# 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 explanationYes 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](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r01-quantifier-0001/) 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 explanationYes 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](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r01-quantifier-0002/) 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 explanationNo 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](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r01-quantifier-0003/) 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 explanationNo 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](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r01-quantifier-0004/) 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 explanationNo 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](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r01-quantifier-0005/) 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 explanationNo 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](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r01-quantifier-0006/) 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 explanationNo 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](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r01-quantifier-0007/) 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 explanationNo 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](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r01-quantifier-0008/) 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 explanationNo 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](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r01-quantifier-0009/) 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 explanationNo 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](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r01-quantifier-0010/) 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 explanationNo 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](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r01-quantifier-0011/) 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 explanationNo 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](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r01-quantifier-0012/) 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 explanationNo 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](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r01-quantifier-0013/) 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 explanationNo 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](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r01-quantifier-0014/) 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 explanationNo 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](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r01-quantifier-0015/) 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 explanationNo 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](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r01-quantifier-0016/) 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 explanationNo 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](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r01-quantifier-0017/) 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 explanationNo 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](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r01-quantifier-0018/) 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 explanationNo 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](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r01-quantifier-0019/) 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 explanationNo 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](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r01-quantifier-0020/) 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 explanationNo 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](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r01-quantifier-0021/) 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 explanationNo 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](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r01-quantifier-0022/) 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 explanationNo 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](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r01-quantifier-0023/) 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 explanationNo 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](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r01-quantifier-0024/) 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 explanationNo 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](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r01-quantifier-0025/) 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 explanationNo 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](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r01-quantifier-0026/) 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 explanationYes 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](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r01-quantifier-0027/) 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 explanationNo 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](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r01-quantifier-0028/) 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 explanationYes 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](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r01-quantifier-0029/) 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 explanationYes 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](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r01-quantifier-0030/) 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 explanationYes 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](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r01-quantifier-0031/) 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 explanationYes 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](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r01-quantifier-0032/) 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 explanationYes 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](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r01-quantifier-0033/) 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 explanationYes 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](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r01-quantifier-0034/) 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 explanationYes 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](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r01-quantifier-0035/) 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 explanationYes 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](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r01-quantifier-0036/) 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 explanationYes 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](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r01-quantifier-0037/) 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 explanationYes 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](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r01-quantifier-0038/) 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 explanationYes 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](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r01-quantifier-0039/) 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 explanationYes 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](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r01-quantifier-0040/) 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 explanationYes 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](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r01-quantifier-0041/) 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 explanationYes 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](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r01-quantifier-0042/) 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 explanationYes 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](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r01-quantifier-0043/) 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 explanationYes 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](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r01-quantifier-0044/) 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 explanationYes 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](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r01-quantifier-0045/) 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 explanationYes 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](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r01-quantifier-0046/) 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 explanationYes 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](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r01-quantifier-0047/) 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 explanationYes 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](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r01-quantifier-0048/) 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 explanationYes 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](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r01-quantifier-0049/) 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 explanationYes 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](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r01-quantifier-0050/) 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 explanationNo 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](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r01-quantifier-0051/) 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 explanationNo 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](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r01-quantifier-0052/) 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 explanationYes 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](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r01-quantifier-0053/) 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 explanationYes 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](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r01-quantifier-0054/) 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 explanationYes 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](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r01-quantifier-0055/) 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 explanationYes 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](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r01-quantifier-0056/) 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 explanationYes 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](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r01-quantifier-0057/) 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 explanationYes 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](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r01-quantifier-0058/) 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 explanationYes 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](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r01-quantifier-0059/) 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 explanationYes 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](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r01-quantifier-0060/) 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 explanationYes 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](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r01-quantifier-0061/) 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 explanationYes 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](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r01-quantifier-0062/) 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 explanationYes 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](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r01-quantifier-0063/) 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 explanationYes 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](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r01-quantifier-0064/) 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 explanationYes 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](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r01-quantifier-0065/) 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 explanationYes 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](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r01-quantifier-0066/) 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 explanationYes 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](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r01-quantifier-0067/) 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 explanationYes 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](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r01-quantifier-0068/) 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 explanationYes 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](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r01-quantifier-0069/) 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 explanationYes 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](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r01-quantifier-0070/) 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 explanationYes 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](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r01-quantifier-0071/) 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 explanationYes 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](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r01-quantifier-0072/) 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 explanationYes 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](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r01-quantifier-0073/) 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 explanationYes 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](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r01-quantifier-0074/) 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 explanationYes 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](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r01-quantifier-0075/) 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 explanationYes 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](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r01-quantifier-0076/) 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 explanationNo 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](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r01-quantifier-0077/) 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 explanationYes 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](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r01-quantifier-0078/) 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 explanationNo 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](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r01-quantifier-0079/) 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 explanationNo 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](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r01-quantifier-0080/) 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 explanationNo 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](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r01-quantifier-0081/) 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 explanationNo 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](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r01-quantifier-0082/) 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 explanationNo 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](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r01-quantifier-0083/) 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 explanationNo 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](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r01-quantifier-0084/) 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 explanationNo 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](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r01-quantifier-0085/) 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 explanationNo 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](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r01-quantifier-0086/) 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 explanationNo 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](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r01-quantifier-0087/) 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 explanationNo 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](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r01-quantifier-0088/) 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 explanationNo 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](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r01-quantifier-0089/) 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 explanationNo 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](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r01-quantifier-0090/) 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 explanationNo 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](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r01-quantifier-0091/) 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 explanationNo 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](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r01-quantifier-0092/) 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 explanationNo 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](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r01-quantifier-0093/) 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 explanationNo 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](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r01-quantifier-0094/) 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 explanationNo 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](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r01-quantifier-0095/) 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 explanationNo 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](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r01-quantifier-0096/) 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 explanationNo 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](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r01-quantifier-0097/) 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 explanationNo 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](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r01-quantifier-0098/) 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 explanationNo 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](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r01-quantifier-0099/) 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 explanationNo 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](https://ocme.evemisslab.com/en/questions/q-20261003-advance-20261004-r01-quantifier-0100/) Quantifiers and explicit domains · qf-quantifier-negated-existential · candidate [Canonical question JSONL](https://ocme.evemisslab.com/data/questions/batches/batch-20261003-advance-20261004-r01-quantifier-foundations/questions.jsonl) · [English JSONL](https://ocme.evemisslab.com/data/i18n/questions/batch-20261003-advance-20261004-r01-quantifier-foundations.en.jsonl) · [Validation report](https://ocme.evemisslab.com/data/questions/batches/batch-20261003-advance-20261004-r01-quantifier-foundations/validation-report.json)