Counterexamples and Domain Eligibility
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This question bank has corresponding bilingual step-by-step derivations. Source candidate and English ai_candidate states are retained. AI review, finite computation and Lean formal proof are labelled separately.
batch-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 explanation
No
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
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 explanation
Yes
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
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 explanation
No
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
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 explanation
No
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
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 explanation
Yes
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
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 explanation
No
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
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 explanation
No
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
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 explanation
Yes
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
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 explanation
No
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
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 explanation
No
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
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 explanation
Yes
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
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 explanation
No
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
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 explanation
No
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
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 explanation
No
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
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 explanation
No
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
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 explanation
No
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
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 explanation
Yes
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
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 explanation
No
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
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 explanation
No
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
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 explanation
Yes
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
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 explanation
No
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
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 explanation
No
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
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 explanation
Yes
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
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 explanation
No
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
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 explanation
No
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
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 explanation
none
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
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 explanation
none
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
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 explanation
1
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
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
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
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
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
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
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
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 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
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
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
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 explanation
0
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
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 explanation
6
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
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 explanation
2
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
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 explanation
3
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
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 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
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 explanation
10
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
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 explanation
6
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
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 explanation
7
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
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 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
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 explanation
14
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
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 explanation
10
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
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 explanation
11
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
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 explanation
12
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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 explanation
Yes
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
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 explanation
No
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
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 explanation
No
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
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 explanation
Yes
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
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 explanation
No
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
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 explanation
No
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
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 explanation
Yes
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
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 explanation
No
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
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 explanation
No
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
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 explanation
Yes
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
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 explanation
No
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
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 explanation
No
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
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 explanation
Yes
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
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 explanation
No
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
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 explanation
No
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
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 explanation
Yes
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
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 explanation
No
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
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 explanation
No
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
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 explanation
Yes
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
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 explanation
No
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
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 explanation
No
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
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 explanation
Yes
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
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 explanation
No
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
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 explanation
No
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
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 explanation
Yes
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
Counterexamples and domain eligibility · qf-counterexample-domain-restriction · candidate
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