Quantifiers and Explicit Finite Domains
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-r01-quantifier-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. Universal Truth on a Finite Domain
Using classical truth semantics, the explicit finite integer domain is U=∅ and P(x) means x≥0. Is ∀x∈U P(x) true? Answer Yes or No.
View answer and explanation
Yes
There are 0 satisfying and 0 failing elements out of 0; the answer is Yes. On the empty domain, universal truth is true and existential truth is false; negation reverses the corresponding truth value. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain.
Reviewed step-by-step derivation
Quantifiers and explicit domains · qf-quantifier-finite-universal · candidate
2. Universal Truth on a Finite Domain
Using classical truth semantics, the explicit finite integer domain is U={-2, -1, 0} and P(x) means x≥-2. Is ∀x∈U P(x) true? Answer Yes or No.
View answer and explanation
Yes
There are 3 satisfying and 0 failing elements out of 3; the answer is Yes. On the empty domain, universal truth is true and existential truth is false; negation reverses the corresponding truth value. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain.
Reviewed step-by-step derivation
Quantifiers and explicit domains · qf-quantifier-finite-universal · candidate
3. Universal Truth on a Finite Domain
Using classical truth semantics, the explicit finite integer domain is U={1, 2, 3} and P(x) means x≥4. Is ∀x∈U P(x) true? Answer Yes or No.
View answer and explanation
No
There are 0 satisfying and 3 failing elements out of 3; the answer is No. On the empty domain, universal truth is true and existential truth is false; negation reverses the corresponding truth value. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain.
Reviewed step-by-step derivation
Quantifiers and explicit domains · qf-quantifier-finite-universal · candidate
4. Universal Truth on 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). Is ∀x∈U P(x) true? Answer Yes or No.
View answer and explanation
No
There are 1 satisfying and 2 failing elements out of 3; the answer is No. On the empty domain, universal truth is true and existential truth is false; negation reverses the corresponding truth value. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain.
Reviewed step-by-step derivation
Quantifiers and explicit domains · qf-quantifier-finite-universal · candidate
5. Universal Truth on a Finite Domain
Using classical truth semantics, the explicit finite integer domain is U={-8, -7, -5, -3} and P(x) means x≥-7. Is ∀x∈U P(x) true? Answer Yes or No.
View answer and explanation
No
There are 3 satisfying and 1 failing elements out of 4; the answer is No. On the empty domain, universal truth is true and existential truth is false; negation reverses the corresponding truth value. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain.
Reviewed step-by-step derivation
Quantifiers and explicit domains · qf-quantifier-finite-universal · candidate
6. Universal Truth on a Finite Domain
Using classical truth semantics, the explicit finite integer domain is U={-7, -6, -4, -2} and P(x) means x≤-4. Is ∀x∈U P(x) true? Answer Yes or No.
View answer and explanation
No
There are 3 satisfying and 1 failing elements out of 4; the answer is No. On the empty domain, universal truth is true and existential truth is false; negation reverses the corresponding truth value. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain.
Reviewed step-by-step derivation
Quantifiers and explicit domains · qf-quantifier-finite-universal · candidate
7. Universal Truth on a Finite Domain
Using classical truth semantics, the explicit finite integer domain is U={-6, -5, -3, -1} and P(x) means x=-5. Is ∀x∈U P(x) true? Answer Yes or No.
View answer and explanation
No
There are 1 satisfying and 3 failing elements out of 4; the answer is No. On the empty domain, universal truth is true and existential truth is false; negation reverses the corresponding truth value. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain.
Reviewed step-by-step derivation
Quantifiers and explicit domains · qf-quantifier-finite-universal · candidate
8. Universal Truth on 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). Is ∀x∈U P(x) true? Answer Yes or No.
View answer and explanation
No
There are 1 satisfying and 3 failing elements out of 4; the answer is No. On the empty domain, universal truth is true and existential truth is false; negation reverses the corresponding truth value. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain.
Reviewed step-by-step derivation
Quantifiers and explicit domains · qf-quantifier-finite-universal · candidate
9. Universal Truth on a Finite Domain
Using classical truth semantics, the explicit finite integer domain is U={-4, -3, -1, 1} and P(x) means x≥-3. Is ∀x∈U P(x) true? Answer Yes or No.
View answer and explanation
No
There are 3 satisfying and 1 failing elements out of 4; the answer is No. On the empty domain, universal truth is true and existential truth is false; negation reverses the corresponding truth value. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain.
Reviewed step-by-step derivation
Quantifiers and explicit domains · qf-quantifier-finite-universal · candidate
10. Universal Truth on a Finite Domain
Using classical truth semantics, the explicit finite integer domain is U={-3, -2, 0, 2} and P(x) means x≤0. Is ∀x∈U P(x) true? Answer Yes or No.
View answer and explanation
No
There are 3 satisfying and 1 failing elements out of 4; the answer is No. On the empty domain, universal truth is true and existential truth is false; negation reverses the corresponding truth value. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain.
Reviewed step-by-step derivation
Quantifiers and explicit domains · qf-quantifier-finite-universal · candidate
11. Universal Truth on a Finite Domain
Using classical truth semantics, the explicit finite integer domain is U={-2, -1, 1, 3} and P(x) means x=-1. Is ∀x∈U P(x) true? Answer Yes or No.
View answer and explanation
No
There are 1 satisfying and 3 failing elements out of 4; the answer is No. On the empty domain, universal truth is true and existential truth is false; negation reverses the corresponding truth value. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain.
Reviewed step-by-step derivation
Quantifiers and explicit domains · qf-quantifier-finite-universal · candidate
12. Universal Truth on 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). Is ∀x∈U P(x) true? Answer Yes or No.
View answer and explanation
No
There are 2 satisfying and 2 failing elements out of 4; the answer is No. On the empty domain, universal truth is true and existential truth is false; negation reverses the corresponding truth value. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain.
Reviewed step-by-step derivation
Quantifiers and explicit domains · qf-quantifier-finite-universal · candidate
13. Universal Truth on a Finite Domain
Using classical truth semantics, the explicit finite integer domain is U={0, 1, 3, 5} and P(x) means x≥1. Is ∀x∈U P(x) true? Answer Yes or No.
View answer and explanation
No
There are 3 satisfying and 1 failing elements out of 4; the answer is No. On the empty domain, universal truth is true and existential truth is false; negation reverses the corresponding truth value. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain.
Reviewed step-by-step derivation
Quantifiers and explicit domains · qf-quantifier-finite-universal · candidate
14. Universal Truth on a Finite Domain
Using classical truth semantics, the explicit finite integer domain is U={1, 2, 4, 6} and P(x) means x≤4. Is ∀x∈U P(x) true? Answer Yes or No.
View answer and explanation
No
There are 3 satisfying and 1 failing elements out of 4; the answer is No. On the empty domain, universal truth is true and existential truth is false; negation reverses the corresponding truth value. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain.
Reviewed step-by-step derivation
Quantifiers and explicit domains · qf-quantifier-finite-universal · candidate
15. Universal Truth on a Finite Domain
Using classical truth semantics, the explicit finite integer domain is U={2, 3, 5, 7} and P(x) means x=3. Is ∀x∈U P(x) true? Answer Yes or No.
View answer and explanation
No
There are 1 satisfying and 3 failing elements out of 4; the answer is No. On the empty domain, universal truth is true and existential truth is false; negation reverses the corresponding truth value. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain.
Reviewed step-by-step derivation
Quantifiers and explicit domains · qf-quantifier-finite-universal · candidate
16. Universal Truth on 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). Is ∀x∈U P(x) true? Answer Yes or No.
View answer and explanation
No
There are 3 satisfying and 1 failing elements out of 4; the answer is No. On the empty domain, universal truth is true and existential truth is false; negation reverses the corresponding truth value. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain.
Reviewed step-by-step derivation
Quantifiers and explicit domains · qf-quantifier-finite-universal · candidate
17. Universal Truth on a Finite Domain
Using classical truth semantics, the explicit finite integer domain is U={4, 5, 7, 9} and P(x) means x≥5. Is ∀x∈U P(x) true? Answer Yes or No.
View answer and explanation
No
There are 3 satisfying and 1 failing elements out of 4; the answer is No. On the empty domain, universal truth is true and existential truth is false; negation reverses the corresponding truth value. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain.
Reviewed step-by-step derivation
Quantifiers and explicit domains · qf-quantifier-finite-universal · candidate
18. Universal Truth on a Finite Domain
Using classical truth semantics, the explicit finite integer domain is U={5, 6, 8, 10} and P(x) means x≤8. Is ∀x∈U P(x) true? Answer Yes or No.
View answer and explanation
No
There are 3 satisfying and 1 failing elements out of 4; the answer is No. On the empty domain, universal truth is true and existential truth is false; negation reverses the corresponding truth value. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain.
Reviewed step-by-step derivation
Quantifiers and explicit domains · qf-quantifier-finite-universal · candidate
19. Universal Truth on a Finite Domain
Using classical truth semantics, the explicit finite integer domain is U={6, 7, 9, 11} and P(x) means x=7. Is ∀x∈U P(x) true? Answer Yes or No.
View answer and explanation
No
There are 1 satisfying and 3 failing elements out of 4; the answer is No. On the empty domain, universal truth is true and existential truth is false; negation reverses the corresponding truth value. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain.
Reviewed step-by-step derivation
Quantifiers and explicit domains · qf-quantifier-finite-universal · candidate
20. Universal Truth on 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). Is ∀x∈U P(x) true? Answer Yes or No.
View answer and explanation
No
There are 1 satisfying and 3 failing elements out of 4; the answer is No. On the empty domain, universal truth is true and existential truth is false; negation reverses the corresponding truth value. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain.
Reviewed step-by-step derivation
Quantifiers and explicit domains · qf-quantifier-finite-universal · candidate
21. Universal Truth on a Finite Domain
Using classical truth semantics, the explicit finite integer domain is U={8, 9, 11, 13} and P(x) means x≥9. Is ∀x∈U P(x) true? Answer Yes or No.
View answer and explanation
No
There are 3 satisfying and 1 failing elements out of 4; the answer is No. On the empty domain, universal truth is true and existential truth is false; negation reverses the corresponding truth value. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain.
Reviewed step-by-step derivation
Quantifiers and explicit domains · qf-quantifier-finite-universal · candidate
22. Universal Truth on a Finite Domain
Using classical truth semantics, the explicit finite integer domain is U={9, 10, 12, 14} and P(x) means x≤12. Is ∀x∈U P(x) true? Answer Yes or No.
View answer and explanation
No
There are 3 satisfying and 1 failing elements out of 4; the answer is No. On the empty domain, universal truth is true and existential truth is false; negation reverses the corresponding truth value. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain.
Reviewed step-by-step derivation
Quantifiers and explicit domains · qf-quantifier-finite-universal · candidate
23. Universal Truth on a Finite Domain
Using classical truth semantics, the explicit finite integer domain is U={10, 11, 13, 15} and P(x) means x=11. Is ∀x∈U P(x) true? Answer Yes or No.
View answer and explanation
No
There are 1 satisfying and 3 failing elements out of 4; the answer is No. On the empty domain, universal truth is true and existential truth is false; negation reverses the corresponding truth value. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain.
Reviewed step-by-step derivation
Quantifiers and explicit domains · qf-quantifier-finite-universal · candidate
24. Universal Truth on 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). Is ∀x∈U P(x) true? Answer Yes or No.
View answer and explanation
No
There are 2 satisfying and 2 failing elements out of 4; the answer is No. On the empty domain, universal truth is true and existential truth is false; negation reverses the corresponding truth value. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain.
Reviewed step-by-step derivation
Quantifiers and explicit domains · qf-quantifier-finite-universal · candidate
25. Universal Truth on a Finite Domain
Using classical truth semantics, the explicit finite integer domain is U={12, 13, 15, 17} and P(x) means x≥13. Is ∀x∈U P(x) true? Answer Yes or No.
View answer and explanation
No
There are 3 satisfying and 1 failing elements out of 4; the answer is No. On the empty domain, universal truth is true and existential truth is false; negation reverses the corresponding truth value. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain.
Reviewed step-by-step derivation
Quantifiers and explicit domains · qf-quantifier-finite-universal · candidate
26. Existential Truth on a Finite Domain
Using classical truth semantics, the explicit finite integer domain is U=∅ and P(x) means x≥0. Is ∃x∈U P(x) true? Answer Yes or No.
View answer and explanation
No
There are 0 satisfying and 0 failing elements out of 0; the answer is No. On the empty domain, universal truth is true and existential truth is false; negation reverses the corresponding truth value. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain.
Reviewed step-by-step derivation
Quantifiers and explicit domains · qf-quantifier-finite-existential · candidate
27. Existential Truth on a Finite Domain
Using classical truth semantics, the explicit finite integer domain is U={-2, -1, 0} and P(x) means x≥-2. Is ∃x∈U P(x) true? Answer Yes or No.
View answer and explanation
Yes
There are 3 satisfying and 0 failing elements out of 3; the answer is Yes. On the empty domain, universal truth is true and existential truth is false; negation reverses the corresponding truth value. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain.
Reviewed step-by-step derivation
Quantifiers and explicit domains · qf-quantifier-finite-existential · candidate
28. Existential Truth on a Finite Domain
Using classical truth semantics, the explicit finite integer domain is U={1, 2, 3} and P(x) means x≥4. Is ∃x∈U P(x) true? Answer Yes or No.
View answer and explanation
No
There are 0 satisfying and 3 failing elements out of 3; the answer is No. On the empty domain, universal truth is true and existential truth is false; negation reverses the corresponding truth value. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain.
Reviewed step-by-step derivation
Quantifiers and explicit domains · qf-quantifier-finite-existential · candidate
29. Existential Truth on 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). Is ∃x∈U P(x) true? Answer Yes or No.
View answer and explanation
Yes
There are 1 satisfying and 2 failing elements out of 3; the answer is Yes. On the empty domain, universal truth is true and existential truth is false; negation reverses the corresponding truth value. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain.
Reviewed step-by-step derivation
Quantifiers and explicit domains · qf-quantifier-finite-existential · candidate
30. Existential Truth on a Finite Domain
Using classical truth semantics, the explicit finite integer domain is U={-8, -7, -5, -3} and P(x) means x≥-7. Is ∃x∈U P(x) true? Answer Yes or No.
View answer and explanation
Yes
There are 3 satisfying and 1 failing elements out of 4; the answer is Yes. On the empty domain, universal truth is true and existential truth is false; negation reverses the corresponding truth value. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain.
Reviewed step-by-step derivation
Quantifiers and explicit domains · qf-quantifier-finite-existential · candidate
31. Existential Truth on a Finite Domain
Using classical truth semantics, the explicit finite integer domain is U={-7, -6, -4, -2} and P(x) means x≤-4. Is ∃x∈U P(x) true? Answer Yes or No.
View answer and explanation
Yes
There are 3 satisfying and 1 failing elements out of 4; the answer is Yes. On the empty domain, universal truth is true and existential truth is false; negation reverses the corresponding truth value. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain.
Reviewed step-by-step derivation
Quantifiers and explicit domains · qf-quantifier-finite-existential · candidate
32. Existential Truth on a Finite Domain
Using classical truth semantics, the explicit finite integer domain is U={-6, -5, -3, -1} and P(x) means x=-5. Is ∃x∈U P(x) true? Answer Yes or No.
View answer and explanation
Yes
There are 1 satisfying and 3 failing elements out of 4; the answer is Yes. On the empty domain, universal truth is true and existential truth is false; negation reverses the corresponding truth value. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain.
Reviewed step-by-step derivation
Quantifiers and explicit domains · qf-quantifier-finite-existential · candidate
33. Existential Truth on 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). Is ∃x∈U P(x) true? Answer Yes or No.
View answer and explanation
Yes
There are 1 satisfying and 3 failing elements out of 4; the answer is Yes. On the empty domain, universal truth is true and existential truth is false; negation reverses the corresponding truth value. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain.
Reviewed step-by-step derivation
Quantifiers and explicit domains · qf-quantifier-finite-existential · candidate
34. Existential Truth on a Finite Domain
Using classical truth semantics, the explicit finite integer domain is U={-4, -3, -1, 1} and P(x) means x≥-3. Is ∃x∈U P(x) true? Answer Yes or No.
View answer and explanation
Yes
There are 3 satisfying and 1 failing elements out of 4; the answer is Yes. On the empty domain, universal truth is true and existential truth is false; negation reverses the corresponding truth value. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain.
Reviewed step-by-step derivation
Quantifiers and explicit domains · qf-quantifier-finite-existential · candidate
35. Existential Truth on a Finite Domain
Using classical truth semantics, the explicit finite integer domain is U={-3, -2, 0, 2} and P(x) means x≤0. Is ∃x∈U P(x) true? Answer Yes or No.
View answer and explanation
Yes
There are 3 satisfying and 1 failing elements out of 4; the answer is Yes. On the empty domain, universal truth is true and existential truth is false; negation reverses the corresponding truth value. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain.
Reviewed step-by-step derivation
Quantifiers and explicit domains · qf-quantifier-finite-existential · candidate
36. Existential Truth on a Finite Domain
Using classical truth semantics, the explicit finite integer domain is U={-2, -1, 1, 3} and P(x) means x=-1. Is ∃x∈U P(x) true? Answer Yes or No.
View answer and explanation
Yes
There are 1 satisfying and 3 failing elements out of 4; the answer is Yes. On the empty domain, universal truth is true and existential truth is false; negation reverses the corresponding truth value. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain.
Reviewed step-by-step derivation
Quantifiers and explicit domains · qf-quantifier-finite-existential · candidate
37. Existential Truth on 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). Is ∃x∈U P(x) true? Answer Yes or No.
View answer and explanation
Yes
There are 2 satisfying and 2 failing elements out of 4; the answer is Yes. On the empty domain, universal truth is true and existential truth is false; negation reverses the corresponding truth value. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain.
Reviewed step-by-step derivation
Quantifiers and explicit domains · qf-quantifier-finite-existential · candidate
38. Existential Truth on a Finite Domain
Using classical truth semantics, the explicit finite integer domain is U={0, 1, 3, 5} and P(x) means x≥1. Is ∃x∈U P(x) true? Answer Yes or No.
View answer and explanation
Yes
There are 3 satisfying and 1 failing elements out of 4; the answer is Yes. On the empty domain, universal truth is true and existential truth is false; negation reverses the corresponding truth value. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain.
Reviewed step-by-step derivation
Quantifiers and explicit domains · qf-quantifier-finite-existential · candidate
39. Existential Truth on a Finite Domain
Using classical truth semantics, the explicit finite integer domain is U={1, 2, 4, 6} and P(x) means x≤4. Is ∃x∈U P(x) true? Answer Yes or No.
View answer and explanation
Yes
There are 3 satisfying and 1 failing elements out of 4; the answer is Yes. On the empty domain, universal truth is true and existential truth is false; negation reverses the corresponding truth value. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain.
Reviewed step-by-step derivation
Quantifiers and explicit domains · qf-quantifier-finite-existential · candidate
40. Existential Truth on a Finite Domain
Using classical truth semantics, the explicit finite integer domain is U={2, 3, 5, 7} and P(x) means x=3. Is ∃x∈U P(x) true? Answer Yes or No.
View answer and explanation
Yes
There are 1 satisfying and 3 failing elements out of 4; the answer is Yes. On the empty domain, universal truth is true and existential truth is false; negation reverses the corresponding truth value. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain.
Reviewed step-by-step derivation
Quantifiers and explicit domains · qf-quantifier-finite-existential · candidate
41. Existential Truth on 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). Is ∃x∈U P(x) true? Answer Yes or No.
View answer and explanation
Yes
There are 3 satisfying and 1 failing elements out of 4; the answer is Yes. On the empty domain, universal truth is true and existential truth is false; negation reverses the corresponding truth value. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain.
Reviewed step-by-step derivation
Quantifiers and explicit domains · qf-quantifier-finite-existential · candidate
42. Existential Truth on a Finite Domain
Using classical truth semantics, the explicit finite integer domain is U={4, 5, 7, 9} and P(x) means x≥5. Is ∃x∈U P(x) true? Answer Yes or No.
View answer and explanation
Yes
There are 3 satisfying and 1 failing elements out of 4; the answer is Yes. On the empty domain, universal truth is true and existential truth is false; negation reverses the corresponding truth value. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain.
Reviewed step-by-step derivation
Quantifiers and explicit domains · qf-quantifier-finite-existential · candidate
43. Existential Truth on a Finite Domain
Using classical truth semantics, the explicit finite integer domain is U={5, 6, 8, 10} and P(x) means x≤8. Is ∃x∈U P(x) true? Answer Yes or No.
View answer and explanation
Yes
There are 3 satisfying and 1 failing elements out of 4; the answer is Yes. On the empty domain, universal truth is true and existential truth is false; negation reverses the corresponding truth value. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain.
Reviewed step-by-step derivation
Quantifiers and explicit domains · qf-quantifier-finite-existential · candidate
44. Existential Truth on a Finite Domain
Using classical truth semantics, the explicit finite integer domain is U={6, 7, 9, 11} and P(x) means x=7. Is ∃x∈U P(x) true? Answer Yes or No.
View answer and explanation
Yes
There are 1 satisfying and 3 failing elements out of 4; the answer is Yes. On the empty domain, universal truth is true and existential truth is false; negation reverses the corresponding truth value. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain.
Reviewed step-by-step derivation
Quantifiers and explicit domains · qf-quantifier-finite-existential · candidate
45. Existential Truth on 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). Is ∃x∈U P(x) true? Answer Yes or No.
View answer and explanation
Yes
There are 1 satisfying and 3 failing elements out of 4; the answer is Yes. On the empty domain, universal truth is true and existential truth is false; negation reverses the corresponding truth value. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain.
Reviewed step-by-step derivation
Quantifiers and explicit domains · qf-quantifier-finite-existential · candidate
46. Existential Truth on a Finite Domain
Using classical truth semantics, the explicit finite integer domain is U={8, 9, 11, 13} and P(x) means x≥9. Is ∃x∈U P(x) true? Answer Yes or No.
View answer and explanation
Yes
There are 3 satisfying and 1 failing elements out of 4; the answer is Yes. On the empty domain, universal truth is true and existential truth is false; negation reverses the corresponding truth value. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain.
Reviewed step-by-step derivation
Quantifiers and explicit domains · qf-quantifier-finite-existential · candidate
47. Existential Truth on a Finite Domain
Using classical truth semantics, the explicit finite integer domain is U={9, 10, 12, 14} and P(x) means x≤12. Is ∃x∈U P(x) true? Answer Yes or No.
View answer and explanation
Yes
There are 3 satisfying and 1 failing elements out of 4; the answer is Yes. On the empty domain, universal truth is true and existential truth is false; negation reverses the corresponding truth value. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain.
Reviewed step-by-step derivation
Quantifiers and explicit domains · qf-quantifier-finite-existential · candidate
48. Existential Truth on a Finite Domain
Using classical truth semantics, the explicit finite integer domain is U={10, 11, 13, 15} and P(x) means x=11. Is ∃x∈U P(x) true? Answer Yes or No.
View answer and explanation
Yes
There are 1 satisfying and 3 failing elements out of 4; the answer is Yes. On the empty domain, universal truth is true and existential truth is false; negation reverses the corresponding truth value. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain.
Reviewed step-by-step derivation
Quantifiers and explicit domains · qf-quantifier-finite-existential · candidate
49. Existential Truth on 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). Is ∃x∈U P(x) true? Answer Yes or No.
View answer and explanation
Yes
There are 2 satisfying and 2 failing elements out of 4; the answer is Yes. On the empty domain, universal truth is true and existential truth is false; negation reverses the corresponding truth value. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain.
Reviewed step-by-step derivation
Quantifiers and explicit domains · qf-quantifier-finite-existential · candidate
50. Existential Truth on a Finite Domain
Using classical truth semantics, the explicit finite integer domain is U={12, 13, 15, 17} and P(x) means x≥13. Is ∃x∈U P(x) true? Answer Yes or No.
View answer and explanation
Yes
There are 3 satisfying and 1 failing elements out of 4; the answer is Yes. On the empty domain, universal truth is true and existential truth is false; negation reverses the corresponding truth value. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain.
Reviewed step-by-step derivation
Quantifiers and explicit domains · qf-quantifier-finite-existential · candidate
51. Negation of a Finite Universal Statement
Using classical truth semantics, the explicit finite integer domain is U=∅ and P(x) means x≥0. Is ¬(∀x∈U P(x)) true? Answer Yes or No.
View answer and explanation
No
There are 0 satisfying and 0 failing elements out of 0; the answer is No. On the empty domain, universal truth is true and existential truth is false; negation reverses the corresponding truth value. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain.
Reviewed step-by-step derivation
Quantifiers and explicit domains · qf-quantifier-negated-universal · candidate
52. Negation of a Finite Universal Statement
Using classical truth semantics, the explicit finite integer domain is U={-2, -1, 0} and P(x) means x≥-2. Is ¬(∀x∈U P(x)) true? Answer Yes or No.
View answer and explanation
No
There are 3 satisfying and 0 failing elements out of 3; the answer is No. On the empty domain, universal truth is true and existential truth is false; negation reverses the corresponding truth value. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain.
Reviewed step-by-step derivation
Quantifiers and explicit domains · qf-quantifier-negated-universal · candidate
53. Negation of a Finite Universal Statement
Using classical truth semantics, the explicit finite integer domain is U={1, 2, 3} and P(x) means x≥4. Is ¬(∀x∈U P(x)) true? Answer Yes or No.
View answer and explanation
Yes
There are 0 satisfying and 3 failing elements out of 3; the answer is Yes. On the empty domain, universal truth is true and existential truth is false; negation reverses the corresponding truth value. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain.
Reviewed step-by-step derivation
Quantifiers and explicit domains · qf-quantifier-negated-universal · candidate
54. Negation of a Finite Universal Statement
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). Is ¬(∀x∈U P(x)) true? Answer Yes or No.
View answer and explanation
Yes
There are 1 satisfying and 2 failing elements out of 3; the answer is Yes. On the empty domain, universal truth is true and existential truth is false; negation reverses the corresponding truth value. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain.
Reviewed step-by-step derivation
Quantifiers and explicit domains · qf-quantifier-negated-universal · candidate
55. Negation of a Finite Universal Statement
Using classical truth semantics, the explicit finite integer domain is U={-8, -7, -5, -3} and P(x) means x≥-7. Is ¬(∀x∈U P(x)) true? Answer Yes or No.
View answer and explanation
Yes
There are 3 satisfying and 1 failing elements out of 4; the answer is Yes. On the empty domain, universal truth is true and existential truth is false; negation reverses the corresponding truth value. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain.
Reviewed step-by-step derivation
Quantifiers and explicit domains · qf-quantifier-negated-universal · candidate
56. Negation of a Finite Universal Statement
Using classical truth semantics, the explicit finite integer domain is U={-7, -6, -4, -2} and P(x) means x≤-4. Is ¬(∀x∈U P(x)) true? Answer Yes or No.
View answer and explanation
Yes
There are 3 satisfying and 1 failing elements out of 4; the answer is Yes. On the empty domain, universal truth is true and existential truth is false; negation reverses the corresponding truth value. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain.
Reviewed step-by-step derivation
Quantifiers and explicit domains · qf-quantifier-negated-universal · candidate
57. Negation of a Finite Universal Statement
Using classical truth semantics, the explicit finite integer domain is U={-6, -5, -3, -1} and P(x) means x=-5. Is ¬(∀x∈U P(x)) true? Answer Yes or No.
View answer and explanation
Yes
There are 1 satisfying and 3 failing elements out of 4; the answer is Yes. On the empty domain, universal truth is true and existential truth is false; negation reverses the corresponding truth value. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain.
Reviewed step-by-step derivation
Quantifiers and explicit domains · qf-quantifier-negated-universal · candidate
58. Negation of a Finite Universal Statement
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). Is ¬(∀x∈U P(x)) true? Answer Yes or No.
View answer and explanation
Yes
There are 1 satisfying and 3 failing elements out of 4; the answer is Yes. On the empty domain, universal truth is true and existential truth is false; negation reverses the corresponding truth value. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain.
Reviewed step-by-step derivation
Quantifiers and explicit domains · qf-quantifier-negated-universal · candidate
59. Negation of a Finite Universal Statement
Using classical truth semantics, the explicit finite integer domain is U={-4, -3, -1, 1} and P(x) means x≥-3. Is ¬(∀x∈U P(x)) true? Answer Yes or No.
View answer and explanation
Yes
There are 3 satisfying and 1 failing elements out of 4; the answer is Yes. On the empty domain, universal truth is true and existential truth is false; negation reverses the corresponding truth value. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain.
Reviewed step-by-step derivation
Quantifiers and explicit domains · qf-quantifier-negated-universal · candidate
60. Negation of a Finite Universal Statement
Using classical truth semantics, the explicit finite integer domain is U={-3, -2, 0, 2} and P(x) means x≤0. Is ¬(∀x∈U P(x)) true? Answer Yes or No.
View answer and explanation
Yes
There are 3 satisfying and 1 failing elements out of 4; the answer is Yes. On the empty domain, universal truth is true and existential truth is false; negation reverses the corresponding truth value. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain.
Reviewed step-by-step derivation
Quantifiers and explicit domains · qf-quantifier-negated-universal · candidate
61. Negation of a Finite Universal Statement
Using classical truth semantics, the explicit finite integer domain is U={-2, -1, 1, 3} and P(x) means x=-1. Is ¬(∀x∈U P(x)) true? Answer Yes or No.
View answer and explanation
Yes
There are 1 satisfying and 3 failing elements out of 4; the answer is Yes. On the empty domain, universal truth is true and existential truth is false; negation reverses the corresponding truth value. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain.
Reviewed step-by-step derivation
Quantifiers and explicit domains · qf-quantifier-negated-universal · candidate
62. Negation of a Finite Universal Statement
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). Is ¬(∀x∈U P(x)) true? Answer Yes or No.
View answer and explanation
Yes
There are 2 satisfying and 2 failing elements out of 4; the answer is Yes. On the empty domain, universal truth is true and existential truth is false; negation reverses the corresponding truth value. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain.
Reviewed step-by-step derivation
Quantifiers and explicit domains · qf-quantifier-negated-universal · candidate
63. Negation of a Finite Universal Statement
Using classical truth semantics, the explicit finite integer domain is U={0, 1, 3, 5} and P(x) means x≥1. Is ¬(∀x∈U P(x)) true? Answer Yes or No.
View answer and explanation
Yes
There are 3 satisfying and 1 failing elements out of 4; the answer is Yes. On the empty domain, universal truth is true and existential truth is false; negation reverses the corresponding truth value. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain.
Reviewed step-by-step derivation
Quantifiers and explicit domains · qf-quantifier-negated-universal · candidate
64. Negation of a Finite Universal Statement
Using classical truth semantics, the explicit finite integer domain is U={1, 2, 4, 6} and P(x) means x≤4. Is ¬(∀x∈U P(x)) true? Answer Yes or No.
View answer and explanation
Yes
There are 3 satisfying and 1 failing elements out of 4; the answer is Yes. On the empty domain, universal truth is true and existential truth is false; negation reverses the corresponding truth value. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain.
Reviewed step-by-step derivation
Quantifiers and explicit domains · qf-quantifier-negated-universal · candidate
65. Negation of a Finite Universal Statement
Using classical truth semantics, the explicit finite integer domain is U={2, 3, 5, 7} and P(x) means x=3. Is ¬(∀x∈U P(x)) true? Answer Yes or No.
View answer and explanation
Yes
There are 1 satisfying and 3 failing elements out of 4; the answer is Yes. On the empty domain, universal truth is true and existential truth is false; negation reverses the corresponding truth value. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain.
Reviewed step-by-step derivation
Quantifiers and explicit domains · qf-quantifier-negated-universal · candidate
66. Negation of a Finite Universal Statement
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). Is ¬(∀x∈U P(x)) true? Answer Yes or No.
View answer and explanation
Yes
There are 3 satisfying and 1 failing elements out of 4; the answer is Yes. On the empty domain, universal truth is true and existential truth is false; negation reverses the corresponding truth value. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain.
Reviewed step-by-step derivation
Quantifiers and explicit domains · qf-quantifier-negated-universal · candidate
67. Negation of a Finite Universal Statement
Using classical truth semantics, the explicit finite integer domain is U={4, 5, 7, 9} and P(x) means x≥5. Is ¬(∀x∈U P(x)) true? Answer Yes or No.
View answer and explanation
Yes
There are 3 satisfying and 1 failing elements out of 4; the answer is Yes. On the empty domain, universal truth is true and existential truth is false; negation reverses the corresponding truth value. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain.
Reviewed step-by-step derivation
Quantifiers and explicit domains · qf-quantifier-negated-universal · candidate
68. Negation of a Finite Universal Statement
Using classical truth semantics, the explicit finite integer domain is U={5, 6, 8, 10} and P(x) means x≤8. Is ¬(∀x∈U P(x)) true? Answer Yes or No.
View answer and explanation
Yes
There are 3 satisfying and 1 failing elements out of 4; the answer is Yes. On the empty domain, universal truth is true and existential truth is false; negation reverses the corresponding truth value. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain.
Reviewed step-by-step derivation
Quantifiers and explicit domains · qf-quantifier-negated-universal · candidate
69. Negation of a Finite Universal Statement
Using classical truth semantics, the explicit finite integer domain is U={6, 7, 9, 11} and P(x) means x=7. Is ¬(∀x∈U P(x)) true? Answer Yes or No.
View answer and explanation
Yes
There are 1 satisfying and 3 failing elements out of 4; the answer is Yes. On the empty domain, universal truth is true and existential truth is false; negation reverses the corresponding truth value. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain.
Reviewed step-by-step derivation
Quantifiers and explicit domains · qf-quantifier-negated-universal · candidate
70. Negation of a Finite Universal Statement
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). Is ¬(∀x∈U P(x)) true? Answer Yes or No.
View answer and explanation
Yes
There are 1 satisfying and 3 failing elements out of 4; the answer is Yes. On the empty domain, universal truth is true and existential truth is false; negation reverses the corresponding truth value. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain.
Reviewed step-by-step derivation
Quantifiers and explicit domains · qf-quantifier-negated-universal · candidate
71. Negation of a Finite Universal Statement
Using classical truth semantics, the explicit finite integer domain is U={8, 9, 11, 13} and P(x) means x≥9. Is ¬(∀x∈U P(x)) true? Answer Yes or No.
View answer and explanation
Yes
There are 3 satisfying and 1 failing elements out of 4; the answer is Yes. On the empty domain, universal truth is true and existential truth is false; negation reverses the corresponding truth value. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain.
Reviewed step-by-step derivation
Quantifiers and explicit domains · qf-quantifier-negated-universal · candidate
72. Negation of a Finite Universal Statement
Using classical truth semantics, the explicit finite integer domain is U={9, 10, 12, 14} and P(x) means x≤12. Is ¬(∀x∈U P(x)) true? Answer Yes or No.
View answer and explanation
Yes
There are 3 satisfying and 1 failing elements out of 4; the answer is Yes. On the empty domain, universal truth is true and existential truth is false; negation reverses the corresponding truth value. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain.
Reviewed step-by-step derivation
Quantifiers and explicit domains · qf-quantifier-negated-universal · candidate
73. Negation of a Finite Universal Statement
Using classical truth semantics, the explicit finite integer domain is U={10, 11, 13, 15} and P(x) means x=11. Is ¬(∀x∈U P(x)) true? Answer Yes or No.
View answer and explanation
Yes
There are 1 satisfying and 3 failing elements out of 4; the answer is Yes. On the empty domain, universal truth is true and existential truth is false; negation reverses the corresponding truth value. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain.
Reviewed step-by-step derivation
Quantifiers and explicit domains · qf-quantifier-negated-universal · candidate
74. Negation of a Finite Universal Statement
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). Is ¬(∀x∈U P(x)) true? Answer Yes or No.
View answer and explanation
Yes
There are 2 satisfying and 2 failing elements out of 4; the answer is Yes. On the empty domain, universal truth is true and existential truth is false; negation reverses the corresponding truth value. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain.
Reviewed step-by-step derivation
Quantifiers and explicit domains · qf-quantifier-negated-universal · candidate
75. Negation of a Finite Universal Statement
Using classical truth semantics, the explicit finite integer domain is U={12, 13, 15, 17} and P(x) means x≥13. Is ¬(∀x∈U P(x)) true? Answer Yes or No.
View answer and explanation
Yes
There are 3 satisfying and 1 failing elements out of 4; the answer is Yes. On the empty domain, universal truth is true and existential truth is false; negation reverses the corresponding truth value. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain.
Reviewed step-by-step derivation
Quantifiers and explicit domains · qf-quantifier-negated-universal · candidate
76. Negation of a Finite Existential Statement
Using classical truth semantics, the explicit finite integer domain is U=∅ and P(x) means x≥0. Is ¬(∃x∈U P(x)) true? Answer Yes or No.
View answer and explanation
Yes
There are 0 satisfying and 0 failing elements out of 0; the answer is Yes. On the empty domain, universal truth is true and existential truth is false; negation reverses the corresponding truth value. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain.
Reviewed step-by-step derivation
Quantifiers and explicit domains · qf-quantifier-negated-existential · candidate
77. Negation of a Finite Existential Statement
Using classical truth semantics, the explicit finite integer domain is U={-2, -1, 0} and P(x) means x≥-2. Is ¬(∃x∈U P(x)) true? Answer Yes or No.
View answer and explanation
No
There are 3 satisfying and 0 failing elements out of 3; the answer is No. On the empty domain, universal truth is true and existential truth is false; negation reverses the corresponding truth value. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain.
Reviewed step-by-step derivation
Quantifiers and explicit domains · qf-quantifier-negated-existential · candidate
78. Negation of a Finite Existential Statement
Using classical truth semantics, the explicit finite integer domain is U={1, 2, 3} and P(x) means x≥4. Is ¬(∃x∈U P(x)) true? Answer Yes or No.
View answer and explanation
Yes
There are 0 satisfying and 3 failing elements out of 3; the answer is Yes. On the empty domain, universal truth is true and existential truth is false; negation reverses the corresponding truth value. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain.
Reviewed step-by-step derivation
Quantifiers and explicit domains · qf-quantifier-negated-existential · candidate
79. Negation of a Finite Existential Statement
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). Is ¬(∃x∈U P(x)) true? Answer Yes or No.
View answer and explanation
No
There are 1 satisfying and 2 failing elements out of 3; the answer is No. On the empty domain, universal truth is true and existential truth is false; negation reverses the corresponding truth value. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain.
Reviewed step-by-step derivation
Quantifiers and explicit domains · qf-quantifier-negated-existential · candidate
80. Negation of a Finite Existential Statement
Using classical truth semantics, the explicit finite integer domain is U={-8, -7, -5, -3} and P(x) means x≥-7. Is ¬(∃x∈U P(x)) true? Answer Yes or No.
View answer and explanation
No
There are 3 satisfying and 1 failing elements out of 4; the answer is No. On the empty domain, universal truth is true and existential truth is false; negation reverses the corresponding truth value. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain.
Reviewed step-by-step derivation
Quantifiers and explicit domains · qf-quantifier-negated-existential · candidate
81. Negation of a Finite Existential Statement
Using classical truth semantics, the explicit finite integer domain is U={-7, -6, -4, -2} and P(x) means x≤-4. Is ¬(∃x∈U P(x)) true? Answer Yes or No.
View answer and explanation
No
There are 3 satisfying and 1 failing elements out of 4; the answer is No. On the empty domain, universal truth is true and existential truth is false; negation reverses the corresponding truth value. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain.
Reviewed step-by-step derivation
Quantifiers and explicit domains · qf-quantifier-negated-existential · candidate
82. Negation of a Finite Existential Statement
Using classical truth semantics, the explicit finite integer domain is U={-6, -5, -3, -1} and P(x) means x=-5. Is ¬(∃x∈U P(x)) true? Answer Yes or No.
View answer and explanation
No
There are 1 satisfying and 3 failing elements out of 4; the answer is No. On the empty domain, universal truth is true and existential truth is false; negation reverses the corresponding truth value. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain.
Reviewed step-by-step derivation
Quantifiers and explicit domains · qf-quantifier-negated-existential · candidate
83. Negation of a Finite Existential Statement
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). Is ¬(∃x∈U P(x)) true? Answer Yes or No.
View answer and explanation
No
There are 1 satisfying and 3 failing elements out of 4; the answer is No. On the empty domain, universal truth is true and existential truth is false; negation reverses the corresponding truth value. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain.
Reviewed step-by-step derivation
Quantifiers and explicit domains · qf-quantifier-negated-existential · candidate
84. Negation of a Finite Existential Statement
Using classical truth semantics, the explicit finite integer domain is U={-4, -3, -1, 1} and P(x) means x≥-3. Is ¬(∃x∈U P(x)) true? Answer Yes or No.
View answer and explanation
No
There are 3 satisfying and 1 failing elements out of 4; the answer is No. On the empty domain, universal truth is true and existential truth is false; negation reverses the corresponding truth value. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain.
Reviewed step-by-step derivation
Quantifiers and explicit domains · qf-quantifier-negated-existential · candidate
85. Negation of a Finite Existential Statement
Using classical truth semantics, the explicit finite integer domain is U={-3, -2, 0, 2} and P(x) means x≤0. Is ¬(∃x∈U P(x)) true? Answer Yes or No.
View answer and explanation
No
There are 3 satisfying and 1 failing elements out of 4; the answer is No. On the empty domain, universal truth is true and existential truth is false; negation reverses the corresponding truth value. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain.
Reviewed step-by-step derivation
Quantifiers and explicit domains · qf-quantifier-negated-existential · candidate
86. Negation of a Finite Existential Statement
Using classical truth semantics, the explicit finite integer domain is U={-2, -1, 1, 3} and P(x) means x=-1. Is ¬(∃x∈U P(x)) true? Answer Yes or No.
View answer and explanation
No
There are 1 satisfying and 3 failing elements out of 4; the answer is No. On the empty domain, universal truth is true and existential truth is false; negation reverses the corresponding truth value. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain.
Reviewed step-by-step derivation
Quantifiers and explicit domains · qf-quantifier-negated-existential · candidate
87. Negation of a Finite Existential Statement
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). Is ¬(∃x∈U P(x)) true? Answer Yes or No.
View answer and explanation
No
There are 2 satisfying and 2 failing elements out of 4; the answer is No. On the empty domain, universal truth is true and existential truth is false; negation reverses the corresponding truth value. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain.
Reviewed step-by-step derivation
Quantifiers and explicit domains · qf-quantifier-negated-existential · candidate
88. Negation of a Finite Existential Statement
Using classical truth semantics, the explicit finite integer domain is U={0, 1, 3, 5} and P(x) means x≥1. Is ¬(∃x∈U P(x)) true? Answer Yes or No.
View answer and explanation
No
There are 3 satisfying and 1 failing elements out of 4; the answer is No. On the empty domain, universal truth is true and existential truth is false; negation reverses the corresponding truth value. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain.
Reviewed step-by-step derivation
Quantifiers and explicit domains · qf-quantifier-negated-existential · candidate
89. Negation of a Finite Existential Statement
Using classical truth semantics, the explicit finite integer domain is U={1, 2, 4, 6} and P(x) means x≤4. Is ¬(∃x∈U P(x)) true? Answer Yes or No.
View answer and explanation
No
There are 3 satisfying and 1 failing elements out of 4; the answer is No. On the empty domain, universal truth is true and existential truth is false; negation reverses the corresponding truth value. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain.
Reviewed step-by-step derivation
Quantifiers and explicit domains · qf-quantifier-negated-existential · candidate
90. Negation of a Finite Existential Statement
Using classical truth semantics, the explicit finite integer domain is U={2, 3, 5, 7} and P(x) means x=3. Is ¬(∃x∈U P(x)) true? Answer Yes or No.
View answer and explanation
No
There are 1 satisfying and 3 failing elements out of 4; the answer is No. On the empty domain, universal truth is true and existential truth is false; negation reverses the corresponding truth value. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain.
Reviewed step-by-step derivation
Quantifiers and explicit domains · qf-quantifier-negated-existential · candidate
91. Negation of a Finite Existential Statement
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). Is ¬(∃x∈U P(x)) true? Answer Yes or No.
View answer and explanation
No
There are 3 satisfying and 1 failing elements out of 4; the answer is No. On the empty domain, universal truth is true and existential truth is false; negation reverses the corresponding truth value. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain.
Reviewed step-by-step derivation
Quantifiers and explicit domains · qf-quantifier-negated-existential · candidate
92. Negation of a Finite Existential Statement
Using classical truth semantics, the explicit finite integer domain is U={4, 5, 7, 9} and P(x) means x≥5. Is ¬(∃x∈U P(x)) true? Answer Yes or No.
View answer and explanation
No
There are 3 satisfying and 1 failing elements out of 4; the answer is No. On the empty domain, universal truth is true and existential truth is false; negation reverses the corresponding truth value. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain.
Reviewed step-by-step derivation
Quantifiers and explicit domains · qf-quantifier-negated-existential · candidate
93. Negation of a Finite Existential Statement
Using classical truth semantics, the explicit finite integer domain is U={5, 6, 8, 10} and P(x) means x≤8. Is ¬(∃x∈U P(x)) true? Answer Yes or No.
View answer and explanation
No
There are 3 satisfying and 1 failing elements out of 4; the answer is No. On the empty domain, universal truth is true and existential truth is false; negation reverses the corresponding truth value. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain.
Reviewed step-by-step derivation
Quantifiers and explicit domains · qf-quantifier-negated-existential · candidate
94. Negation of a Finite Existential Statement
Using classical truth semantics, the explicit finite integer domain is U={6, 7, 9, 11} and P(x) means x=7. Is ¬(∃x∈U P(x)) true? Answer Yes or No.
View answer and explanation
No
There are 1 satisfying and 3 failing elements out of 4; the answer is No. On the empty domain, universal truth is true and existential truth is false; negation reverses the corresponding truth value. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain.
Reviewed step-by-step derivation
Quantifiers and explicit domains · qf-quantifier-negated-existential · candidate
95. Negation of a Finite Existential Statement
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). Is ¬(∃x∈U P(x)) true? Answer Yes or No.
View answer and explanation
No
There are 1 satisfying and 3 failing elements out of 4; the answer is No. On the empty domain, universal truth is true and existential truth is false; negation reverses the corresponding truth value. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain.
Reviewed step-by-step derivation
Quantifiers and explicit domains · qf-quantifier-negated-existential · candidate
96. Negation of a Finite Existential Statement
Using classical truth semantics, the explicit finite integer domain is U={8, 9, 11, 13} and P(x) means x≥9. Is ¬(∃x∈U P(x)) true? Answer Yes or No.
View answer and explanation
No
There are 3 satisfying and 1 failing elements out of 4; the answer is No. On the empty domain, universal truth is true and existential truth is false; negation reverses the corresponding truth value. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain.
Reviewed step-by-step derivation
Quantifiers and explicit domains · qf-quantifier-negated-existential · candidate
97. Negation of a Finite Existential Statement
Using classical truth semantics, the explicit finite integer domain is U={9, 10, 12, 14} and P(x) means x≤12. Is ¬(∃x∈U P(x)) true? Answer Yes or No.
View answer and explanation
No
There are 3 satisfying and 1 failing elements out of 4; the answer is No. On the empty domain, universal truth is true and existential truth is false; negation reverses the corresponding truth value. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain.
Reviewed step-by-step derivation
Quantifiers and explicit domains · qf-quantifier-negated-existential · candidate
98. Negation of a Finite Existential Statement
Using classical truth semantics, the explicit finite integer domain is U={10, 11, 13, 15} and P(x) means x=11. Is ¬(∃x∈U P(x)) true? Answer Yes or No.
View answer and explanation
No
There are 1 satisfying and 3 failing elements out of 4; the answer is No. On the empty domain, universal truth is true and existential truth is false; negation reverses the corresponding truth value. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain.
Reviewed step-by-step derivation
Quantifiers and explicit domains · qf-quantifier-negated-existential · candidate
99. Negation of a Finite Existential Statement
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). Is ¬(∃x∈U P(x)) true? Answer Yes or No.
View answer and explanation
No
There are 2 satisfying and 2 failing elements out of 4; the answer is No. On the empty domain, universal truth is true and existential truth is false; negation reverses the corresponding truth value. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain.
Reviewed step-by-step derivation
Quantifiers and explicit domains · qf-quantifier-negated-existential · candidate
100. Negation of a Finite Existential Statement
Using classical truth semantics, the explicit finite integer domain is U={12, 13, 15, 17} and P(x) means x≥13. Is ¬(∃x∈U P(x)) true? Answer Yes or No.
View answer and explanation
No
There are 3 satisfying and 1 failing elements out of 4; the answer is No. On the empty domain, universal truth is true and existential truth is false; negation reverses the corresponding truth value. This exhaustive result concerns only the declared finite domain and does not extend to an infinite domain.
Reviewed step-by-step derivation
Quantifiers and explicit domains · qf-quantifier-negated-existential · candidate
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