{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r01-quantifier-0001","batch_id":"batch-20261003-advance-20261004-r01-quantifier-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"082adbab57659e28440c8eda2aad3e1ff6599e075883bd7a8bb7901224941227","family_id":"qf-quantifier-finite-universal","seed":202610041001,"semantic_template":"qf-quantifier-finite-universal/{\"U\":[],\"operation\":\"forall\",\"predicate\":{\"kind\":\"ge\",\"value\":0}}"},"fields":{"title":"Universal Truth on a Finite Domain","stem":"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.","answer":{"kind":"short_text","value":"Yes","explanation":"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."},"skills":["Universal Truth on a Finite Domain"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r01-quantifier-0002","batch_id":"batch-20261003-advance-20261004-r01-quantifier-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7117413d1337065167e80ba89f55c33bf2f9c635327b721fc796dd1401f6308e","family_id":"qf-quantifier-finite-universal","seed":202610041002,"semantic_template":"qf-quantifier-finite-universal/{\"U\":[-2,-1,0],\"operation\":\"forall\",\"predicate\":{\"kind\":\"ge\",\"value\":-2}}"},"fields":{"title":"Universal Truth on a Finite Domain","stem":"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.","answer":{"kind":"short_text","value":"Yes","explanation":"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."},"skills":["Universal Truth on a Finite Domain"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r01-quantifier-0003","batch_id":"batch-20261003-advance-20261004-r01-quantifier-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e7ae6c14d647dba43bcf5b4011af1a10916c7c5f9e33e2387dd58b49c181a9b8","family_id":"qf-quantifier-finite-universal","seed":202610041003,"semantic_template":"qf-quantifier-finite-universal/{\"U\":[1,2,3],\"operation\":\"forall\",\"predicate\":{\"kind\":\"ge\",\"value\":4}}"},"fields":{"title":"Universal Truth on a Finite Domain","stem":"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.","answer":{"kind":"short_text","value":"No","explanation":"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."},"skills":["Universal Truth on a Finite Domain"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r01-quantifier-0004","batch_id":"batch-20261003-advance-20261004-r01-quantifier-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d451b4e265d4de72904b6244906515b3550156e58ddb9a71967b14920a60b872","family_id":"qf-quantifier-finite-universal","seed":202610041004,"semantic_template":"qf-quantifier-finite-universal/{\"U\":[-1,0,1],\"operation\":\"forall\",\"predicate\":{\"kind\":\"multiple\",\"value\":2}}"},"fields":{"title":"Universal Truth on a Finite Domain","stem":"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.","answer":{"kind":"short_text","value":"No","explanation":"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."},"skills":["Universal Truth on a Finite Domain"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r01-quantifier-0005","batch_id":"batch-20261003-advance-20261004-r01-quantifier-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"11d99e8491e3d319df28df21c9e63fb9d28287b99dee47a450bc018ed4033d12","family_id":"qf-quantifier-finite-universal","seed":202610041005,"semantic_template":"qf-quantifier-finite-universal/{\"U\":[-8,-7,-5,-3],\"operation\":\"forall\",\"predicate\":{\"kind\":\"ge\",\"value\":-7}}"},"fields":{"title":"Universal Truth on a Finite Domain","stem":"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.","answer":{"kind":"short_text","value":"No","explanation":"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."},"skills":["Universal Truth on a Finite Domain"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r01-quantifier-0006","batch_id":"batch-20261003-advance-20261004-r01-quantifier-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d9cbb7643e515be598f54bc9e88d2be78f346289b990b56b678ebe002031f90a","family_id":"qf-quantifier-finite-universal","seed":202610041006,"semantic_template":"qf-quantifier-finite-universal/{\"U\":[-7,-6,-4,-2],\"operation\":\"forall\",\"predicate\":{\"kind\":\"le\",\"value\":-4}}"},"fields":{"title":"Universal Truth on a Finite Domain","stem":"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.","answer":{"kind":"short_text","value":"No","explanation":"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."},"skills":["Universal Truth on a Finite Domain"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r01-quantifier-0007","batch_id":"batch-20261003-advance-20261004-r01-quantifier-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"62d48956585a9228834fdd39d1ff153816f496dd87b5e200255c76e714cfdc16","family_id":"qf-quantifier-finite-universal","seed":202610041007,"semantic_template":"qf-quantifier-finite-universal/{\"U\":[-6,-5,-3,-1],\"operation\":\"forall\",\"predicate\":{\"kind\":\"eq\",\"value\":-5}}"},"fields":{"title":"Universal Truth on a Finite Domain","stem":"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.","answer":{"kind":"short_text","value":"No","explanation":"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."},"skills":["Universal Truth on a Finite Domain"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r01-quantifier-0008","batch_id":"batch-20261003-advance-20261004-r01-quantifier-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5ce55149c62cc557f14db9bab28ea1d6cf3d35588749327cacc07b6e0bb114f3","family_id":"qf-quantifier-finite-universal","seed":202610041008,"semantic_template":"qf-quantifier-finite-universal/{\"U\":[-5,-4,-2,0],\"operation\":\"forall\",\"predicate\":{\"kind\":\"multiple\",\"value\":3}}"},"fields":{"title":"Universal Truth on a Finite Domain","stem":"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.","answer":{"kind":"short_text","value":"No","explanation":"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."},"skills":["Universal Truth on a Finite Domain"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r01-quantifier-0009","batch_id":"batch-20261003-advance-20261004-r01-quantifier-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ef4ecd25e2fe7faadfdbf9ad96302e489137e2c48740d9afb2205d6b7c1b7dbb","family_id":"qf-quantifier-finite-universal","seed":202610041009,"semantic_template":"qf-quantifier-finite-universal/{\"U\":[-4,-3,-1,1],\"operation\":\"forall\",\"predicate\":{\"kind\":\"ge\",\"value\":-3}}"},"fields":{"title":"Universal Truth on a Finite Domain","stem":"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.","answer":{"kind":"short_text","value":"No","explanation":"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."},"skills":["Universal Truth on a Finite Domain"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r01-quantifier-0010","batch_id":"batch-20261003-advance-20261004-r01-quantifier-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9a39abfd1ffb68f256f2952377c55b5279a0eeed36c3d21b6fd9065975371b18","family_id":"qf-quantifier-finite-universal","seed":202610041010,"semantic_template":"qf-quantifier-finite-universal/{\"U\":[-3,-2,0,2],\"operation\":\"forall\",\"predicate\":{\"kind\":\"le\",\"value\":0}}"},"fields":{"title":"Universal Truth on a Finite Domain","stem":"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.","answer":{"kind":"short_text","value":"No","explanation":"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."},"skills":["Universal Truth on a Finite Domain"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r01-quantifier-0011","batch_id":"batch-20261003-advance-20261004-r01-quantifier-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"61d01b65112a0be53442407fa8ce16645e681ce6ea54028f0523c4554cf5ffd8","family_id":"qf-quantifier-finite-universal","seed":202610041011,"semantic_template":"qf-quantifier-finite-universal/{\"U\":[-2,-1,1,3],\"operation\":\"forall\",\"predicate\":{\"kind\":\"eq\",\"value\":-1}}"},"fields":{"title":"Universal Truth on a Finite Domain","stem":"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.","answer":{"kind":"short_text","value":"No","explanation":"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."},"skills":["Universal Truth on a Finite Domain"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r01-quantifier-0012","batch_id":"batch-20261003-advance-20261004-r01-quantifier-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"140c3e8db66814c757c7aace2861998277366d90c8660284da95e9cfd440ec01","family_id":"qf-quantifier-finite-universal","seed":202610041012,"semantic_template":"qf-quantifier-finite-universal/{\"U\":[-1,0,2,4],\"operation\":\"forall\",\"predicate\":{\"kind\":\"multiple\",\"value\":4}}"},"fields":{"title":"Universal Truth on a Finite Domain","stem":"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.","answer":{"kind":"short_text","value":"No","explanation":"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."},"skills":["Universal Truth on a Finite Domain"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r01-quantifier-0013","batch_id":"batch-20261003-advance-20261004-r01-quantifier-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"47dac2c870f8a9b67d104c2efe4597b842aec32aa9fe4142deb6837e87011cf2","family_id":"qf-quantifier-finite-universal","seed":202610041013,"semantic_template":"qf-quantifier-finite-universal/{\"U\":[0,1,3,5],\"operation\":\"forall\",\"predicate\":{\"kind\":\"ge\",\"value\":1}}"},"fields":{"title":"Universal Truth on a Finite Domain","stem":"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.","answer":{"kind":"short_text","value":"No","explanation":"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."},"skills":["Universal Truth on a Finite Domain"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r01-quantifier-0014","batch_id":"batch-20261003-advance-20261004-r01-quantifier-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"db2d91258d2abaf1f4577514b428d0a8516310feecfadc20ad9500e643b5ce0e","family_id":"qf-quantifier-finite-universal","seed":202610041014,"semantic_template":"qf-quantifier-finite-universal/{\"U\":[1,2,4,6],\"operation\":\"forall\",\"predicate\":{\"kind\":\"le\",\"value\":4}}"},"fields":{"title":"Universal Truth on a Finite Domain","stem":"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.","answer":{"kind":"short_text","value":"No","explanation":"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."},"skills":["Universal Truth on a Finite Domain"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r01-quantifier-0015","batch_id":"batch-20261003-advance-20261004-r01-quantifier-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1e5c0c4a1a671e0e0e7ff01f09f60f73500506cd06fa4a68c897102128c70539","family_id":"qf-quantifier-finite-universal","seed":202610041015,"semantic_template":"qf-quantifier-finite-universal/{\"U\":[2,3,5,7],\"operation\":\"forall\",\"predicate\":{\"kind\":\"eq\",\"value\":3}}"},"fields":{"title":"Universal Truth on a Finite Domain","stem":"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.","answer":{"kind":"short_text","value":"No","explanation":"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."},"skills":["Universal Truth on a Finite Domain"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r01-quantifier-0016","batch_id":"batch-20261003-advance-20261004-r01-quantifier-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a4b9c5c464d89eff68bbf384de753ff0c93780694cc10897b9199636ee43da7f","family_id":"qf-quantifier-finite-universal","seed":202610041016,"semantic_template":"qf-quantifier-finite-universal/{\"U\":[3,4,6,8],\"operation\":\"forall\",\"predicate\":{\"kind\":\"multiple\",\"value\":2}}"},"fields":{"title":"Universal Truth on a Finite Domain","stem":"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.","answer":{"kind":"short_text","value":"No","explanation":"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."},"skills":["Universal Truth on a Finite Domain"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r01-quantifier-0017","batch_id":"batch-20261003-advance-20261004-r01-quantifier-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5528f7345f49ae95ba02fd86248a2b2d98b643748727ba95d2e420c81daa2d57","family_id":"qf-quantifier-finite-universal","seed":202610041017,"semantic_template":"qf-quantifier-finite-universal/{\"U\":[4,5,7,9],\"operation\":\"forall\",\"predicate\":{\"kind\":\"ge\",\"value\":5}}"},"fields":{"title":"Universal Truth on a Finite Domain","stem":"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.","answer":{"kind":"short_text","value":"No","explanation":"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."},"skills":["Universal Truth on a Finite Domain"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r01-quantifier-0018","batch_id":"batch-20261003-advance-20261004-r01-quantifier-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"53bbe6640f7e4fad8a8cd9f8b50eada17e9a45da2efc2d83ddf18877fa7fd0d3","family_id":"qf-quantifier-finite-universal","seed":202610041018,"semantic_template":"qf-quantifier-finite-universal/{\"U\":[5,6,8,10],\"operation\":\"forall\",\"predicate\":{\"kind\":\"le\",\"value\":8}}"},"fields":{"title":"Universal Truth on a Finite Domain","stem":"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.","answer":{"kind":"short_text","value":"No","explanation":"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."},"skills":["Universal Truth on a Finite Domain"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r01-quantifier-0019","batch_id":"batch-20261003-advance-20261004-r01-quantifier-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9c9591b1ccc5c9d3695c3701caf6507be51e0c9e09e141c1b3db641571694510","family_id":"qf-quantifier-finite-universal","seed":202610041019,"semantic_template":"qf-quantifier-finite-universal/{\"U\":[6,7,9,11],\"operation\":\"forall\",\"predicate\":{\"kind\":\"eq\",\"value\":7}}"},"fields":{"title":"Universal Truth on a Finite Domain","stem":"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.","answer":{"kind":"short_text","value":"No","explanation":"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."},"skills":["Universal Truth on a Finite Domain"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r01-quantifier-0020","batch_id":"batch-20261003-advance-20261004-r01-quantifier-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3aafd4ab64c5377e190cf611b71b4b07bfe1923a3b346fe526b656f6c3031c0a","family_id":"qf-quantifier-finite-universal","seed":202610041020,"semantic_template":"qf-quantifier-finite-universal/{\"U\":[7,8,10,12],\"operation\":\"forall\",\"predicate\":{\"kind\":\"multiple\",\"value\":3}}"},"fields":{"title":"Universal Truth on a Finite Domain","stem":"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.","answer":{"kind":"short_text","value":"No","explanation":"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."},"skills":["Universal Truth on a Finite Domain"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r01-quantifier-0021","batch_id":"batch-20261003-advance-20261004-r01-quantifier-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b7edfd2f0fd83433f598cd444c5545cd177ffe4ea214ea245099f2e20e430145","family_id":"qf-quantifier-finite-universal","seed":202610041021,"semantic_template":"qf-quantifier-finite-universal/{\"U\":[8,9,11,13],\"operation\":\"forall\",\"predicate\":{\"kind\":\"ge\",\"value\":9}}"},"fields":{"title":"Universal Truth on a Finite Domain","stem":"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.","answer":{"kind":"short_text","value":"No","explanation":"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."},"skills":["Universal Truth on a Finite Domain"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r01-quantifier-0022","batch_id":"batch-20261003-advance-20261004-r01-quantifier-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f8b8e55f1b202b0d157fb0e1706abb79e0f3de4e6aee1cfa29d7ec9143d19849","family_id":"qf-quantifier-finite-universal","seed":202610041022,"semantic_template":"qf-quantifier-finite-universal/{\"U\":[9,10,12,14],\"operation\":\"forall\",\"predicate\":{\"kind\":\"le\",\"value\":12}}"},"fields":{"title":"Universal Truth on a Finite Domain","stem":"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.","answer":{"kind":"short_text","value":"No","explanation":"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."},"skills":["Universal Truth on a Finite Domain"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r01-quantifier-0023","batch_id":"batch-20261003-advance-20261004-r01-quantifier-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"297fe046502f7ce30d2fa9b025527703fe3948b8b927e276b65268b0d5bba0de","family_id":"qf-quantifier-finite-universal","seed":202610041023,"semantic_template":"qf-quantifier-finite-universal/{\"U\":[10,11,13,15],\"operation\":\"forall\",\"predicate\":{\"kind\":\"eq\",\"value\":11}}"},"fields":{"title":"Universal Truth on a Finite Domain","stem":"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.","answer":{"kind":"short_text","value":"No","explanation":"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."},"skills":["Universal Truth on a Finite Domain"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r01-quantifier-0024","batch_id":"batch-20261003-advance-20261004-r01-quantifier-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e4b2382f19c5e52e73a797eac92a147bf3f74fcbc8be3c3b7a875649b00bd17d","family_id":"qf-quantifier-finite-universal","seed":202610041024,"semantic_template":"qf-quantifier-finite-universal/{\"U\":[11,12,14,16],\"operation\":\"forall\",\"predicate\":{\"kind\":\"multiple\",\"value\":4}}"},"fields":{"title":"Universal Truth on a Finite Domain","stem":"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.","answer":{"kind":"short_text","value":"No","explanation":"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."},"skills":["Universal Truth on a Finite Domain"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r01-quantifier-0025","batch_id":"batch-20261003-advance-20261004-r01-quantifier-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"67840cff09fdd972bab53a0321d47556ba012ee87dcf453f50ccd80f6fa91d56","family_id":"qf-quantifier-finite-universal","seed":202610041025,"semantic_template":"qf-quantifier-finite-universal/{\"U\":[12,13,15,17],\"operation\":\"forall\",\"predicate\":{\"kind\":\"ge\",\"value\":13}}"},"fields":{"title":"Universal Truth on a Finite Domain","stem":"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.","answer":{"kind":"short_text","value":"No","explanation":"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."},"skills":["Universal Truth on a Finite Domain"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r01-quantifier-0026","batch_id":"batch-20261003-advance-20261004-r01-quantifier-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6391fc571da3ac841df7bfaf927bcd7e5239ee7f182eefe125b714fa1629e488","family_id":"qf-quantifier-finite-existential","seed":202610041026,"semantic_template":"qf-quantifier-finite-existential/{\"U\":[],\"operation\":\"exists\",\"predicate\":{\"kind\":\"ge\",\"value\":0}}"},"fields":{"title":"Existential Truth on a Finite Domain","stem":"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.","answer":{"kind":"short_text","value":"No","explanation":"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."},"skills":["Existential Truth on a Finite Domain"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r01-quantifier-0027","batch_id":"batch-20261003-advance-20261004-r01-quantifier-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"20d4cbc9fc6ff33ef6e1858b0562b5308f5e81febfcfee83240b5b1c1eba7cb4","family_id":"qf-quantifier-finite-existential","seed":202610041027,"semantic_template":"qf-quantifier-finite-existential/{\"U\":[-2,-1,0],\"operation\":\"exists\",\"predicate\":{\"kind\":\"ge\",\"value\":-2}}"},"fields":{"title":"Existential Truth on a Finite Domain","stem":"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.","answer":{"kind":"short_text","value":"Yes","explanation":"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."},"skills":["Existential Truth on a Finite Domain"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r01-quantifier-0028","batch_id":"batch-20261003-advance-20261004-r01-quantifier-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"da116f9f51c792b805db48880c1630c49b3f1e55825fe429527e93aa0809aa1d","family_id":"qf-quantifier-finite-existential","seed":202610041028,"semantic_template":"qf-quantifier-finite-existential/{\"U\":[1,2,3],\"operation\":\"exists\",\"predicate\":{\"kind\":\"ge\",\"value\":4}}"},"fields":{"title":"Existential Truth on a Finite Domain","stem":"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.","answer":{"kind":"short_text","value":"No","explanation":"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."},"skills":["Existential Truth on a Finite Domain"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r01-quantifier-0029","batch_id":"batch-20261003-advance-20261004-r01-quantifier-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"86a4b4f70cdef4c92eb8a91adf2df50b71717901d57dd753d25f04aa9bb9ff99","family_id":"qf-quantifier-finite-existential","seed":202610041029,"semantic_template":"qf-quantifier-finite-existential/{\"U\":[-1,0,1],\"operation\":\"exists\",\"predicate\":{\"kind\":\"multiple\",\"value\":2}}"},"fields":{"title":"Existential Truth on a Finite Domain","stem":"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.","answer":{"kind":"short_text","value":"Yes","explanation":"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."},"skills":["Existential Truth on a Finite Domain"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r01-quantifier-0030","batch_id":"batch-20261003-advance-20261004-r01-quantifier-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f766c808e4c530b4e7a175ec4359477aeba809a5dc2c62d69ad79e6d59a2ccf2","family_id":"qf-quantifier-finite-existential","seed":202610041030,"semantic_template":"qf-quantifier-finite-existential/{\"U\":[-8,-7,-5,-3],\"operation\":\"exists\",\"predicate\":{\"kind\":\"ge\",\"value\":-7}}"},"fields":{"title":"Existential Truth on a Finite Domain","stem":"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.","answer":{"kind":"short_text","value":"Yes","explanation":"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."},"skills":["Existential Truth on a Finite Domain"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r01-quantifier-0031","batch_id":"batch-20261003-advance-20261004-r01-quantifier-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c5900b2ea5d1aacb78d15293cecc881d655b22b59a3ef3a31d62fa89dd159a49","family_id":"qf-quantifier-finite-existential","seed":202610041031,"semantic_template":"qf-quantifier-finite-existential/{\"U\":[-7,-6,-4,-2],\"operation\":\"exists\",\"predicate\":{\"kind\":\"le\",\"value\":-4}}"},"fields":{"title":"Existential Truth on a Finite Domain","stem":"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.","answer":{"kind":"short_text","value":"Yes","explanation":"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."},"skills":["Existential Truth on a Finite Domain"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r01-quantifier-0032","batch_id":"batch-20261003-advance-20261004-r01-quantifier-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3503fb389b7d48218f07e24e848a55af123c818488d598fc77b5cf5701bdce15","family_id":"qf-quantifier-finite-existential","seed":202610041032,"semantic_template":"qf-quantifier-finite-existential/{\"U\":[-6,-5,-3,-1],\"operation\":\"exists\",\"predicate\":{\"kind\":\"eq\",\"value\":-5}}"},"fields":{"title":"Existential Truth on a Finite Domain","stem":"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.","answer":{"kind":"short_text","value":"Yes","explanation":"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."},"skills":["Existential Truth on a Finite Domain"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r01-quantifier-0033","batch_id":"batch-20261003-advance-20261004-r01-quantifier-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3ecaf741ac8944cdb793a3348af1f863e54cba56b7bac5a8cf91065a58b95d8c","family_id":"qf-quantifier-finite-existential","seed":202610041033,"semantic_template":"qf-quantifier-finite-existential/{\"U\":[-5,-4,-2,0],\"operation\":\"exists\",\"predicate\":{\"kind\":\"multiple\",\"value\":3}}"},"fields":{"title":"Existential Truth on a Finite Domain","stem":"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.","answer":{"kind":"short_text","value":"Yes","explanation":"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."},"skills":["Existential Truth on a Finite Domain"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r01-quantifier-0034","batch_id":"batch-20261003-advance-20261004-r01-quantifier-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"459e45156166437c2aecb8088c5cbc1fa3943061f811acb2d1388918dc058584","family_id":"qf-quantifier-finite-existential","seed":202610041034,"semantic_template":"qf-quantifier-finite-existential/{\"U\":[-4,-3,-1,1],\"operation\":\"exists\",\"predicate\":{\"kind\":\"ge\",\"value\":-3}}"},"fields":{"title":"Existential Truth on a Finite Domain","stem":"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.","answer":{"kind":"short_text","value":"Yes","explanation":"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."},"skills":["Existential Truth on a Finite Domain"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r01-quantifier-0035","batch_id":"batch-20261003-advance-20261004-r01-quantifier-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d460b3f69d2606de5e4229ae9939b067513eae23c3e5dfe77b7938f8db813cb8","family_id":"qf-quantifier-finite-existential","seed":202610041035,"semantic_template":"qf-quantifier-finite-existential/{\"U\":[-3,-2,0,2],\"operation\":\"exists\",\"predicate\":{\"kind\":\"le\",\"value\":0}}"},"fields":{"title":"Existential Truth on a Finite Domain","stem":"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.","answer":{"kind":"short_text","value":"Yes","explanation":"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."},"skills":["Existential Truth on a Finite Domain"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r01-quantifier-0036","batch_id":"batch-20261003-advance-20261004-r01-quantifier-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3f3509a5db52d8e6fa822bf5517b7cfed1af6ba59cdd187d48cd682c3bda50ad","family_id":"qf-quantifier-finite-existential","seed":202610041036,"semantic_template":"qf-quantifier-finite-existential/{\"U\":[-2,-1,1,3],\"operation\":\"exists\",\"predicate\":{\"kind\":\"eq\",\"value\":-1}}"},"fields":{"title":"Existential Truth on a Finite Domain","stem":"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.","answer":{"kind":"short_text","value":"Yes","explanation":"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."},"skills":["Existential Truth on a Finite Domain"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r01-quantifier-0037","batch_id":"batch-20261003-advance-20261004-r01-quantifier-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f7802990fd7fa13344281bf9f78ada75c533f2e6a610ca473d4f84529d641c13","family_id":"qf-quantifier-finite-existential","seed":202610041037,"semantic_template":"qf-quantifier-finite-existential/{\"U\":[-1,0,2,4],\"operation\":\"exists\",\"predicate\":{\"kind\":\"multiple\",\"value\":4}}"},"fields":{"title":"Existential Truth on a Finite Domain","stem":"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.","answer":{"kind":"short_text","value":"Yes","explanation":"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."},"skills":["Existential Truth on a Finite Domain"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r01-quantifier-0038","batch_id":"batch-20261003-advance-20261004-r01-quantifier-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9d7b6e65856088716e90c85d14c4dde1055f7b75819d8c9e5d9184e729c7c461","family_id":"qf-quantifier-finite-existential","seed":202610041038,"semantic_template":"qf-quantifier-finite-existential/{\"U\":[0,1,3,5],\"operation\":\"exists\",\"predicate\":{\"kind\":\"ge\",\"value\":1}}"},"fields":{"title":"Existential Truth on a Finite Domain","stem":"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.","answer":{"kind":"short_text","value":"Yes","explanation":"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."},"skills":["Existential Truth on a Finite Domain"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r01-quantifier-0039","batch_id":"batch-20261003-advance-20261004-r01-quantifier-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"abe1d1e78dbc04aa98c7061d67e7798472023192a1c48ae26de351f3dcfb55ea","family_id":"qf-quantifier-finite-existential","seed":202610041039,"semantic_template":"qf-quantifier-finite-existential/{\"U\":[1,2,4,6],\"operation\":\"exists\",\"predicate\":{\"kind\":\"le\",\"value\":4}}"},"fields":{"title":"Existential Truth on a Finite Domain","stem":"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.","answer":{"kind":"short_text","value":"Yes","explanation":"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."},"skills":["Existential Truth on a Finite Domain"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r01-quantifier-0040","batch_id":"batch-20261003-advance-20261004-r01-quantifier-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a84d9bf0b719e8be1a747f87ff5b596e2bc06411c8d51b2441d8131adc9725fd","family_id":"qf-quantifier-finite-existential","seed":202610041040,"semantic_template":"qf-quantifier-finite-existential/{\"U\":[2,3,5,7],\"operation\":\"exists\",\"predicate\":{\"kind\":\"eq\",\"value\":3}}"},"fields":{"title":"Existential Truth on a Finite Domain","stem":"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.","answer":{"kind":"short_text","value":"Yes","explanation":"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."},"skills":["Existential Truth on a Finite Domain"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r01-quantifier-0041","batch_id":"batch-20261003-advance-20261004-r01-quantifier-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1ad88f88ca5232b9b2e9efa6e6e6591e5d85d8f1b768fe510ad955bb61c2bda9","family_id":"qf-quantifier-finite-existential","seed":202610041041,"semantic_template":"qf-quantifier-finite-existential/{\"U\":[3,4,6,8],\"operation\":\"exists\",\"predicate\":{\"kind\":\"multiple\",\"value\":2}}"},"fields":{"title":"Existential Truth on a Finite Domain","stem":"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.","answer":{"kind":"short_text","value":"Yes","explanation":"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."},"skills":["Existential Truth on a Finite Domain"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r01-quantifier-0042","batch_id":"batch-20261003-advance-20261004-r01-quantifier-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"111a725c7d9b937f20c0003c46a5d415972f395439212865323b3d5747f94305","family_id":"qf-quantifier-finite-existential","seed":202610041042,"semantic_template":"qf-quantifier-finite-existential/{\"U\":[4,5,7,9],\"operation\":\"exists\",\"predicate\":{\"kind\":\"ge\",\"value\":5}}"},"fields":{"title":"Existential Truth on a Finite Domain","stem":"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.","answer":{"kind":"short_text","value":"Yes","explanation":"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."},"skills":["Existential Truth on a Finite Domain"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r01-quantifier-0043","batch_id":"batch-20261003-advance-20261004-r01-quantifier-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7f46bbabe6bc19acbcee6017b43c22e26752e7005655d5faecc647638babb957","family_id":"qf-quantifier-finite-existential","seed":202610041043,"semantic_template":"qf-quantifier-finite-existential/{\"U\":[5,6,8,10],\"operation\":\"exists\",\"predicate\":{\"kind\":\"le\",\"value\":8}}"},"fields":{"title":"Existential Truth on a Finite Domain","stem":"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.","answer":{"kind":"short_text","value":"Yes","explanation":"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."},"skills":["Existential Truth on a Finite Domain"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r01-quantifier-0044","batch_id":"batch-20261003-advance-20261004-r01-quantifier-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e7a954114e2ed11f527c6e0757e7e1b1d6c3e79e4bae2dfa33c95a9866794680","family_id":"qf-quantifier-finite-existential","seed":202610041044,"semantic_template":"qf-quantifier-finite-existential/{\"U\":[6,7,9,11],\"operation\":\"exists\",\"predicate\":{\"kind\":\"eq\",\"value\":7}}"},"fields":{"title":"Existential Truth on a Finite Domain","stem":"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.","answer":{"kind":"short_text","value":"Yes","explanation":"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."},"skills":["Existential Truth on a Finite Domain"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r01-quantifier-0045","batch_id":"batch-20261003-advance-20261004-r01-quantifier-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0695e2fe5ea2a93825238a125c2f28622b31d748ad54a67c9e30c0d69f5f012e","family_id":"qf-quantifier-finite-existential","seed":202610041045,"semantic_template":"qf-quantifier-finite-existential/{\"U\":[7,8,10,12],\"operation\":\"exists\",\"predicate\":{\"kind\":\"multiple\",\"value\":3}}"},"fields":{"title":"Existential Truth on a Finite Domain","stem":"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.","answer":{"kind":"short_text","value":"Yes","explanation":"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."},"skills":["Existential Truth on a Finite Domain"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r01-quantifier-0046","batch_id":"batch-20261003-advance-20261004-r01-quantifier-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"acbf8f28140503526072a0b2d5b674f20f6d1c9dc8bd21915e73b43c496cb290","family_id":"qf-quantifier-finite-existential","seed":202610041046,"semantic_template":"qf-quantifier-finite-existential/{\"U\":[8,9,11,13],\"operation\":\"exists\",\"predicate\":{\"kind\":\"ge\",\"value\":9}}"},"fields":{"title":"Existential Truth on a Finite Domain","stem":"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.","answer":{"kind":"short_text","value":"Yes","explanation":"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."},"skills":["Existential Truth on a Finite Domain"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r01-quantifier-0047","batch_id":"batch-20261003-advance-20261004-r01-quantifier-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"461fd019980581e5a20f9ec9c61e233afed20390b00941632d6602af980b6722","family_id":"qf-quantifier-finite-existential","seed":202610041047,"semantic_template":"qf-quantifier-finite-existential/{\"U\":[9,10,12,14],\"operation\":\"exists\",\"predicate\":{\"kind\":\"le\",\"value\":12}}"},"fields":{"title":"Existential Truth on a Finite Domain","stem":"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.","answer":{"kind":"short_text","value":"Yes","explanation":"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."},"skills":["Existential Truth on a Finite Domain"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r01-quantifier-0048","batch_id":"batch-20261003-advance-20261004-r01-quantifier-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6364ca2019d92531aacef301a814233805402d364412125bd39290844d0ce667","family_id":"qf-quantifier-finite-existential","seed":202610041048,"semantic_template":"qf-quantifier-finite-existential/{\"U\":[10,11,13,15],\"operation\":\"exists\",\"predicate\":{\"kind\":\"eq\",\"value\":11}}"},"fields":{"title":"Existential Truth on a Finite Domain","stem":"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.","answer":{"kind":"short_text","value":"Yes","explanation":"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."},"skills":["Existential Truth on a Finite Domain"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r01-quantifier-0049","batch_id":"batch-20261003-advance-20261004-r01-quantifier-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bc413c5120262ceab3275bc4c8f5dd56ccbbb3f5f459f9e4428a092c89750b5c","family_id":"qf-quantifier-finite-existential","seed":202610041049,"semantic_template":"qf-quantifier-finite-existential/{\"U\":[11,12,14,16],\"operation\":\"exists\",\"predicate\":{\"kind\":\"multiple\",\"value\":4}}"},"fields":{"title":"Existential Truth on a Finite Domain","stem":"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.","answer":{"kind":"short_text","value":"Yes","explanation":"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."},"skills":["Existential Truth on a Finite Domain"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r01-quantifier-0050","batch_id":"batch-20261003-advance-20261004-r01-quantifier-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e24343fbffdbfb0a101e4a9810e32e6d13b0ca5794976e75cee5796a448a82d7","family_id":"qf-quantifier-finite-existential","seed":202610041050,"semantic_template":"qf-quantifier-finite-existential/{\"U\":[12,13,15,17],\"operation\":\"exists\",\"predicate\":{\"kind\":\"ge\",\"value\":13}}"},"fields":{"title":"Existential Truth on a Finite Domain","stem":"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.","answer":{"kind":"short_text","value":"Yes","explanation":"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."},"skills":["Existential Truth on a Finite Domain"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r01-quantifier-0051","batch_id":"batch-20261003-advance-20261004-r01-quantifier-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4b6fd6da9c74796cdd44a55a7d813cb9247884b77869801e55495ade1ee6a34a","family_id":"qf-quantifier-negated-universal","seed":202610041051,"semantic_template":"qf-quantifier-negated-universal/{\"U\":[],\"operation\":\"not_forall\",\"predicate\":{\"kind\":\"ge\",\"value\":0}}"},"fields":{"title":"Negation of a Finite Universal Statement","stem":"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.","answer":{"kind":"short_text","value":"No","explanation":"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."},"skills":["Negation of a Finite Universal Statement"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r01-quantifier-0052","batch_id":"batch-20261003-advance-20261004-r01-quantifier-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"31ae13ff33196ef7faf99897fe5508568e5efa1fa982b1465f05e7d2b3054805","family_id":"qf-quantifier-negated-universal","seed":202610041052,"semantic_template":"qf-quantifier-negated-universal/{\"U\":[-2,-1,0],\"operation\":\"not_forall\",\"predicate\":{\"kind\":\"ge\",\"value\":-2}}"},"fields":{"title":"Negation of a Finite Universal Statement","stem":"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.","answer":{"kind":"short_text","value":"No","explanation":"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."},"skills":["Negation of a Finite Universal Statement"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r01-quantifier-0053","batch_id":"batch-20261003-advance-20261004-r01-quantifier-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3531b0303ff2c141a3d3c3f0d10f700ad6170b07419d360337132c9d2d369f04","family_id":"qf-quantifier-negated-universal","seed":202610041053,"semantic_template":"qf-quantifier-negated-universal/{\"U\":[1,2,3],\"operation\":\"not_forall\",\"predicate\":{\"kind\":\"ge\",\"value\":4}}"},"fields":{"title":"Negation of a Finite Universal Statement","stem":"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.","answer":{"kind":"short_text","value":"Yes","explanation":"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."},"skills":["Negation of a Finite Universal Statement"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r01-quantifier-0054","batch_id":"batch-20261003-advance-20261004-r01-quantifier-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"316a281822982cea8d01f8e64a5b78f27c92ac7a4efcd99769bebc0b6e90302c","family_id":"qf-quantifier-negated-universal","seed":202610041054,"semantic_template":"qf-quantifier-negated-universal/{\"U\":[-1,0,1],\"operation\":\"not_forall\",\"predicate\":{\"kind\":\"multiple\",\"value\":2}}"},"fields":{"title":"Negation of a Finite Universal Statement","stem":"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.","answer":{"kind":"short_text","value":"Yes","explanation":"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."},"skills":["Negation of a Finite Universal Statement"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r01-quantifier-0055","batch_id":"batch-20261003-advance-20261004-r01-quantifier-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7a14d35a011fbc51b3c407ae9ab0351430d94a9997afdea03055114eaeeda153","family_id":"qf-quantifier-negated-universal","seed":202610041055,"semantic_template":"qf-quantifier-negated-universal/{\"U\":[-8,-7,-5,-3],\"operation\":\"not_forall\",\"predicate\":{\"kind\":\"ge\",\"value\":-7}}"},"fields":{"title":"Negation of a Finite Universal Statement","stem":"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.","answer":{"kind":"short_text","value":"Yes","explanation":"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."},"skills":["Negation of a Finite Universal Statement"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r01-quantifier-0056","batch_id":"batch-20261003-advance-20261004-r01-quantifier-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"534f68903de0f3676514523bc2734925e96c94defeb7736c1922abd53acc66f6","family_id":"qf-quantifier-negated-universal","seed":202610041056,"semantic_template":"qf-quantifier-negated-universal/{\"U\":[-7,-6,-4,-2],\"operation\":\"not_forall\",\"predicate\":{\"kind\":\"le\",\"value\":-4}}"},"fields":{"title":"Negation of a Finite Universal Statement","stem":"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.","answer":{"kind":"short_text","value":"Yes","explanation":"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."},"skills":["Negation of a Finite Universal Statement"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r01-quantifier-0057","batch_id":"batch-20261003-advance-20261004-r01-quantifier-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1dc77bfa08e891fb34aa48732b2728101336a771d2ae5c86da2f870fca607496","family_id":"qf-quantifier-negated-universal","seed":202610041057,"semantic_template":"qf-quantifier-negated-universal/{\"U\":[-6,-5,-3,-1],\"operation\":\"not_forall\",\"predicate\":{\"kind\":\"eq\",\"value\":-5}}"},"fields":{"title":"Negation of a Finite Universal Statement","stem":"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.","answer":{"kind":"short_text","value":"Yes","explanation":"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."},"skills":["Negation of a Finite Universal Statement"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r01-quantifier-0058","batch_id":"batch-20261003-advance-20261004-r01-quantifier-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f19354006b9a903152be23259dd7d2aaa142f7104e8d936a3d3240590ee3ab27","family_id":"qf-quantifier-negated-universal","seed":202610041058,"semantic_template":"qf-quantifier-negated-universal/{\"U\":[-5,-4,-2,0],\"operation\":\"not_forall\",\"predicate\":{\"kind\":\"multiple\",\"value\":3}}"},"fields":{"title":"Negation of a Finite Universal Statement","stem":"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.","answer":{"kind":"short_text","value":"Yes","explanation":"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."},"skills":["Negation of a Finite Universal Statement"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r01-quantifier-0059","batch_id":"batch-20261003-advance-20261004-r01-quantifier-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"805a286fc0462cfc70a92101bdc1f1ee03d3470a92252865eee4e2e0c1a3bd7f","family_id":"qf-quantifier-negated-universal","seed":202610041059,"semantic_template":"qf-quantifier-negated-universal/{\"U\":[-4,-3,-1,1],\"operation\":\"not_forall\",\"predicate\":{\"kind\":\"ge\",\"value\":-3}}"},"fields":{"title":"Negation of a Finite Universal Statement","stem":"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.","answer":{"kind":"short_text","value":"Yes","explanation":"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."},"skills":["Negation of a Finite Universal Statement"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r01-quantifier-0060","batch_id":"batch-20261003-advance-20261004-r01-quantifier-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4504fc0c159ef43671969f304f997f946f47b8a2f82c1d92007dc836bdd95181","family_id":"qf-quantifier-negated-universal","seed":202610041060,"semantic_template":"qf-quantifier-negated-universal/{\"U\":[-3,-2,0,2],\"operation\":\"not_forall\",\"predicate\":{\"kind\":\"le\",\"value\":0}}"},"fields":{"title":"Negation of a Finite Universal Statement","stem":"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.","answer":{"kind":"short_text","value":"Yes","explanation":"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."},"skills":["Negation of a Finite Universal Statement"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r01-quantifier-0061","batch_id":"batch-20261003-advance-20261004-r01-quantifier-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f3b6a7b91d2c6adb3ac31cfd35b00c43f6b583f71c1e1ef9782d708e2094dc1a","family_id":"qf-quantifier-negated-universal","seed":202610041061,"semantic_template":"qf-quantifier-negated-universal/{\"U\":[-2,-1,1,3],\"operation\":\"not_forall\",\"predicate\":{\"kind\":\"eq\",\"value\":-1}}"},"fields":{"title":"Negation of a Finite Universal Statement","stem":"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.","answer":{"kind":"short_text","value":"Yes","explanation":"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."},"skills":["Negation of a Finite Universal Statement"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r01-quantifier-0062","batch_id":"batch-20261003-advance-20261004-r01-quantifier-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c2b550982a89f28f520051225012669a6c6aedea4794e8bd64b3dac9e9d8989e","family_id":"qf-quantifier-negated-universal","seed":202610041062,"semantic_template":"qf-quantifier-negated-universal/{\"U\":[-1,0,2,4],\"operation\":\"not_forall\",\"predicate\":{\"kind\":\"multiple\",\"value\":4}}"},"fields":{"title":"Negation of a Finite Universal Statement","stem":"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.","answer":{"kind":"short_text","value":"Yes","explanation":"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."},"skills":["Negation of a Finite Universal Statement"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r01-quantifier-0063","batch_id":"batch-20261003-advance-20261004-r01-quantifier-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ad4b38e6987bdc7705f75d2631d490820c5b281372e62340ed829c6ac0fe83d3","family_id":"qf-quantifier-negated-universal","seed":202610041063,"semantic_template":"qf-quantifier-negated-universal/{\"U\":[0,1,3,5],\"operation\":\"not_forall\",\"predicate\":{\"kind\":\"ge\",\"value\":1}}"},"fields":{"title":"Negation of a Finite Universal Statement","stem":"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.","answer":{"kind":"short_text","value":"Yes","explanation":"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."},"skills":["Negation of a Finite Universal Statement"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r01-quantifier-0064","batch_id":"batch-20261003-advance-20261004-r01-quantifier-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6047229d6c060c48765775a5db9f33bd167c14eeaf2ab02177008ff228684d7e","family_id":"qf-quantifier-negated-universal","seed":202610041064,"semantic_template":"qf-quantifier-negated-universal/{\"U\":[1,2,4,6],\"operation\":\"not_forall\",\"predicate\":{\"kind\":\"le\",\"value\":4}}"},"fields":{"title":"Negation of a Finite Universal Statement","stem":"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.","answer":{"kind":"short_text","value":"Yes","explanation":"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."},"skills":["Negation of a Finite Universal Statement"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r01-quantifier-0065","batch_id":"batch-20261003-advance-20261004-r01-quantifier-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9b88a77e2146911bb6e428eb35c1bf037f2a4bb6c94ed648d767b09a40c10a9c","family_id":"qf-quantifier-negated-universal","seed":202610041065,"semantic_template":"qf-quantifier-negated-universal/{\"U\":[2,3,5,7],\"operation\":\"not_forall\",\"predicate\":{\"kind\":\"eq\",\"value\":3}}"},"fields":{"title":"Negation of a Finite Universal Statement","stem":"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.","answer":{"kind":"short_text","value":"Yes","explanation":"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."},"skills":["Negation of a Finite Universal Statement"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r01-quantifier-0066","batch_id":"batch-20261003-advance-20261004-r01-quantifier-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"aa784a128a7f5ae55255733f19fd00b633c06bf80b25b6f6e6c5129522ec319b","family_id":"qf-quantifier-negated-universal","seed":202610041066,"semantic_template":"qf-quantifier-negated-universal/{\"U\":[3,4,6,8],\"operation\":\"not_forall\",\"predicate\":{\"kind\":\"multiple\",\"value\":2}}"},"fields":{"title":"Negation of a Finite Universal Statement","stem":"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.","answer":{"kind":"short_text","value":"Yes","explanation":"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."},"skills":["Negation of a Finite Universal Statement"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r01-quantifier-0067","batch_id":"batch-20261003-advance-20261004-r01-quantifier-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8d370b816961e420a40b40249e37fd639221c21bba4f0737ae6048081389eb86","family_id":"qf-quantifier-negated-universal","seed":202610041067,"semantic_template":"qf-quantifier-negated-universal/{\"U\":[4,5,7,9],\"operation\":\"not_forall\",\"predicate\":{\"kind\":\"ge\",\"value\":5}}"},"fields":{"title":"Negation of a Finite Universal Statement","stem":"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.","answer":{"kind":"short_text","value":"Yes","explanation":"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."},"skills":["Negation of a Finite Universal Statement"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r01-quantifier-0068","batch_id":"batch-20261003-advance-20261004-r01-quantifier-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fcec1d6d5f2c3812a4d188c1444e984d1c15ea32a685061a10a2fc6df02ddb68","family_id":"qf-quantifier-negated-universal","seed":202610041068,"semantic_template":"qf-quantifier-negated-universal/{\"U\":[5,6,8,10],\"operation\":\"not_forall\",\"predicate\":{\"kind\":\"le\",\"value\":8}}"},"fields":{"title":"Negation of a Finite Universal Statement","stem":"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.","answer":{"kind":"short_text","value":"Yes","explanation":"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."},"skills":["Negation of a Finite Universal Statement"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r01-quantifier-0069","batch_id":"batch-20261003-advance-20261004-r01-quantifier-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f483a0cfca6d89aab12fe35bbdb4593e011577b723de994bf92e8019830cc58b","family_id":"qf-quantifier-negated-universal","seed":202610041069,"semantic_template":"qf-quantifier-negated-universal/{\"U\":[6,7,9,11],\"operation\":\"not_forall\",\"predicate\":{\"kind\":\"eq\",\"value\":7}}"},"fields":{"title":"Negation of a Finite Universal Statement","stem":"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.","answer":{"kind":"short_text","value":"Yes","explanation":"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."},"skills":["Negation of a Finite Universal Statement"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r01-quantifier-0070","batch_id":"batch-20261003-advance-20261004-r01-quantifier-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"380ee02ae84148567b5eaa23d051b376c5dc546490e919a629043ba7a43ff2f5","family_id":"qf-quantifier-negated-universal","seed":202610041070,"semantic_template":"qf-quantifier-negated-universal/{\"U\":[7,8,10,12],\"operation\":\"not_forall\",\"predicate\":{\"kind\":\"multiple\",\"value\":3}}"},"fields":{"title":"Negation of a Finite Universal Statement","stem":"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.","answer":{"kind":"short_text","value":"Yes","explanation":"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."},"skills":["Negation of a Finite Universal Statement"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r01-quantifier-0071","batch_id":"batch-20261003-advance-20261004-r01-quantifier-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0219582edb705fb6849f1d083c008d21941591363979ec1ee4a828f6ce0e2c1e","family_id":"qf-quantifier-negated-universal","seed":202610041071,"semantic_template":"qf-quantifier-negated-universal/{\"U\":[8,9,11,13],\"operation\":\"not_forall\",\"predicate\":{\"kind\":\"ge\",\"value\":9}}"},"fields":{"title":"Negation of a Finite Universal Statement","stem":"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.","answer":{"kind":"short_text","value":"Yes","explanation":"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."},"skills":["Negation of a Finite Universal Statement"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r01-quantifier-0072","batch_id":"batch-20261003-advance-20261004-r01-quantifier-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4ba80be1c88612083e73a2b6b887f6aae9518c9d4ae68223be42e73f6bfd2913","family_id":"qf-quantifier-negated-universal","seed":202610041072,"semantic_template":"qf-quantifier-negated-universal/{\"U\":[9,10,12,14],\"operation\":\"not_forall\",\"predicate\":{\"kind\":\"le\",\"value\":12}}"},"fields":{"title":"Negation of a Finite Universal Statement","stem":"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.","answer":{"kind":"short_text","value":"Yes","explanation":"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."},"skills":["Negation of a Finite Universal Statement"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r01-quantifier-0073","batch_id":"batch-20261003-advance-20261004-r01-quantifier-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f52de8562b27d76963af07d9823811800608df1910d98c9cdc302277bb5c2824","family_id":"qf-quantifier-negated-universal","seed":202610041073,"semantic_template":"qf-quantifier-negated-universal/{\"U\":[10,11,13,15],\"operation\":\"not_forall\",\"predicate\":{\"kind\":\"eq\",\"value\":11}}"},"fields":{"title":"Negation of a Finite Universal Statement","stem":"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.","answer":{"kind":"short_text","value":"Yes","explanation":"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."},"skills":["Negation of a Finite Universal Statement"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r01-quantifier-0074","batch_id":"batch-20261003-advance-20261004-r01-quantifier-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ad131b3357ab6e82b3260de5b36b26ca8e7ff916e56627eae88b736615ff9878","family_id":"qf-quantifier-negated-universal","seed":202610041074,"semantic_template":"qf-quantifier-negated-universal/{\"U\":[11,12,14,16],\"operation\":\"not_forall\",\"predicate\":{\"kind\":\"multiple\",\"value\":4}}"},"fields":{"title":"Negation of a Finite Universal Statement","stem":"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.","answer":{"kind":"short_text","value":"Yes","explanation":"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."},"skills":["Negation of a Finite Universal Statement"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r01-quantifier-0075","batch_id":"batch-20261003-advance-20261004-r01-quantifier-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"37098fd0cd1a9f0aa218618bcdfca18fa370b8c35550917139cd3e9ea30d06dc","family_id":"qf-quantifier-negated-universal","seed":202610041075,"semantic_template":"qf-quantifier-negated-universal/{\"U\":[12,13,15,17],\"operation\":\"not_forall\",\"predicate\":{\"kind\":\"ge\",\"value\":13}}"},"fields":{"title":"Negation of a Finite Universal Statement","stem":"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.","answer":{"kind":"short_text","value":"Yes","explanation":"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."},"skills":["Negation of a Finite Universal Statement"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r01-quantifier-0076","batch_id":"batch-20261003-advance-20261004-r01-quantifier-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9ded018e80f2da1667b93798b7aab8d72bf3609132abc809c37031ffe732e31f","family_id":"qf-quantifier-negated-existential","seed":202610041076,"semantic_template":"qf-quantifier-negated-existential/{\"U\":[],\"operation\":\"not_exists\",\"predicate\":{\"kind\":\"ge\",\"value\":0}}"},"fields":{"title":"Negation of a Finite Existential Statement","stem":"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.","answer":{"kind":"short_text","value":"Yes","explanation":"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."},"skills":["Negation of a Finite Existential Statement"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r01-quantifier-0077","batch_id":"batch-20261003-advance-20261004-r01-quantifier-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"140354b7aa9e74fa53e4cafbfca228be6321a8f02ebdbba7c2c606c8e97a66bb","family_id":"qf-quantifier-negated-existential","seed":202610041077,"semantic_template":"qf-quantifier-negated-existential/{\"U\":[-2,-1,0],\"operation\":\"not_exists\",\"predicate\":{\"kind\":\"ge\",\"value\":-2}}"},"fields":{"title":"Negation of a Finite Existential Statement","stem":"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.","answer":{"kind":"short_text","value":"No","explanation":"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."},"skills":["Negation of a Finite Existential Statement"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r01-quantifier-0078","batch_id":"batch-20261003-advance-20261004-r01-quantifier-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1ba624cae6fecf113658d528e6a31433dd2d1600317a13483fee17a81e6edb7e","family_id":"qf-quantifier-negated-existential","seed":202610041078,"semantic_template":"qf-quantifier-negated-existential/{\"U\":[1,2,3],\"operation\":\"not_exists\",\"predicate\":{\"kind\":\"ge\",\"value\":4}}"},"fields":{"title":"Negation of a Finite Existential Statement","stem":"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.","answer":{"kind":"short_text","value":"Yes","explanation":"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."},"skills":["Negation of a Finite Existential Statement"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r01-quantifier-0079","batch_id":"batch-20261003-advance-20261004-r01-quantifier-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e92d1b27db6a07fdb7d008f9c18d9e90402064cc8b4d9b7491a1feaed00e9a9d","family_id":"qf-quantifier-negated-existential","seed":202610041079,"semantic_template":"qf-quantifier-negated-existential/{\"U\":[-1,0,1],\"operation\":\"not_exists\",\"predicate\":{\"kind\":\"multiple\",\"value\":2}}"},"fields":{"title":"Negation of a Finite Existential Statement","stem":"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.","answer":{"kind":"short_text","value":"No","explanation":"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."},"skills":["Negation of a Finite Existential Statement"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r01-quantifier-0080","batch_id":"batch-20261003-advance-20261004-r01-quantifier-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a3abe0aa5a9e782dbe905e79b7c2fb3341f3c7f5f64ce439f334a3997d032531","family_id":"qf-quantifier-negated-existential","seed":202610041080,"semantic_template":"qf-quantifier-negated-existential/{\"U\":[-8,-7,-5,-3],\"operation\":\"not_exists\",\"predicate\":{\"kind\":\"ge\",\"value\":-7}}"},"fields":{"title":"Negation of a Finite Existential Statement","stem":"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.","answer":{"kind":"short_text","value":"No","explanation":"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."},"skills":["Negation of a Finite Existential Statement"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r01-quantifier-0081","batch_id":"batch-20261003-advance-20261004-r01-quantifier-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"90d2813e8b8f965a4811ba782f0ae1e0b2b238f527a564157fe91d87a5b20d8f","family_id":"qf-quantifier-negated-existential","seed":202610041081,"semantic_template":"qf-quantifier-negated-existential/{\"U\":[-7,-6,-4,-2],\"operation\":\"not_exists\",\"predicate\":{\"kind\":\"le\",\"value\":-4}}"},"fields":{"title":"Negation of a Finite Existential Statement","stem":"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.","answer":{"kind":"short_text","value":"No","explanation":"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."},"skills":["Negation of a Finite Existential Statement"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r01-quantifier-0082","batch_id":"batch-20261003-advance-20261004-r01-quantifier-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c5e986e17658ae0fb04b8544edb7c04b2f01dda1aaff265142f4c490284fa007","family_id":"qf-quantifier-negated-existential","seed":202610041082,"semantic_template":"qf-quantifier-negated-existential/{\"U\":[-6,-5,-3,-1],\"operation\":\"not_exists\",\"predicate\":{\"kind\":\"eq\",\"value\":-5}}"},"fields":{"title":"Negation of a Finite Existential Statement","stem":"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.","answer":{"kind":"short_text","value":"No","explanation":"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."},"skills":["Negation of a Finite Existential Statement"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r01-quantifier-0083","batch_id":"batch-20261003-advance-20261004-r01-quantifier-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"18e5a6dc6d3e37357d8b445cd4808b98b750066c3892254c3a9e6f4c33cf8ea2","family_id":"qf-quantifier-negated-existential","seed":202610041083,"semantic_template":"qf-quantifier-negated-existential/{\"U\":[-5,-4,-2,0],\"operation\":\"not_exists\",\"predicate\":{\"kind\":\"multiple\",\"value\":3}}"},"fields":{"title":"Negation of a Finite Existential Statement","stem":"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.","answer":{"kind":"short_text","value":"No","explanation":"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."},"skills":["Negation of a Finite Existential Statement"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r01-quantifier-0084","batch_id":"batch-20261003-advance-20261004-r01-quantifier-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9c7183b1f01c7b173089d3d65d9460cafbf82e1e73062398fadd90db10142282","family_id":"qf-quantifier-negated-existential","seed":202610041084,"semantic_template":"qf-quantifier-negated-existential/{\"U\":[-4,-3,-1,1],\"operation\":\"not_exists\",\"predicate\":{\"kind\":\"ge\",\"value\":-3}}"},"fields":{"title":"Negation of a Finite Existential Statement","stem":"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.","answer":{"kind":"short_text","value":"No","explanation":"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."},"skills":["Negation of a Finite Existential Statement"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r01-quantifier-0085","batch_id":"batch-20261003-advance-20261004-r01-quantifier-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0d04111605b238ba106ca3c7db20291a379043ec7e657e2e4876d3ff261e1850","family_id":"qf-quantifier-negated-existential","seed":202610041085,"semantic_template":"qf-quantifier-negated-existential/{\"U\":[-3,-2,0,2],\"operation\":\"not_exists\",\"predicate\":{\"kind\":\"le\",\"value\":0}}"},"fields":{"title":"Negation of a Finite Existential Statement","stem":"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.","answer":{"kind":"short_text","value":"No","explanation":"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."},"skills":["Negation of a Finite Existential Statement"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r01-quantifier-0086","batch_id":"batch-20261003-advance-20261004-r01-quantifier-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"52cf93d4cd7a1e407a5cb81bae72b33a8a3ce081e406c34f19606946974f4ef5","family_id":"qf-quantifier-negated-existential","seed":202610041086,"semantic_template":"qf-quantifier-negated-existential/{\"U\":[-2,-1,1,3],\"operation\":\"not_exists\",\"predicate\":{\"kind\":\"eq\",\"value\":-1}}"},"fields":{"title":"Negation of a Finite Existential Statement","stem":"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.","answer":{"kind":"short_text","value":"No","explanation":"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."},"skills":["Negation of a Finite Existential Statement"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r01-quantifier-0087","batch_id":"batch-20261003-advance-20261004-r01-quantifier-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5da02c14d1e5ae9d784acc908407767a58caaf5b91f07d52e22c7e05b190fed4","family_id":"qf-quantifier-negated-existential","seed":202610041087,"semantic_template":"qf-quantifier-negated-existential/{\"U\":[-1,0,2,4],\"operation\":\"not_exists\",\"predicate\":{\"kind\":\"multiple\",\"value\":4}}"},"fields":{"title":"Negation of a Finite Existential Statement","stem":"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.","answer":{"kind":"short_text","value":"No","explanation":"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."},"skills":["Negation of a Finite Existential Statement"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r01-quantifier-0088","batch_id":"batch-20261003-advance-20261004-r01-quantifier-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e6a00670bc13d5b906270b0c1fe2ad7ba69252234509aed812317352ffa6c816","family_id":"qf-quantifier-negated-existential","seed":202610041088,"semantic_template":"qf-quantifier-negated-existential/{\"U\":[0,1,3,5],\"operation\":\"not_exists\",\"predicate\":{\"kind\":\"ge\",\"value\":1}}"},"fields":{"title":"Negation of a Finite Existential Statement","stem":"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.","answer":{"kind":"short_text","value":"No","explanation":"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."},"skills":["Negation of a Finite Existential Statement"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r01-quantifier-0089","batch_id":"batch-20261003-advance-20261004-r01-quantifier-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e5f203dad1b2cee2257b7d59fcb4a118e73bf768236fa35a794857f17e1083c7","family_id":"qf-quantifier-negated-existential","seed":202610041089,"semantic_template":"qf-quantifier-negated-existential/{\"U\":[1,2,4,6],\"operation\":\"not_exists\",\"predicate\":{\"kind\":\"le\",\"value\":4}}"},"fields":{"title":"Negation of a Finite Existential Statement","stem":"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.","answer":{"kind":"short_text","value":"No","explanation":"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."},"skills":["Negation of a Finite Existential Statement"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r01-quantifier-0090","batch_id":"batch-20261003-advance-20261004-r01-quantifier-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"657a096864a4f013fab31eec1df03f28c3baace0987d0c0f049e8fb1432c8e73","family_id":"qf-quantifier-negated-existential","seed":202610041090,"semantic_template":"qf-quantifier-negated-existential/{\"U\":[2,3,5,7],\"operation\":\"not_exists\",\"predicate\":{\"kind\":\"eq\",\"value\":3}}"},"fields":{"title":"Negation of a Finite Existential Statement","stem":"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.","answer":{"kind":"short_text","value":"No","explanation":"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."},"skills":["Negation of a Finite Existential Statement"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r01-quantifier-0091","batch_id":"batch-20261003-advance-20261004-r01-quantifier-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b3e834181a6558c52ab5d674ff26491d7a1ab2d113ec0f6745efa7ae3f2540b5","family_id":"qf-quantifier-negated-existential","seed":202610041091,"semantic_template":"qf-quantifier-negated-existential/{\"U\":[3,4,6,8],\"operation\":\"not_exists\",\"predicate\":{\"kind\":\"multiple\",\"value\":2}}"},"fields":{"title":"Negation of a Finite Existential Statement","stem":"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.","answer":{"kind":"short_text","value":"No","explanation":"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."},"skills":["Negation of a Finite Existential Statement"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r01-quantifier-0092","batch_id":"batch-20261003-advance-20261004-r01-quantifier-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8af5a9fb43f56cf3ee4faa163bb7b87c62c49110c422f1649b7af174210b9878","family_id":"qf-quantifier-negated-existential","seed":202610041092,"semantic_template":"qf-quantifier-negated-existential/{\"U\":[4,5,7,9],\"operation\":\"not_exists\",\"predicate\":{\"kind\":\"ge\",\"value\":5}}"},"fields":{"title":"Negation of a Finite Existential Statement","stem":"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.","answer":{"kind":"short_text","value":"No","explanation":"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."},"skills":["Negation of a Finite Existential Statement"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r01-quantifier-0093","batch_id":"batch-20261003-advance-20261004-r01-quantifier-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"18fdd1588b1170d316e54ba73430138c61558142c723e4d5042eef581ce05c73","family_id":"qf-quantifier-negated-existential","seed":202610041093,"semantic_template":"qf-quantifier-negated-existential/{\"U\":[5,6,8,10],\"operation\":\"not_exists\",\"predicate\":{\"kind\":\"le\",\"value\":8}}"},"fields":{"title":"Negation of a Finite Existential Statement","stem":"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.","answer":{"kind":"short_text","value":"No","explanation":"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."},"skills":["Negation of a Finite Existential Statement"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r01-quantifier-0094","batch_id":"batch-20261003-advance-20261004-r01-quantifier-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"026738d9170cad7e778e0576feedef92b5042e421450f2ce0a1a4676e0c367c2","family_id":"qf-quantifier-negated-existential","seed":202610041094,"semantic_template":"qf-quantifier-negated-existential/{\"U\":[6,7,9,11],\"operation\":\"not_exists\",\"predicate\":{\"kind\":\"eq\",\"value\":7}}"},"fields":{"title":"Negation of a Finite Existential Statement","stem":"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.","answer":{"kind":"short_text","value":"No","explanation":"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."},"skills":["Negation of a Finite Existential Statement"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r01-quantifier-0095","batch_id":"batch-20261003-advance-20261004-r01-quantifier-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1394b52443d8c505103735da11617b01d248e0f530e6762078417a94ffafbb3f","family_id":"qf-quantifier-negated-existential","seed":202610041095,"semantic_template":"qf-quantifier-negated-existential/{\"U\":[7,8,10,12],\"operation\":\"not_exists\",\"predicate\":{\"kind\":\"multiple\",\"value\":3}}"},"fields":{"title":"Negation of a Finite Existential Statement","stem":"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.","answer":{"kind":"short_text","value":"No","explanation":"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."},"skills":["Negation of a Finite Existential Statement"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r01-quantifier-0096","batch_id":"batch-20261003-advance-20261004-r01-quantifier-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"51cbb3db3488b4fd1f38f0ba9bee23cdd59fa1989c5a164ef0f0511a154fc730","family_id":"qf-quantifier-negated-existential","seed":202610041096,"semantic_template":"qf-quantifier-negated-existential/{\"U\":[8,9,11,13],\"operation\":\"not_exists\",\"predicate\":{\"kind\":\"ge\",\"value\":9}}"},"fields":{"title":"Negation of a Finite Existential Statement","stem":"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.","answer":{"kind":"short_text","value":"No","explanation":"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."},"skills":["Negation of a Finite Existential Statement"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r01-quantifier-0097","batch_id":"batch-20261003-advance-20261004-r01-quantifier-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"601ed3585a5620cf5a3fea564e028f9aa40f4c48787e7bad1102b7e20b5f33e0","family_id":"qf-quantifier-negated-existential","seed":202610041097,"semantic_template":"qf-quantifier-negated-existential/{\"U\":[9,10,12,14],\"operation\":\"not_exists\",\"predicate\":{\"kind\":\"le\",\"value\":12}}"},"fields":{"title":"Negation of a Finite Existential Statement","stem":"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.","answer":{"kind":"short_text","value":"No","explanation":"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."},"skills":["Negation of a Finite Existential Statement"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r01-quantifier-0098","batch_id":"batch-20261003-advance-20261004-r01-quantifier-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d432656c6a33d7ea16b88ea3a1f68d7fe904714cfa4ec1522bc3514b3e0212de","family_id":"qf-quantifier-negated-existential","seed":202610041098,"semantic_template":"qf-quantifier-negated-existential/{\"U\":[10,11,13,15],\"operation\":\"not_exists\",\"predicate\":{\"kind\":\"eq\",\"value\":11}}"},"fields":{"title":"Negation of a Finite Existential Statement","stem":"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.","answer":{"kind":"short_text","value":"No","explanation":"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."},"skills":["Negation of a Finite Existential Statement"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r01-quantifier-0099","batch_id":"batch-20261003-advance-20261004-r01-quantifier-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a4d695d0d43090d148c97d2b851c039f69e16792d5fb18391c8ea66a9a519aa6","family_id":"qf-quantifier-negated-existential","seed":202610041099,"semantic_template":"qf-quantifier-negated-existential/{\"U\":[11,12,14,16],\"operation\":\"not_exists\",\"predicate\":{\"kind\":\"multiple\",\"value\":4}}"},"fields":{"title":"Negation of a Finite Existential Statement","stem":"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.","answer":{"kind":"short_text","value":"No","explanation":"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."},"skills":["Negation of a Finite Existential Statement"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r01-quantifier-0100","batch_id":"batch-20261003-advance-20261004-r01-quantifier-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"31cc046e57b850d4a0f56a23759544432c92bc0aa2038d6ef15cbb4c13107135","family_id":"qf-quantifier-negated-existential","seed":202610041100,"semantic_template":"qf-quantifier-negated-existential/{\"U\":[12,13,15,17],\"operation\":\"not_exists\",\"predicate\":{\"kind\":\"ge\",\"value\":13}}"},"fields":{"title":"Negation of a Finite Existential Statement","stem":"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.","answer":{"kind":"short_text","value":"No","explanation":"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."},"skills":["Negation of a Finite Existential Statement"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}