P(x) means x is an integer multiple of (4). For the claim ∀x∈U P(x), is c=0 a valid counterexample?

View answer and explanationNo

c belongs to U and its integer predicate value is true, so the answer is No. An outside value cannot refute this restricted claim even when its integer predicate fails. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r02-counterexample-0012/)

Counterexamples and domain eligibility · qf-counterexample-witness-validity · candidate





## 13. Domain Eligibility and Predicate Failure

Using classical truth semantics, the explicit finite integer domain is U={0, 1, 2} and P(x) means x≥1. For the claim ∀x∈U P(x), is c=4 a valid counterexample?

View answer and explanationNo

c does not belong to U and its integer predicate value is true, so the answer is No. An outside value cannot refute this restricted claim even when its integer predicate fails. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r02-counterexample-0013/)

Counterexamples and domain eligibility · qf-counterexample-witness-validity · candidate





## 14. Domain Eligibility and Predicate Failure

Using classical truth semantics, the explicit finite integer domain is U={1, 2, 3} and P(x) means x≤2. For the claim ∀x∈U P(x), is c=1 a valid counterexample?

View answer and explanationNo

c belongs to U and its integer predicate value is true, so the answer is No. An outside value cannot refute this restricted claim even when its integer predicate fails. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r02-counterexample-0014/)

Counterexamples and domain eligibility · qf-counterexample-witness-validity · candidate





## 15. Domain Eligibility and Predicate Failure

Using classical truth semantics, the explicit finite integer domain is U={2, 3, 4} and P(x) means x=3. For the claim ∀x∈U P(x), is c=3 a valid counterexample?

View answer and explanationNo

c belongs to U and its integer predicate value is true, so the answer is No. An outside value cannot refute this restricted claim even when its integer predicate fails. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r02-counterexample-0015/)

Counterexamples and domain eligibility · qf-counterexample-witness-validity · candidate





## 16. Domain Eligibility and Predicate Failure

Using classical truth semantics, the explicit finite integer domain is U={3, 4, 5} and P(x) means x is an integer multiple of (2). For the claim ∀x∈U P(x), is c=7 a valid counterexample?

View answer and explanationNo

c does not belong to U and its integer predicate value is false, so the answer is No. An outside value cannot refute this restricted claim even when its integer predicate fails. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r02-counterexample-0016/)

Counterexamples and domain eligibility · qf-counterexample-witness-validity · candidate





## 17. Domain Eligibility and Predicate Failure

Using classical truth semantics, the explicit finite integer domain is U={4, 5, 6} and P(x) means x≥5. For the claim ∀x∈U P(x), is c=4 a valid counterexample?

View answer and explanationYes

c belongs to U and its integer predicate value is false, so the answer is Yes. An outside value cannot refute this restricted claim even when its integer predicate fails. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r02-counterexample-0017/)

Counterexamples and domain eligibility · qf-counterexample-witness-validity · candidate





## 18. Domain Eligibility and Predicate Failure

Using classical truth semantics, the explicit finite integer domain is U={5, 6, 7} and P(x) means x≤6. For the claim ∀x∈U P(x), is c=6 a valid counterexample?

View answer and explanationNo

c belongs to U and its integer predicate value is true, so the answer is No. An outside value cannot refute this restricted claim even when its integer predicate fails. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r02-counterexample-0018/)

Counterexamples and domain eligibility · qf-counterexample-witness-validity · candidate





## 19. Domain Eligibility and Predicate Failure

Using classical truth semantics, the explicit finite integer domain is U={6, 7, 8} and P(x) means x=7. For the claim ∀x∈U P(x), is c=10 a valid counterexample?

View answer and explanationNo

c does not belong to U and its integer predicate value is false, so the answer is No. An outside value cannot refute this restricted claim even when its integer predicate fails. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r02-counterexample-0019/)

Counterexamples and domain eligibility · qf-counterexample-witness-validity · candidate





## 20. Domain Eligibility and Predicate Failure

Using classical truth semantics, the explicit finite integer domain is U={7, 8, 9} and P(x) means x is an integer multiple of (3). For the claim ∀x∈U P(x), is c=7 a valid counterexample?

View answer and explanationYes

c belongs to U and its integer predicate value is false, so the answer is Yes. An outside value cannot refute this restricted claim even when its integer predicate fails. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r02-counterexample-0020/)

Counterexamples and domain eligibility · qf-counterexample-witness-validity · candidate





## 21. Domain Eligibility and Predicate Failure

Using classical truth semantics, the explicit finite integer domain is U={8, 9, 10} and P(x) means x≥9. For the claim ∀x∈U P(x), is c=9 a valid counterexample?

View answer and explanationNo

c belongs to U and its integer predicate value is true, so the answer is No. An outside value cannot refute this restricted claim even when its integer predicate fails. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r02-counterexample-0021/)

Counterexamples and domain eligibility · qf-counterexample-witness-validity · candidate





## 22. Domain Eligibility and Predicate Failure

Using classical truth semantics, the explicit finite integer domain is U={9, 10, 11} and P(x) means x≤10. For the claim ∀x∈U P(x), is c=13 a valid counterexample?

View answer and explanationNo

c does not belong to U and its integer predicate value is false, so the answer is No. An outside value cannot refute this restricted claim even when its integer predicate fails. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r02-counterexample-0022/)

Counterexamples and domain eligibility · qf-counterexample-witness-validity · candidate





## 23. Domain Eligibility and Predicate Failure

Using classical truth semantics, the explicit finite integer domain is U={10, 11, 12} and P(x) means x=11. For the claim ∀x∈U P(x), is c=10 a valid counterexample?

View answer and explanationYes

c belongs to U and its integer predicate value is false, so the answer is Yes. An outside value cannot refute this restricted claim even when its integer predicate fails. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r02-counterexample-0023/)

Counterexamples and domain eligibility · qf-counterexample-witness-validity · candidate





## 24. Domain Eligibility and Predicate Failure

Using classical truth semantics, the explicit finite integer domain is U={11, 12, 13} and P(x) means x is an integer multiple of (4). For the claim ∀x∈U P(x), is c=12 a valid counterexample?

View answer and explanationNo

c belongs to U and its integer predicate value is true, so the answer is No. An outside value cannot refute this restricted claim even when its integer predicate fails. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r02-counterexample-0024/)

Counterexamples and domain eligibility · qf-counterexample-witness-validity · candidate





## 25. Domain Eligibility and Predicate Failure

Using classical truth semantics, the explicit finite integer domain is U={12, 13, 14} and P(x) means x≥13. For the claim ∀x∈U P(x), is c=16 a valid counterexample?

View answer and explanationNo

c does not belong to U and its integer predicate value is true, so the answer is No. An outside value cannot refute this restricted claim even when its integer predicate fails. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r02-counterexample-0025/)

Counterexamples and domain eligibility · qf-counterexample-witness-validity · candidate





## 26. Least Counterexample in a Finite Domain

Using classical truth semantics, the explicit finite integer domain is U=∅ and P(x) means x≥0. Find the numerically least valid counterexample to ∀x∈U P(x), answering none if there is no counterexample.

View answer and explanationnone

The complete counterexample set is ∅. It is empty, so there is no counterexample in U. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r02-counterexample-0026/)

Counterexamples and domain eligibility · qf-counterexample-first-witness · candidate





## 27. Least Counterexample in a Finite Domain

Using classical truth semantics, the explicit finite integer domain is U={-2, -1, 0} and P(x) means x≥-2. Find the numerically least valid counterexample to ∀x∈U P(x), answering none if there is no counterexample.

View answer and explanationnone

The complete counterexample set is ∅. It is empty, so there is no counterexample in U. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r02-counterexample-0027/)

Counterexamples and domain eligibility · qf-counterexample-first-witness · candidate





## 28. Least Counterexample in a Finite Domain

Using classical truth semantics, the explicit finite integer domain is U={1, 2, 3} and P(x) means x≥4. Find the numerically least valid counterexample to ∀x∈U P(x), answering none if there is no counterexample.

View answer and explanation1

The complete counterexample set is {1, 2, 3}. Its least element is 1. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r02-counterexample-0028/)

Counterexamples and domain eligibility · qf-counterexample-first-witness · candidate





## 29. Least Counterexample in a Finite Domain

Using classical truth semantics, the explicit finite integer domain is U={-1, 0, 1} and P(x) means x is an integer multiple of (2). Find the numerically least valid counterexample to ∀x∈U P(x), answering none if there is no counterexample.

View answer and explanation-1

The complete counterexample set is {-1, 1}. Its least element is -1. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r02-counterexample-0029/)

Counterexamples and domain eligibility · qf-counterexample-first-witness · candidate





## 30. Least Counterexample in a Finite Domain

Using classical truth semantics, the explicit finite integer domain is U={-8, -7, -5, -3} and P(x) means x≥-7. Find the numerically least valid counterexample to ∀x∈U P(x), answering none if there is no counterexample.

View answer and explanation-8

The complete counterexample set is {-8}. Its least element is -8. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r02-counterexample-0030/)

Counterexamples and domain eligibility · qf-counterexample-first-witness · candidate





## 31. Least Counterexample in a Finite Domain

Using classical truth semantics, the explicit finite integer domain is U={-7, -6, -4, -2} and P(x) means x≤-4. Find the numerically least valid counterexample to ∀x∈U P(x), answering none if there is no counterexample.

View answer and explanation-2

The complete counterexample set is {-2}. Its least element is -2. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r02-counterexample-0031/)

Counterexamples and domain eligibility · qf-counterexample-first-witness · candidate





## 32. Least Counterexample in a Finite Domain

Using classical truth semantics, the explicit finite integer domain is U={-6, -5, -3, -1} and P(x) means x=-5. Find the numerically least valid counterexample to ∀x∈U P(x), answering none if there is no counterexample.

View answer and explanation-6

The complete counterexample set is {-6, -3, -1}. Its least element is -6. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r02-counterexample-0032/)

Counterexamples and domain eligibility · qf-counterexample-first-witness · candidate





## 33. Least Counterexample in a Finite Domain

Using classical truth semantics, the explicit finite integer domain is U={-5, -4, -2, 0} and P(x) means x is an integer multiple of (3). Find the numerically least valid counterexample to ∀x∈U P(x), answering none if there is no counterexample.

View answer and explanation-5

The complete counterexample set is {-5, -4, -2}. Its least element is -5. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r02-counterexample-0033/)

Counterexamples and domain eligibility · qf-counterexample-first-witness · candidate





## 34. Least Counterexample in a Finite Domain

Using classical truth semantics, the explicit finite integer domain is U={-4, -3, -1, 1} and P(x) means x≥-3. Find the numerically least valid counterexample to ∀x∈U P(x), answering none if there is no counterexample.

View answer and explanation-4

The complete counterexample set is {-4}. Its least element is -4. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r02-counterexample-0034/)

Counterexamples and domain eligibility · qf-counterexample-first-witness · candidate





## 35. Least Counterexample in a Finite Domain

Using classical truth semantics, the explicit finite integer domain is U={-3, -2, 0, 2} and P(x) means x≤0. Find the numerically least valid counterexample to ∀x∈U P(x), answering none if there is no counterexample.

View answer and explanation2

The complete counterexample set is {2}. Its least element is 2. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r02-counterexample-0035/)

Counterexamples and domain eligibility · qf-counterexample-first-witness · candidate





## 36. Least Counterexample in a Finite Domain

Using classical truth semantics, the explicit finite integer domain is U={-2, -1, 1, 3} and P(x) means x=-1. Find the numerically least valid counterexample to ∀x∈U P(x), answering none if there is no counterexample.

View answer and explanation-2

The complete counterexample set is {-2, 1, 3}. Its least element is -2. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r02-counterexample-0036/)

Counterexamples and domain eligibility · qf-counterexample-first-witness · candidate





## 37. Least Counterexample in a Finite Domain

Using classical truth semantics, the explicit finite integer domain is U={-1, 0, 2, 4} and P(x) means x is an integer multiple of (4). Find the numerically least valid counterexample to ∀x∈U P(x), answering none if there is no counterexample.

View answer and explanation-1

The complete counterexample set is {-1, 2}. Its least element is -1. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r02-counterexample-0037/)

Counterexamples and domain eligibility · qf-counterexample-first-witness · candidate





## 38. Least Counterexample in a Finite Domain

Using classical truth semantics, the explicit finite integer domain is U={0, 1, 3, 5} and P(x) means x≥1. Find the numerically least valid counterexample to ∀x∈U P(x), answering none if there is no counterexample.

View answer and explanation0

The complete counterexample set is {0}. Its least element is 0. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r02-counterexample-0038/)

Counterexamples and domain eligibility · qf-counterexample-first-witness · candidate





## 39. Least Counterexample in a Finite Domain

Using classical truth semantics, the explicit finite integer domain is U={1, 2, 4, 6} and P(x) means x≤4. Find the numerically least valid counterexample to ∀x∈U P(x), answering none if there is no counterexample.

View answer and explanation6

The complete counterexample set is {6}. Its least element is 6. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r02-counterexample-0039/)

Counterexamples and domain eligibility · qf-counterexample-first-witness · candidate





## 40. Least Counterexample in a Finite Domain

Using classical truth semantics, the explicit finite integer domain is U={2, 3, 5, 7} and P(x) means x=3. Find the numerically least valid counterexample to ∀x∈U P(x), answering none if there is no counterexample.

View answer and explanation2

The complete counterexample set is {2, 5, 7}. Its least element is 2. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r02-counterexample-0040/)

Counterexamples and domain eligibility · qf-counterexample-first-witness · candidate





## 41. Least Counterexample in a Finite Domain

Using classical truth semantics, the explicit finite integer domain is U={3, 4, 6, 8} and P(x) means x is an integer multiple of (2). Find the numerically least valid counterexample to ∀x∈U P(x), answering none if there is no counterexample.

View answer and explanation3

The complete counterexample set is {3}. Its least element is 3. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r02-counterexample-0041/)

Counterexamples and domain eligibility · qf-counterexample-first-witness · candidate





## 42. Least Counterexample in a Finite Domain

Using classical truth semantics, the explicit finite integer domain is U={4, 5, 7, 9} and P(x) means x≥5. Find the numerically least valid counterexample to ∀x∈U P(x), answering none if there is no counterexample.

View answer and explanation4

The complete counterexample set is {4}. Its least element is 4. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r02-counterexample-0042/)

Counterexamples and domain eligibility · qf-counterexample-first-witness · candidate





## 43. Least Counterexample in a Finite Domain

Using classical truth semantics, the explicit finite integer domain is U={5, 6, 8, 10} and P(x) means x≤8. Find the numerically least valid counterexample to ∀x∈U P(x), answering none if there is no counterexample.

View answer and explanation10

The complete counterexample set is {10}. Its least element is 10. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r02-counterexample-0043/)

Counterexamples and domain eligibility · qf-counterexample-first-witness · candidate





## 44. Least Counterexample in a Finite Domain

Using classical truth semantics, the explicit finite integer domain is U={6, 7, 9, 11} and P(x) means x=7. Find the numerically least valid counterexample to ∀x∈U P(x), answering none if there is no counterexample.

View answer and explanation6

The complete counterexample set is {6, 9, 11}. Its least element is 6. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r02-counterexample-0044/)

Counterexamples and domain eligibility · qf-counterexample-first-witness · candidate





## 45. Least Counterexample in a Finite Domain

Using classical truth semantics, the explicit finite integer domain is U={7, 8, 10, 12} and P(x) means x is an integer multiple of (3). Find the numerically least valid counterexample to ∀x∈U P(x), answering none if there is no counterexample.

View answer and explanation7

The complete counterexample set is {7, 8, 10}. Its least element is 7. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r02-counterexample-0045/)

Counterexamples and domain eligibility · qf-counterexample-first-witness · candidate





## 46. Least Counterexample in a Finite Domain

Using classical truth semantics, the explicit finite integer domain is U={8, 9, 11, 13} and P(x) means x≥9. Find the numerically least valid counterexample to ∀x∈U P(x), answering none if there is no counterexample.

View answer and explanation8

The complete counterexample set is {8}. Its least element is 8. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r02-counterexample-0046/)

Counterexamples and domain eligibility · qf-counterexample-first-witness · candidate





## 47. Least Counterexample in a Finite Domain

Using classical truth semantics, the explicit finite integer domain is U={9, 10, 12, 14} and P(x) means x≤12. Find the numerically least valid counterexample to ∀x∈U P(x), answering none if there is no counterexample.

View answer and explanation14

The complete counterexample set is {14}. Its least element is 14. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r02-counterexample-0047/)

Counterexamples and domain eligibility · qf-counterexample-first-witness · candidate





## 48. Least Counterexample in a Finite Domain

Using classical truth semantics, the explicit finite integer domain is U={10, 11, 13, 15} and P(x) means x=11. Find the numerically least valid counterexample to ∀x∈U P(x), answering none if there is no counterexample.

View answer and explanation10

The complete counterexample set is {10, 13, 15}. Its least element is 10. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r02-counterexample-0048/)

Counterexamples and domain eligibility · qf-counterexample-first-witness · candidate





## 49. Least Counterexample in a Finite Domain

Using classical truth semantics, the explicit finite integer domain is U={11, 12, 14, 16} and P(x) means x is an integer multiple of (4). Find the numerically least valid counterexample to ∀x∈U P(x), answering none if there is no counterexample.

View answer and explanation11

The complete counterexample set is {11, 14}. Its least element is 11. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r02-counterexample-0049/)

Counterexamples and domain eligibility · qf-counterexample-first-witness · candidate





## 50. Least Counterexample in a Finite Domain

Using classical truth semantics, the explicit finite integer domain is U={12, 13, 15, 17} and P(x) means x≥13. Find the numerically least valid counterexample to ∀x∈U P(x), answering none if there is no counterexample.

View answer and explanation12

The complete counterexample set is {12}. Its least element is 12. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r02-counterexample-0050/)

Counterexamples and domain eligibility · qf-counterexample-first-witness · candidate





## 51. Valid Counterexamples in a Candidate Set

Using classical truth semantics, the explicit finite integer domain is U=∅ and P(x) means x=0. Which members of C={-2, 0, 4} are valid counterexamples to ∀x∈U P(x)? Return the set in increasing order with no duplicates.

View answer and explanation∅

Require membership in both U and C and failure of P. The valid candidate counterexample set is ∅; outside values are ineligible. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r02-counterexample-0051/)

Counterexamples and domain eligibility · qf-counterexample-candidate-set · candidate





## 52. Valid Counterexamples in a Candidate Set

Using classical truth semantics, the explicit finite integer domain is U={-2, -1, 0} and P(x) means x≥-2. Which members of C={-3, -2, 0, 5} are valid counterexamples to ∀x∈U P(x)? Return the set in increasing order with no duplicates.

View answer and explanation∅

Require membership in both U and C and failure of P. The valid candidate counterexample set is ∅; outside values are ineligible. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r02-counterexample-0052/)

Counterexamples and domain eligibility · qf-counterexample-candidate-set · candidate





## 53. Valid Counterexamples in a Candidate Set

Using classical truth semantics, the explicit finite integer domain is U={-10, -9, -7} and P(x) means x=-9. Which members of C={-12, -10, -9, -7, -3} are valid counterexamples to ∀x∈U P(x)? Return the set in increasing order with no duplicates.

View answer and explanation{-10, -7}

Require membership in both U and C and failure of P. The valid candidate counterexample set is {-10, -7}; outside values are ineligible. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r02-counterexample-0053/)

Counterexamples and domain eligibility · qf-counterexample-candidate-set · candidate





## 54. Valid Counterexamples in a Candidate Set

Using classical truth semantics, the explicit finite integer domain is U={-9, -8, -6} and P(x) means x is an integer multiple of (2). Which members of C={-11, -9, -8, -6, -2} are valid counterexamples to ∀x∈U P(x)? Return the set in increasing order with no duplicates.

View answer and explanation{-9}

Require membership in both U and C and failure of P. The valid candidate counterexample set is {-9}; outside values are ineligible. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r02-counterexample-0054/)

Counterexamples and domain eligibility · qf-counterexample-candidate-set · candidate





## 55. Valid Counterexamples in a Candidate Set

Using classical truth semantics, the explicit finite integer domain is U={-8, -7, -5} and P(x) means x≥-7. Which members of C={-10, -8, -7, -5, -1} are valid counterexamples to ∀x∈U P(x)? Return the set in increasing order with no duplicates.

View answer and explanation{-8}

Require membership in both U and C and failure of P. The valid candidate counterexample set is {-8}; outside values are ineligible. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r02-counterexample-0055/)

Counterexamples and domain eligibility · qf-counterexample-candidate-set · candidate





## 56. Valid Counterexamples in a Candidate Set

Using classical truth semantics, the explicit finite integer domain is U={-7, -6, -4} and P(x) means x≤-6. Which members of C={-9, -7, -6, -4, 0} are valid counterexamples to ∀x∈U P(x)? Return the set in increasing order with no duplicates.

View answer and explanation{-4}

Require membership in both U and C and failure of P. The valid candidate counterexample set is {-4}; outside values are ineligible. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r02-counterexample-0056/)

Counterexamples and domain eligibility · qf-counterexample-candidate-set · candidate





## 57. Valid Counterexamples in a Candidate Set

Using classical truth semantics, the explicit finite integer domain is U={-6, -5, -3} and P(x) means x=-5. Which members of C={-8, -6, -5, -3, 1} are valid counterexamples to ∀x∈U P(x)? Return the set in increasing order with no duplicates.

View answer and explanation{-6, -3}

Require membership in both U and C and failure of P. The valid candidate counterexample set is {-6, -3}; outside values are ineligible. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r02-counterexample-0057/)

Counterexamples and domain eligibility · qf-counterexample-candidate-set · candidate





## 58. Valid Counterexamples in a Candidate Set

Using classical truth semantics, the explicit finite integer domain is U={-5, -4, -2} and P(x) means x is an integer multiple of (3). Which members of C={-7, -5, -4, -2, 2} are valid counterexamples to ∀x∈U P(x)? Return the set in increasing order with no duplicates.

View answer and explanation{-5, -4, -2}

Require membership in both U and C and failure of P. The valid candidate counterexample set is {-5, -4, -2}; outside values are ineligible. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r02-counterexample-0058/)

Counterexamples and domain eligibility · qf-counterexample-candidate-set · candidate





## 59. Valid Counterexamples in a Candidate Set

Using classical truth semantics, the explicit finite integer domain is U={-4, -3, -1} and P(x) means x≥-3. Which members of C={-6, -4, -3, -1, 3} are valid counterexamples to ∀x∈U P(x)? Return the set in increasing order with no duplicates.

View answer and explanation{-4}

Require membership in both U and C and failure of P. The valid candidate counterexample set is {-4}; outside values are ineligible. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r02-counterexample-0059/)

Counterexamples and domain eligibility · qf-counterexample-candidate-set · candidate





## 60. Valid Counterexamples in a Candidate Set

Using classical truth semantics, the explicit finite integer domain is U={-3, -2, 0} and P(x) means x≤-2. Which members of C={-5, -3, -2, 0, 4} are valid counterexamples to ∀x∈U P(x)? Return the set in increasing order with no duplicates.

View answer and explanation{0}

Require membership in both U and C and failure of P. The valid candidate counterexample set is {0}; outside values are ineligible. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r02-counterexample-0060/)

Counterexamples and domain eligibility · qf-counterexample-candidate-set · candidate





## 61. Valid Counterexamples in a Candidate Set

Using classical truth semantics, the explicit finite integer domain is U={-2, -1, 1} and P(x) means x=-1. Which members of C={-4, -2, -1, 1, 5} are valid counterexamples to ∀x∈U P(x)? Return the set in increasing order with no duplicates.

View answer and explanation{-2, 1}

Require membership in both U and C and failure of P. The valid candidate counterexample set is {-2, 1}; outside values are ineligible. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r02-counterexample-0061/)

Counterexamples and domain eligibility · qf-counterexample-candidate-set · candidate





## 62. Valid Counterexamples in a Candidate Set

Using classical truth semantics, the explicit finite integer domain is U={-1, 0, 2} and P(x) means x is an integer multiple of (4). Which members of C={-3, -1, 0, 2, 6} are valid counterexamples to ∀x∈U P(x)? Return the set in increasing order with no duplicates.

View answer and explanation{-1, 2}

Require membership in both U and C and failure of P. The valid candidate counterexample set is {-1, 2}; outside values are ineligible. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r02-counterexample-0062/)

Counterexamples and domain eligibility · qf-counterexample-candidate-set · candidate





## 63. Valid Counterexamples in a Candidate Set

Using classical truth semantics, the explicit finite integer domain is U={0, 1, 3} and P(x) means x≥1. Which members of C={-2, 0, 1, 3, 7} are valid counterexamples to ∀x∈U P(x)? Return the set in increasing order with no duplicates.

View answer and explanation{0}

Require membership in both U and C and failure of P. The valid candidate counterexample set is {0}; outside values are ineligible. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r02-counterexample-0063/)

Counterexamples and domain eligibility · qf-counterexample-candidate-set · candidate





## 64. Valid Counterexamples in a Candidate Set

Using classical truth semantics, the explicit finite integer domain is U={1, 2, 4} and P(x) means x≤2. Which members of C={-1, 1, 2, 4, 8} are valid counterexamples to ∀x∈U P(x)? Return the set in increasing order with no duplicates.

View answer and explanation{4}

Require membership in both U and C and failure of P. The valid candidate counterexample set is {4}; outside values are ineligible. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r02-counterexample-0064/)

Counterexamples and domain eligibility · qf-counterexample-candidate-set · candidate





## 65. Valid Counterexamples in a Candidate Set

Using classical truth semantics, the explicit finite integer domain is U={2, 3, 5} and P(x) means x=3. Which members of C={0, 2, 3, 5, 9} are valid counterexamples to ∀x∈U P(x)? Return the set in increasing order with no duplicates.

View answer and explanation{2, 5}

Require membership in both U and C and failure of P. The valid candidate counterexample set is {2, 5}; outside values are ineligible. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r02-counterexample-0065/)

Counterexamples and domain eligibility · qf-counterexample-candidate-set · candidate





## 66. Valid Counterexamples in a Candidate Set

Using classical truth semantics, the explicit finite integer domain is U={3, 4, 6} and P(x) means x is an integer multiple of (2). Which members of C={1, 3, 4, 6, 10} are valid counterexamples to ∀x∈U P(x)? Return the set in increasing order with no duplicates.

View answer and explanation{3}

Require membership in both U and C and failure of P. The valid candidate counterexample set is {3}; outside values are ineligible. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r02-counterexample-0066/)

Counterexamples and domain eligibility · qf-counterexample-candidate-set · candidate





## 67. Valid Counterexamples in a Candidate Set

Using classical truth semantics, the explicit finite integer domain is U={4, 5, 7} and P(x) means x≥5. Which members of C={2, 4, 5, 7, 11} are valid counterexamples to ∀x∈U P(x)? Return the set in increasing order with no duplicates.

View answer and explanation{4}

Require membership in both U and C and failure of P. The valid candidate counterexample set is {4}; outside values are ineligible. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r02-counterexample-0067/)

Counterexamples and domain eligibility · qf-counterexample-candidate-set · candidate





## 68. Valid Counterexamples in a Candidate Set

Using classical truth semantics, the explicit finite integer domain is U={5, 6, 8} and P(x) means x≤6. Which members of C={3, 5, 6, 8, 12} are valid counterexamples to ∀x∈U P(x)? Return the set in increasing order with no duplicates.

View answer and explanation{8}

Require membership in both U and C and failure of P. The valid candidate counterexample set is {8}; outside values are ineligible. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r02-counterexample-0068/)

Counterexamples and domain eligibility · qf-counterexample-candidate-set · candidate





## 69. Valid Counterexamples in a Candidate Set

Using classical truth semantics, the explicit finite integer domain is U={6, 7, 9} and P(x) means x=7. Which members of C={4, 6, 7, 9, 13} are valid counterexamples to ∀x∈U P(x)? Return the set in increasing order with no duplicates.

View answer and explanation{6, 9}

Require membership in both U and C and failure of P. The valid candidate counterexample set is {6, 9}; outside values are ineligible. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r02-counterexample-0069/)

Counterexamples and domain eligibility · qf-counterexample-candidate-set · candidate





## 70. Valid Counterexamples in a Candidate Set

Using classical truth semantics, the explicit finite integer domain is U={7, 8, 10} and P(x) means x is an integer multiple of (3). Which members of C={5, 7, 8, 10, 14} are valid counterexamples to ∀x∈U P(x)? Return the set in increasing order with no duplicates.

View answer and explanation{7, 8, 10}

Require membership in both U and C and failure of P. The valid candidate counterexample set is {7, 8, 10}; outside values are ineligible. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r02-counterexample-0070/)

Counterexamples and domain eligibility · qf-counterexample-candidate-set · candidate





## 71. Valid Counterexamples in a Candidate Set

Using classical truth semantics, the explicit finite integer domain is U={8, 9, 11} and P(x) means x≥9. Which members of C={6, 8, 9, 11, 15} are valid counterexamples to ∀x∈U P(x)? Return the set in increasing order with no duplicates.

View answer and explanation{8}

Require membership in both U and C and failure of P. The valid candidate counterexample set is {8}; outside values are ineligible. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r02-counterexample-0071/)

Counterexamples and domain eligibility · qf-counterexample-candidate-set · candidate





## 72. Valid Counterexamples in a Candidate Set

Using classical truth semantics, the explicit finite integer domain is U={9, 10, 12} and P(x) means x≤10. Which members of C={7, 9, 10, 12, 16} are valid counterexamples to ∀x∈U P(x)? Return the set in increasing order with no duplicates.

View answer and explanation{12}

Require membership in both U and C and failure of P. The valid candidate counterexample set is {12}; outside values are ineligible. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r02-counterexample-0072/)

Counterexamples and domain eligibility · qf-counterexample-candidate-set · candidate





## 73. Valid Counterexamples in a Candidate Set

Using classical truth semantics, the explicit finite integer domain is U={10, 11, 13} and P(x) means x=11. Which members of C={8, 10, 11, 13, 17} are valid counterexamples to ∀x∈U P(x)? Return the set in increasing order with no duplicates.

View answer and explanation{10, 13}

Require membership in both U and C and failure of P. The valid candidate counterexample set is {10, 13}; outside values are ineligible. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r02-counterexample-0073/)

Counterexamples and domain eligibility · qf-counterexample-candidate-set · candidate





## 74. Valid Counterexamples in a Candidate Set

Using classical truth semantics, the explicit finite integer domain is U={11, 12, 14} and P(x) means x is an integer multiple of (4). Which members of C={9, 11, 12, 14, 18} are valid counterexamples to ∀x∈U P(x)? Return the set in increasing order with no duplicates.

View answer and explanation{11, 14}

Require membership in both U and C and failure of P. The valid candidate counterexample set is {11, 14}; outside values are ineligible. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r02-counterexample-0074/)

Counterexamples and domain eligibility · qf-counterexample-candidate-set · candidate





## 75. Valid Counterexamples in a Candidate Set

Using classical truth semantics, the explicit finite integer domain is U={12, 13, 15} and P(x) means x≥13. Which members of C={10, 12, 13, 15, 19} are valid counterexamples to ∀x∈U P(x)? Return the set in increasing order with no duplicates.

View answer and explanation{12}

Require membership in both U and C and failure of P. The valid candidate counterexample set is {12}; outside values are ineligible. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r02-counterexample-0075/)

Counterexamples and domain eligibility · qf-counterexample-candidate-set · candidate





## 76. Counterexample Eligibility after Domain Restriction

Using classical truth semantics, the explicit finite integer domain is U={-12, -11, -10} and P(x) means x≥-11. The value c=-12 is a valid counterexample to the original universal claim. After restricting to V={-12}⊆U, is c still a valid counterexample?

View answer and explanationYes

The original predicate at c is false, and c belongs to V, giving Yes. Loss of this witness does not itself establish truth of the new universal claim. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r02-counterexample-0076/)

Counterexamples and domain eligibility · qf-counterexample-domain-restriction · candidate





## 77. Counterexample Eligibility after Domain Restriction

Using classical truth semantics, the explicit finite integer domain is U={-11, -10, -9} and P(x) means x≥-9. The value c=-11 is a valid counterexample to the original universal claim. After restricting to V={-10, -9}⊆U, is c still a valid counterexample?

View answer and explanationNo

The original predicate at c is false, and c does not belong to V, giving No. Loss of this witness does not itself establish truth of the new universal claim. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r02-counterexample-0077/)

Counterexamples and domain eligibility · qf-counterexample-domain-restriction · candidate





## 78. Counterexample Eligibility after Domain Restriction

Using classical truth semantics, the explicit finite integer domain is U={-10, -9, -8} and P(x) means x≥-9. The value c=-10 is a valid counterexample to the original universal claim. After restricting to V=∅⊆U, is c still a valid counterexample?

View answer and explanationNo

The original predicate at c is false, and c does not belong to V, giving No. Loss of this witness does not itself establish truth of the new universal claim. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r02-counterexample-0078/)

Counterexamples and domain eligibility · qf-counterexample-domain-restriction · candidate





## 79. Counterexample Eligibility after Domain Restriction

Using classical truth semantics, the explicit finite integer domain is U={-9, -8, -7} and P(x) means x≥-7. The value c=-9 is a valid counterexample to the original universal claim. After restricting to V={-9}⊆U, is c still a valid counterexample?

View answer and explanationYes

The original predicate at c is false, and c belongs to V, giving Yes. Loss of this witness does not itself establish truth of the new universal claim. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r02-counterexample-0079/)

Counterexamples and domain eligibility · qf-counterexample-domain-restriction · candidate





## 80. Counterexample Eligibility after Domain Restriction

Using classical truth semantics, the explicit finite integer domain is U={-8, -7, -6} and P(x) means x≥-7. The value c=-8 is a valid counterexample to the original universal claim. After restricting to V={-7, -6}⊆U, is c still a valid counterexample?

View answer and explanationNo

The original predicate at c is false, and c does not belong to V, giving No. Loss of this witness does not itself establish truth of the new universal claim. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r02-counterexample-0080/)

Counterexamples and domain eligibility · qf-counterexample-domain-restriction · candidate





## 81. Counterexample Eligibility after Domain Restriction

Using classical truth semantics, the explicit finite integer domain is U={-7, -6, -5} and P(x) means x≥-5. The value c=-7 is a valid counterexample to the original universal claim. After restricting to V=∅⊆U, is c still a valid counterexample?

View answer and explanationNo

The original predicate at c is false, and c does not belong to V, giving No. Loss of this witness does not itself establish truth of the new universal claim. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r02-counterexample-0081/)

Counterexamples and domain eligibility · qf-counterexample-domain-restriction · candidate





## 82. Counterexample Eligibility after Domain Restriction

Using classical truth semantics, the explicit finite integer domain is U={-6, -5, -4} and P(x) means x≥-5. The value c=-6 is a valid counterexample to the original universal claim. After restricting to V={-6}⊆U, is c still a valid counterexample?

View answer and explanationYes

The original predicate at c is false, and c belongs to V, giving Yes. Loss of this witness does not itself establish truth of the new universal claim. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r02-counterexample-0082/)

Counterexamples and domain eligibility · qf-counterexample-domain-restriction · candidate





## 83. Counterexample Eligibility after Domain Restriction

Using classical truth semantics, the explicit finite integer domain is U={-5, -4, -3} and P(x) means x≥-3. The value c=-5 is a valid counterexample to the original universal claim. After restricting to V={-4, -3}⊆U, is c still a valid counterexample?

View answer and explanationNo

The original predicate at c is false, and c does not belong to V, giving No. Loss of this witness does not itself establish truth of the new universal claim. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r02-counterexample-0083/)

Counterexamples and domain eligibility · qf-counterexample-domain-restriction · candidate





## 84. Counterexample Eligibility after Domain Restriction

Using classical truth semantics, the explicit finite integer domain is U={-4, -3, -2} and P(x) means x≥-3. The value c=-4 is a valid counterexample to the original universal claim. After restricting to V=∅⊆U, is c still a valid counterexample?

View answer and explanationNo

The original predicate at c is false, and c does not belong to V, giving No. Loss of this witness does not itself establish truth of the new universal claim. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r02-counterexample-0084/)

Counterexamples and domain eligibility · qf-counterexample-domain-restriction · candidate





## 85. Counterexample Eligibility after Domain Restriction

Using classical truth semantics, the explicit finite integer domain is U={-3, -2, -1} and P(x) means x≥-1. The value c=-3 is a valid counterexample to the original universal claim. After restricting to V={-3}⊆U, is c still a valid counterexample?

View answer and explanationYes

The original predicate at c is false, and c belongs to V, giving Yes. Loss of this witness does not itself establish truth of the new universal claim. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r02-counterexample-0085/)

Counterexamples and domain eligibility · qf-counterexample-domain-restriction · candidate





## 86. Counterexample Eligibility after Domain Restriction

Using classical truth semantics, the explicit finite integer domain is U={-2, -1, 0} and P(x) means x≥-1. The value c=-2 is a valid counterexample to the original universal claim. After restricting to V={-1, 0}⊆U, is c still a valid counterexample?

View answer and explanationNo

The original predicate at c is false, and c does not belong to V, giving No. Loss of this witness does not itself establish truth of the new universal claim. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r02-counterexample-0086/)

Counterexamples and domain eligibility · qf-counterexample-domain-restriction · candidate





## 87. Counterexample Eligibility after Domain Restriction

Using classical truth semantics, the explicit finite integer domain is U={-1, 0, 1} and P(x) means x≥1. The value c=-1 is a valid counterexample to the original universal claim. After restricting to V=∅⊆U, is c still a valid counterexample?

View answer and explanationNo

The original predicate at c is false, and c does not belong to V, giving No. Loss of this witness does not itself establish truth of the new universal claim. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r02-counterexample-0087/)

Counterexamples and domain eligibility · qf-counterexample-domain-restriction · candidate





## 88. Counterexample Eligibility after Domain Restriction

Using classical truth semantics, the explicit finite integer domain is U={0, 1, 2} and P(x) means x≥1. The value c=0 is a valid counterexample to the original universal claim. After restricting to V={0}⊆U, is c still a valid counterexample?

View answer and explanationYes

The original predicate at c is false, and c belongs to V, giving Yes. Loss of this witness does not itself establish truth of the new universal claim. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r02-counterexample-0088/)

Counterexamples and domain eligibility · qf-counterexample-domain-restriction · candidate





## 89. Counterexample Eligibility after Domain Restriction

Using classical truth semantics, the explicit finite integer domain is U={1, 2, 3} and P(x) means x≥3. The value c=1 is a valid counterexample to the original universal claim. After restricting to V={2, 3}⊆U, is c still a valid counterexample?

View answer and explanationNo

The original predicate at c is false, and c does not belong to V, giving No. Loss of this witness does not itself establish truth of the new universal claim. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r02-counterexample-0089/)

Counterexamples and domain eligibility · qf-counterexample-domain-restriction · candidate





## 90. Counterexample Eligibility after Domain Restriction

Using classical truth semantics, the explicit finite integer domain is U={2, 3, 4} and P(x) means x≥3. The value c=2 is a valid counterexample to the original universal claim. After restricting to V=∅⊆U, is c still a valid counterexample?

View answer and explanationNo

The original predicate at c is false, and c does not belong to V, giving No. Loss of this witness does not itself establish truth of the new universal claim. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r02-counterexample-0090/)

Counterexamples and domain eligibility · qf-counterexample-domain-restriction · candidate





## 91. Counterexample Eligibility after Domain Restriction

Using classical truth semantics, the explicit finite integer domain is U={3, 4, 5} and P(x) means x≥5. The value c=3 is a valid counterexample to the original universal claim. After restricting to V={3}⊆U, is c still a valid counterexample?

View answer and explanationYes

The original predicate at c is false, and c belongs to V, giving Yes. Loss of this witness does not itself establish truth of the new universal claim. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r02-counterexample-0091/)

Counterexamples and domain eligibility · qf-counterexample-domain-restriction · candidate





## 92. Counterexample Eligibility after Domain Restriction

Using classical truth semantics, the explicit finite integer domain is U={4, 5, 6} and P(x) means x≥5. The value c=4 is a valid counterexample to the original universal claim. After restricting to V={5, 6}⊆U, is c still a valid counterexample?

View answer and explanationNo

The original predicate at c is false, and c does not belong to V, giving No. Loss of this witness does not itself establish truth of the new universal claim. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r02-counterexample-0092/)

Counterexamples and domain eligibility · qf-counterexample-domain-restriction · candidate





## 93. Counterexample Eligibility after Domain Restriction

Using classical truth semantics, the explicit finite integer domain is U={5, 6, 7} and P(x) means x≥7. The value c=5 is a valid counterexample to the original universal claim. After restricting to V=∅⊆U, is c still a valid counterexample?

View answer and explanationNo

The original predicate at c is false, and c does not belong to V, giving No. Loss of this witness does not itself establish truth of the new universal claim. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r02-counterexample-0093/)

Counterexamples and domain eligibility · qf-counterexample-domain-restriction · candidate





## 94. Counterexample Eligibility after Domain Restriction

Using classical truth semantics, the explicit finite integer domain is U={6, 7, 8} and P(x) means x≥7. The value c=6 is a valid counterexample to the original universal claim. After restricting to V={6}⊆U, is c still a valid counterexample?

View answer and explanationYes

The original predicate at c is false, and c belongs to V, giving Yes. Loss of this witness does not itself establish truth of the new universal claim. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r02-counterexample-0094/)

Counterexamples and domain eligibility · qf-counterexample-domain-restriction · candidate





## 95. Counterexample Eligibility after Domain Restriction

Using classical truth semantics, the explicit finite integer domain is U={7, 8, 9} and P(x) means x≥9. The value c=7 is a valid counterexample to the original universal claim. After restricting to V={8, 9}⊆U, is c still a valid counterexample?

View answer and explanationNo

The original predicate at c is false, and c does not belong to V, giving No. Loss of this witness does not itself establish truth of the new universal claim. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r02-counterexample-0095/)

Counterexamples and domain eligibility · qf-counterexample-domain-restriction · candidate





## 96. Counterexample Eligibility after Domain Restriction

Using classical truth semantics, the explicit finite integer domain is U={8, 9, 10} and P(x) means x≥9. The value c=8 is a valid counterexample to the original universal claim. After restricting to V=∅⊆U, is c still a valid counterexample?

View answer and explanationNo

The original predicate at c is false, and c does not belong to V, giving No. Loss of this witness does not itself establish truth of the new universal claim. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r02-counterexample-0096/)

Counterexamples and domain eligibility · qf-counterexample-domain-restriction · candidate





## 97. Counterexample Eligibility after Domain Restriction

Using classical truth semantics, the explicit finite integer domain is U={9, 10, 11} and P(x) means x≥11. The value c=9 is a valid counterexample to the original universal claim. After restricting to V={9}⊆U, is c still a valid counterexample?

View answer and explanationYes

The original predicate at c is false, and c belongs to V, giving Yes. Loss of this witness does not itself establish truth of the new universal claim. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r02-counterexample-0097/)

Counterexamples and domain eligibility · qf-counterexample-domain-restriction · candidate





## 98. Counterexample Eligibility after Domain Restriction

Using classical truth semantics, the explicit finite integer domain is U={10, 11, 12} and P(x) means x≥11. The value c=10 is a valid counterexample to the original universal claim. After restricting to V={11, 12}⊆U, is c still a valid counterexample?

View answer and explanationNo

The original predicate at c is false, and c does not belong to V, giving No. Loss of this witness does not itself establish truth of the new universal claim. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r02-counterexample-0098/)

Counterexamples and domain eligibility · qf-counterexample-domain-restriction · candidate





## 99. Counterexample Eligibility after Domain Restriction

Using classical truth semantics, the explicit finite integer domain is U={11, 12, 13} and P(x) means x≥13. The value c=11 is a valid counterexample to the original universal claim. After restricting to V=∅⊆U, is c still a valid counterexample?

View answer and explanationNo

The original predicate at c is false, and c does not belong to V, giving No. Loss of this witness does not itself establish truth of the new universal claim. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r02-counterexample-0099/)

Counterexamples and domain eligibility · qf-counterexample-domain-restriction · candidate





## 100. Counterexample Eligibility after Domain Restriction

Using classical truth semantics, the explicit finite integer domain is U={12, 13, 14} and P(x) means x≥13. The value c=12 is a valid counterexample to the original universal claim. After restricting to V={12}⊆U, is c still a valid counterexample?

View answer and explanationYes

The original predicate at c is false, and c belongs to V, giving Yes. Loss of this witness does not itself establish truth of the new universal claim. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r02-counterexample-0100/)

Counterexamples and domain eligibility · qf-counterexample-domain-restriction · candidate



[Canonical question JSONL](https://ocme.evemisslab.com/data/questions/batches/batch-20261003-advance-20261004-r02-counterexample-foundations/questions.jsonl) · [English JSONL](https://ocme.evemisslab.com/data/i18n/questions/batch-20261003-advance-20261004-r02-counterexample-foundations.en.jsonl) · [Validation report](https://ocme.evemisslab.com/data/questions/batches/batch-20261003-advance-20261004-r02-counterexample-foundations/validation-report.json)