{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261019-r01-open-closed-set-0001","batch_id":"batch-20261013-advance-20261019-r01-open-closed-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2701074fb7e218e7451687272ebb22e3fad29cd0148d1eff28f2d879bd633ae7","family_id":"qf-ext19-openclosed-status","seed":202610191001,"semantic_template":"qf-ext19-openclosed-status/{\"n\":4,\"opens\":[[],[0],[0,1],[0,1,2],[0,1,2,3]],\"subset\":[]}"},"fields":{"title":"Open, Closed, Both, or Neither in One Fixed Topology","stem":"Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3]]. Normalize subsets and repeated members as sets; order families by binary subset masks. Complements are relative to this same X. Closed does not mean not open. Candidate=[]. Give (n, normalized candidate, open?, closed?, clopen?, clopen/open/closed/neither).","answer":{"kind":"expression","value":"(4, [], Yes, Yes, Yes, clopen)","explanation":"Check membership in the open family and openness of the relative complement independently. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [], Yes, Yes, Yes, clopen)."},"skills":["Open, Closed, Both, or Neither in One Fixed Topology"],"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-20261013-advance-20261019-r01-open-closed-set-0002","batch_id":"batch-20261013-advance-20261019-r01-open-closed-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d33e59ca10e0ac0c0ee1979d4d1aa6040b565167884cd6c0aa4aa63b479541a5","family_id":"qf-ext19-openclosed-status","seed":202610191002,"semantic_template":"qf-ext19-openclosed-status/{\"n\":4,\"opens\":[[],[1],[0,1],[0,1,2],[0,1,2,3]],\"subset\":[0]}"},"fields":{"title":"Open, Closed, Both, or Neither in One Fixed Topology","stem":"Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [1], [0, 1], [0, 1, 2], [0, 1, 2, 3]]. Normalize subsets and repeated members as sets; order families by binary subset masks. Complements are relative to this same X. Closed does not mean not open. Candidate=[0]. Give (n, normalized candidate, open?, closed?, clopen?, clopen/open/closed/neither).","answer":{"kind":"expression","value":"(4, [0], No, No, No, neither)","explanation":"Check membership in the open family and openness of the relative complement independently. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [0], No, No, No, neither)."},"skills":["Open, Closed, Both, or Neither in One Fixed Topology"],"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-20261013-advance-20261019-r01-open-closed-set-0003","batch_id":"batch-20261013-advance-20261019-r01-open-closed-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"37ab380227840f08f01e8f88f4e2b5d7ed835184794189d770886ae019b9e8e9","family_id":"qf-ext19-openclosed-status","seed":202610191003,"semantic_template":"qf-ext19-openclosed-status/{\"n\":4,\"opens\":[[],[2],[0,2],[0,1,2],[0,1,2,3]],\"subset\":[1]}"},"fields":{"title":"Open, Closed, Both, or Neither in One Fixed Topology","stem":"Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [2], [0, 2], [0, 1, 2], [0, 1, 2, 3]]. Normalize subsets and repeated members as sets; order families by binary subset masks. Complements are relative to this same X. Closed does not mean not open. Candidate=[1]. Give (n, normalized candidate, open?, closed?, clopen?, clopen/open/closed/neither).","answer":{"kind":"expression","value":"(4, [1], No, No, No, neither)","explanation":"Check membership in the open family and openness of the relative complement independently. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [1], No, No, No, neither)."},"skills":["Open, Closed, Both, or Neither in One Fixed Topology"],"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-20261013-advance-20261019-r01-open-closed-set-0004","batch_id":"batch-20261013-advance-20261019-r01-open-closed-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"82ff0286ce550f75a20e8fd709f148a899e3a5ad8c030bdf445598c1ff9f513d","family_id":"qf-ext19-openclosed-status","seed":202610191004,"semantic_template":"qf-ext19-openclosed-status/{\"n\":4,\"opens\":[[],[3],[0,3],[0,1,3],[0,1,2,3]],\"subset\":[0,1]}"},"fields":{"title":"Open, Closed, Both, or Neither in One Fixed Topology","stem":"Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [3], [0, 3], [0, 1, 3], [0, 1, 2, 3]]. Normalize subsets and repeated members as sets; order families by binary subset masks. Complements are relative to this same X. Closed does not mean not open. Candidate=[0, 1]. Give (n, normalized candidate, open?, closed?, clopen?, clopen/open/closed/neither).","answer":{"kind":"expression","value":"(4, [0, 1], No, No, No, neither)","explanation":"Check membership in the open family and openness of the relative complement independently. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [0, 1], No, No, No, neither)."},"skills":["Open, Closed, Both, or Neither in One Fixed Topology"],"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-20261013-advance-20261019-r01-open-closed-set-0005","batch_id":"batch-20261013-advance-20261019-r01-open-closed-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b4aa8f79e1a25a8c76a9b1e8104fb4a18db42120c90d1d033fae93b237f530be","family_id":"qf-ext19-openclosed-status","seed":202610191005,"semantic_template":"qf-ext19-openclosed-status/{\"n\":4,\"opens\":[[],[0],[0,2],[0,1,2],[0,1,2,3]],\"subset\":[2]}"},"fields":{"title":"Open, Closed, Both, or Neither in One Fixed Topology","stem":"Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [0], [0, 2], [0, 1, 2], [0, 1, 2, 3]]. Normalize subsets and repeated members as sets; order families by binary subset masks. Complements are relative to this same X. Closed does not mean not open. Candidate=[2]. Give (n, normalized candidate, open?, closed?, clopen?, clopen/open/closed/neither).","answer":{"kind":"expression","value":"(4, [2], No, No, No, neither)","explanation":"Check membership in the open family and openness of the relative complement independently. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [2], No, No, No, neither)."},"skills":["Open, Closed, Both, or Neither in One Fixed Topology"],"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-20261013-advance-20261019-r01-open-closed-set-0006","batch_id":"batch-20261013-advance-20261019-r01-open-closed-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c2025b550dc587878481d84ed1689becfe907c9ec2d6f4911ed4b8c200f13f8b","family_id":"qf-ext19-openclosed-status","seed":202610191006,"semantic_template":"qf-ext19-openclosed-status/{\"n\":4,\"opens\":[[],[1],[1,2],[0,1,2],[0,1,2,3]],\"subset\":[0,2]}"},"fields":{"title":"Open, Closed, Both, or Neither in One Fixed Topology","stem":"Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [1], [1, 2], [0, 1, 2], [0, 1, 2, 3]]. Normalize subsets and repeated members as sets; order families by binary subset masks. Complements are relative to this same X. Closed does not mean not open. Candidate=[0, 2]. Give (n, normalized candidate, open?, closed?, clopen?, clopen/open/closed/neither).","answer":{"kind":"expression","value":"(4, [0, 2], No, No, No, neither)","explanation":"Check membership in the open family and openness of the relative complement independently. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [0, 2], No, No, No, neither)."},"skills":["Open, Closed, Both, or Neither in One Fixed Topology"],"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-20261013-advance-20261019-r01-open-closed-set-0007","batch_id":"batch-20261013-advance-20261019-r01-open-closed-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fd5acf25951a063906b237bfb5516a56a2e0d00d2b04cecdfc6e077274515e6c","family_id":"qf-ext19-openclosed-status","seed":202610191007,"semantic_template":"qf-ext19-openclosed-status/{\"n\":4,\"opens\":[[],[2],[1,2],[0,1,2],[0,1,2,3]],\"subset\":[1,2]}"},"fields":{"title":"Open, Closed, Both, or Neither in One Fixed Topology","stem":"Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [2], [1, 2], [0, 1, 2], [0, 1, 2, 3]]. Normalize subsets and repeated members as sets; order families by binary subset masks. Complements are relative to this same X. Closed does not mean not open. Candidate=[1, 2]. Give (n, normalized candidate, open?, closed?, clopen?, clopen/open/closed/neither).","answer":{"kind":"expression","value":"(4, [1, 2], Yes, No, No, open)","explanation":"Check membership in the open family and openness of the relative complement independently. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [1, 2], Yes, No, No, open)."},"skills":["Open, Closed, Both, or Neither in One Fixed Topology"],"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-20261013-advance-20261019-r01-open-closed-set-0008","batch_id":"batch-20261013-advance-20261019-r01-open-closed-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f39ad66ce1519a6498c01cda0cbbab8275a8ff8282c9642cdee84ba4081fe91f","family_id":"qf-ext19-openclosed-status","seed":202610191008,"semantic_template":"qf-ext19-openclosed-status/{\"n\":4,\"opens\":[[],[3],[1,3],[0,1,3],[0,1,2,3]],\"subset\":[0,1,2]}"},"fields":{"title":"Open, Closed, Both, or Neither in One Fixed Topology","stem":"Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [3], [1, 3], [0, 1, 3], [0, 1, 2, 3]]. Normalize subsets and repeated members as sets; order families by binary subset masks. Complements are relative to this same X. Closed does not mean not open. Candidate=[0, 1, 2]. Give (n, normalized candidate, open?, closed?, clopen?, clopen/open/closed/neither).","answer":{"kind":"expression","value":"(4, [0, 1, 2], No, Yes, No, closed)","explanation":"Check membership in the open family and openness of the relative complement independently. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [0, 1, 2], No, Yes, No, closed)."},"skills":["Open, Closed, Both, or Neither in One Fixed Topology"],"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-20261013-advance-20261019-r01-open-closed-set-0009","batch_id":"batch-20261013-advance-20261019-r01-open-closed-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"10fc574820449b6d04af8bbfa241b54bf59e99cf4b7be50acb4c5704ec79167b","family_id":"qf-ext19-openclosed-status","seed":202610191009,"semantic_template":"qf-ext19-openclosed-status/{\"n\":4,\"opens\":[[],[0],[0,3],[0,1,3],[0,1,2,3]],\"subset\":[3]}"},"fields":{"title":"Open, Closed, Both, or Neither in One Fixed Topology","stem":"Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [0], [0, 3], [0, 1, 3], [0, 1, 2, 3]]. Normalize subsets and repeated members as sets; order families by binary subset masks. Complements are relative to this same X. Closed does not mean not open. Candidate=[3]. Give (n, normalized candidate, open?, closed?, clopen?, clopen/open/closed/neither).","answer":{"kind":"expression","value":"(4, [3], No, No, No, neither)","explanation":"Check membership in the open family and openness of the relative complement independently. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [3], No, No, No, neither)."},"skills":["Open, Closed, Both, or Neither in One Fixed Topology"],"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-20261013-advance-20261019-r01-open-closed-set-0010","batch_id":"batch-20261013-advance-20261019-r01-open-closed-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"49203bc98b9f73b532a9f6f59ad423a38283a19e836bbb2c2ea9664f7c28a9bd","family_id":"qf-ext19-openclosed-status","seed":202610191010,"semantic_template":"qf-ext19-openclosed-status/{\"n\":4,\"opens\":[[],[1],[1,3],[0,1,3],[0,1,2,3]],\"subset\":[0,3]}"},"fields":{"title":"Open, Closed, Both, or Neither in One Fixed Topology","stem":"Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [1], [1, 3], [0, 1, 3], [0, 1, 2, 3]]. Normalize subsets and repeated members as sets; order families by binary subset masks. Complements are relative to this same X. Closed does not mean not open. Candidate=[0, 3]. Give (n, normalized candidate, open?, closed?, clopen?, clopen/open/closed/neither).","answer":{"kind":"expression","value":"(4, [0, 3], No, No, No, neither)","explanation":"Check membership in the open family and openness of the relative complement independently. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [0, 3], No, No, No, neither)."},"skills":["Open, Closed, Both, or Neither in One Fixed Topology"],"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-20261013-advance-20261019-r01-open-closed-set-0011","batch_id":"batch-20261013-advance-20261019-r01-open-closed-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c008b3a5ff5506221c517b84a099abd8ac07ad5fe1d74ada7f0611b65bccc912","family_id":"qf-ext19-openclosed-status","seed":202610191011,"semantic_template":"qf-ext19-openclosed-status/{\"n\":4,\"opens\":[[],[2],[2,3],[0,2,3],[0,1,2,3]],\"subset\":[1,3]}"},"fields":{"title":"Open, Closed, Both, or Neither in One Fixed Topology","stem":"Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [2], [2, 3], [0, 2, 3], [0, 1, 2, 3]]. Normalize subsets and repeated members as sets; order families by binary subset masks. Complements are relative to this same X. Closed does not mean not open. Candidate=[1, 3]. Give (n, normalized candidate, open?, closed?, clopen?, clopen/open/closed/neither).","answer":{"kind":"expression","value":"(4, [1, 3], No, No, No, neither)","explanation":"Check membership in the open family and openness of the relative complement independently. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [1, 3], No, No, No, neither)."},"skills":["Open, Closed, Both, or Neither in One Fixed Topology"],"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-20261013-advance-20261019-r01-open-closed-set-0012","batch_id":"batch-20261013-advance-20261019-r01-open-closed-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9bf3c676c407340a6365afaf0d29d8b622e60e5a566c9c47e3a252ee0d009f8f","family_id":"qf-ext19-openclosed-status","seed":202610191012,"semantic_template":"qf-ext19-openclosed-status/{\"n\":4,\"opens\":[[],[3],[2,3],[0,2,3],[0,1,2,3]],\"subset\":[0,1,3]}"},"fields":{"title":"Open, Closed, Both, or Neither in One Fixed Topology","stem":"Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [3], [2, 3], [0, 2, 3], [0, 1, 2, 3]]. Normalize subsets and repeated members as sets; order families by binary subset masks. Complements are relative to this same X. Closed does not mean not open. Candidate=[0, 1, 3]. Give (n, normalized candidate, open?, closed?, clopen?, clopen/open/closed/neither).","answer":{"kind":"expression","value":"(4, [0, 1, 3], No, No, No, neither)","explanation":"Check membership in the open family and openness of the relative complement independently. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [0, 1, 3], No, No, No, neither)."},"skills":["Open, Closed, Both, or Neither in One Fixed Topology"],"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-20261013-advance-20261019-r01-open-closed-set-0013","batch_id":"batch-20261013-advance-20261019-r01-open-closed-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e8162eafd9d24582907b56b3c7dbc9cd843e809f52321fbcd60067dcbc0adbf8","family_id":"qf-ext19-openclosed-status","seed":202610191013,"semantic_template":"qf-ext19-openclosed-status/{\"n\":4,\"opens\":[[],[0],[0,1],[0,1,3],[0,1,2,3]],\"subset\":[2,3]}"},"fields":{"title":"Open, Closed, Both, or Neither in One Fixed Topology","stem":"Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [0], [0, 1], [0, 1, 3], [0, 1, 2, 3]]. Normalize subsets and repeated members as sets; order families by binary subset masks. Complements are relative to this same X. Closed does not mean not open. Candidate=[2, 3]. Give (n, normalized candidate, open?, closed?, clopen?, clopen/open/closed/neither).","answer":{"kind":"expression","value":"(4, [2, 3], No, Yes, No, closed)","explanation":"Check membership in the open family and openness of the relative complement independently. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [2, 3], No, Yes, No, closed)."},"skills":["Open, Closed, Both, or Neither in One Fixed Topology"],"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-20261013-advance-20261019-r01-open-closed-set-0014","batch_id":"batch-20261013-advance-20261019-r01-open-closed-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"80f6d7d0707cbdc24a488dfbcfd1831c7f6a2c5390d69ccaa2443ea04f5317af","family_id":"qf-ext19-openclosed-status","seed":202610191014,"semantic_template":"qf-ext19-openclosed-status/{\"n\":4,\"opens\":[[],[1],[0,1],[0,1,3],[0,1,2,3]],\"subset\":[0,2,3]}"},"fields":{"title":"Open, Closed, Both, or Neither in One Fixed Topology","stem":"Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [1], [0, 1], [0, 1, 3], [0, 1, 2, 3]]. Normalize subsets and repeated members as sets; order families by binary subset masks. Complements are relative to this same X. Closed does not mean not open. Candidate=[0, 2, 3]. Give (n, normalized candidate, open?, closed?, clopen?, clopen/open/closed/neither).","answer":{"kind":"expression","value":"(4, [0, 2, 3], No, Yes, No, closed)","explanation":"Check membership in the open family and openness of the relative complement independently. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [0, 2, 3], No, Yes, No, closed)."},"skills":["Open, Closed, Both, or Neither in One Fixed Topology"],"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-20261013-advance-20261019-r01-open-closed-set-0015","batch_id":"batch-20261013-advance-20261019-r01-open-closed-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"04db3c0950b37309bcf89d5d3f232e4415bbf5467497275d6f9d6f306a0fc3e4","family_id":"qf-ext19-openclosed-status","seed":202610191015,"semantic_template":"qf-ext19-openclosed-status/{\"n\":4,\"opens\":[[],[2],[0,2],[0,2,3],[0,1,2,3]],\"subset\":[1,2,3]}"},"fields":{"title":"Open, Closed, Both, or Neither in One Fixed Topology","stem":"Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [2], [0, 2], [0, 2, 3], [0, 1, 2, 3]]. Normalize subsets and repeated members as sets; order families by binary subset masks. Complements are relative to this same X. Closed does not mean not open. Candidate=[1, 2, 3]. Give (n, normalized candidate, open?, closed?, clopen?, clopen/open/closed/neither).","answer":{"kind":"expression","value":"(4, [1, 2, 3], No, No, No, neither)","explanation":"Check membership in the open family and openness of the relative complement independently. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [1, 2, 3], No, No, No, neither)."},"skills":["Open, Closed, Both, or Neither in One Fixed Topology"],"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-20261013-advance-20261019-r01-open-closed-set-0016","batch_id":"batch-20261013-advance-20261019-r01-open-closed-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d6d79b2ad9423ec2dd4911660e30727ed7d7ba6f396d83c162377eea92c3301b","family_id":"qf-ext19-openclosed-status","seed":202610191016,"semantic_template":"qf-ext19-openclosed-status/{\"n\":4,\"opens\":[[],[3],[0,3],[0,2,3],[0,1,2,3]],\"subset\":[0,1,2,3]}"},"fields":{"title":"Open, Closed, Both, or Neither in One Fixed Topology","stem":"Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [3], [0, 3], [0, 2, 3], [0, 1, 2, 3]]. Normalize subsets and repeated members as sets; order families by binary subset masks. Complements are relative to this same X. Closed does not mean not open. Candidate=[0, 1, 2, 3]. Give (n, normalized candidate, open?, closed?, clopen?, clopen/open/closed/neither).","answer":{"kind":"expression","value":"(4, [0, 1, 2, 3], Yes, Yes, Yes, clopen)","explanation":"Check membership in the open family and openness of the relative complement independently. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [0, 1, 2, 3], Yes, Yes, Yes, clopen)."},"skills":["Open, Closed, Both, or Neither in One Fixed Topology"],"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-20261013-advance-20261019-r01-open-closed-set-0017","batch_id":"batch-20261013-advance-20261019-r01-open-closed-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f824cfca7f29ac0b49fd8db0fcb19ac916badbe345f94a9a0b2417b7c4fe8f1f","family_id":"qf-ext19-openclosed-status","seed":202610191017,"semantic_template":"qf-ext19-openclosed-status/{\"n\":4,\"opens\":[[],[0],[0,2],[0,2,3],[0,1,2,3]],\"subset\":[]}"},"fields":{"title":"Open, Closed, Both, or Neither in One Fixed Topology","stem":"Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [0], [0, 2], [0, 2, 3], [0, 1, 2, 3]]. Normalize subsets and repeated members as sets; order families by binary subset masks. Complements are relative to this same X. Closed does not mean not open. Candidate=[]. Give (n, normalized candidate, open?, closed?, clopen?, clopen/open/closed/neither).","answer":{"kind":"expression","value":"(4, [], Yes, Yes, Yes, clopen)","explanation":"Check membership in the open family and openness of the relative complement independently. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [], Yes, Yes, Yes, clopen)."},"skills":["Open, Closed, Both, or Neither in One Fixed Topology"],"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-20261013-advance-20261019-r01-open-closed-set-0018","batch_id":"batch-20261013-advance-20261019-r01-open-closed-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ddf9ef4a8dd31e9d88ba4cac61aed90920fa70434cb5837916d243aeb7765d0f","family_id":"qf-ext19-openclosed-status","seed":202610191018,"semantic_template":"qf-ext19-openclosed-status/{\"n\":4,\"opens\":[[],[1],[1,2],[1,2,3],[0,1,2,3]],\"subset\":[0]}"},"fields":{"title":"Open, Closed, Both, or Neither in One Fixed Topology","stem":"Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [1], [1, 2], [1, 2, 3], [0, 1, 2, 3]]. Normalize subsets and repeated members as sets; order families by binary subset masks. Complements are relative to this same X. Closed does not mean not open. Candidate=[0]. Give (n, normalized candidate, open?, closed?, clopen?, clopen/open/closed/neither).","answer":{"kind":"expression","value":"(4, [0], No, Yes, No, closed)","explanation":"Check membership in the open family and openness of the relative complement independently. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [0], No, Yes, No, closed)."},"skills":["Open, Closed, Both, or Neither in One Fixed Topology"],"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-20261013-advance-20261019-r01-open-closed-set-0019","batch_id":"batch-20261013-advance-20261019-r01-open-closed-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ffedf649c65018b145e27be6c5df064c0d8ab8695503f0181cfd694c3061ab7b","family_id":"qf-ext19-openclosed-status","seed":202610191019,"semantic_template":"qf-ext19-openclosed-status/{\"n\":4,\"opens\":[[],[2],[1,2],[1,2,3],[0,1,2,3]],\"subset\":[1]}"},"fields":{"title":"Open, Closed, Both, or Neither in One Fixed Topology","stem":"Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [2], [1, 2], [1, 2, 3], [0, 1, 2, 3]]. Normalize subsets and repeated members as sets; order families by binary subset masks. Complements are relative to this same X. Closed does not mean not open. Candidate=[1]. Give (n, normalized candidate, open?, closed?, clopen?, clopen/open/closed/neither).","answer":{"kind":"expression","value":"(4, [1], No, No, No, neither)","explanation":"Check membership in the open family and openness of the relative complement independently. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [1], No, No, No, neither)."},"skills":["Open, Closed, Both, or Neither in One Fixed Topology"],"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-20261013-advance-20261019-r01-open-closed-set-0020","batch_id":"batch-20261013-advance-20261019-r01-open-closed-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7565cbd6e1723f722a6b743ac103f382ad654de5c0f1d03ddd8acf518e7f58eb","family_id":"qf-ext19-openclosed-status","seed":202610191020,"semantic_template":"qf-ext19-openclosed-status/{\"n\":4,\"opens\":[[],[3],[1,3],[1,2,3],[0,1,2,3]],\"subset\":[0,1]}"},"fields":{"title":"Open, Closed, Both, or Neither in One Fixed Topology","stem":"Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [3], [1, 3], [1, 2, 3], [0, 1, 2, 3]]. Normalize subsets and repeated members as sets; order families by binary subset masks. Complements are relative to this same X. Closed does not mean not open. Candidate=[0, 1]. Give (n, normalized candidate, open?, closed?, clopen?, clopen/open/closed/neither).","answer":{"kind":"expression","value":"(4, [0, 1], No, No, No, neither)","explanation":"Check membership in the open family and openness of the relative complement independently. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [0, 1], No, No, No, neither)."},"skills":["Open, Closed, Both, or Neither in One Fixed Topology"],"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-20261013-advance-20261019-r01-open-closed-set-0021","batch_id":"batch-20261013-advance-20261019-r01-open-closed-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b767fa41562bd8fd9bf76ad389442dd6d03674ef59056c6e6a7f6d7c19bddb71","family_id":"qf-ext19-openclosed-status","seed":202610191021,"semantic_template":"qf-ext19-openclosed-status/{\"n\":4,\"opens\":[[],[0],[0,3],[0,2,3],[0,1,2,3]],\"subset\":[2]}"},"fields":{"title":"Open, Closed, Both, or Neither in One Fixed Topology","stem":"Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [0], [0, 3], [0, 2, 3], [0, 1, 2, 3]]. Normalize subsets and repeated members as sets; order families by binary subset masks. Complements are relative to this same X. Closed does not mean not open. Candidate=[2]. Give (n, normalized candidate, open?, closed?, clopen?, clopen/open/closed/neither).","answer":{"kind":"expression","value":"(4, [2], No, No, No, neither)","explanation":"Check membership in the open family and openness of the relative complement independently. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [2], No, No, No, neither)."},"skills":["Open, Closed, Both, or Neither in One Fixed Topology"],"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-20261013-advance-20261019-r01-open-closed-set-0022","batch_id":"batch-20261013-advance-20261019-r01-open-closed-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e1e38e44d34a1c57e4dc5c9e1cba2ab185392770bdf7225d80f2711de20aae80","family_id":"qf-ext19-openclosed-status","seed":202610191022,"semantic_template":"qf-ext19-openclosed-status/{\"n\":4,\"opens\":[[],[1],[1,3],[1,2,3],[0,1,2,3]],\"subset\":[0,2]}"},"fields":{"title":"Open, Closed, Both, or Neither in One Fixed Topology","stem":"Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [1], [1, 3], [1, 2, 3], [0, 1, 2, 3]]. Normalize subsets and repeated members as sets; order families by binary subset masks. Complements are relative to this same X. Closed does not mean not open. Candidate=[0, 2]. Give (n, normalized candidate, open?, closed?, clopen?, clopen/open/closed/neither).","answer":{"kind":"expression","value":"(4, [0, 2], No, Yes, No, closed)","explanation":"Check membership in the open family and openness of the relative complement independently. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [0, 2], No, Yes, No, closed)."},"skills":["Open, Closed, Both, or Neither in One Fixed Topology"],"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-20261013-advance-20261019-r01-open-closed-set-0023","batch_id":"batch-20261013-advance-20261019-r01-open-closed-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"53ddd6243c539d93be66124932d8eac467922b9f66c1f8cb2805eaeabe308127","family_id":"qf-ext19-openclosed-status","seed":202610191023,"semantic_template":"qf-ext19-openclosed-status/{\"n\":4,\"opens\":[[],[2],[2,3],[1,2,3],[0,1,2,3]],\"subset\":[1,2]}"},"fields":{"title":"Open, Closed, Both, or Neither in One Fixed Topology","stem":"Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [2], [2, 3], [1, 2, 3], [0, 1, 2, 3]]. Normalize subsets and repeated members as sets; order families by binary subset masks. Complements are relative to this same X. Closed does not mean not open. Candidate=[1, 2]. Give (n, normalized candidate, open?, closed?, clopen?, clopen/open/closed/neither).","answer":{"kind":"expression","value":"(4, [1, 2], No, No, No, neither)","explanation":"Check membership in the open family and openness of the relative complement independently. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [1, 2], No, No, No, neither)."},"skills":["Open, Closed, Both, or Neither in One Fixed Topology"],"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-20261013-advance-20261019-r01-open-closed-set-0024","batch_id":"batch-20261013-advance-20261019-r01-open-closed-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ae37c82fc3c3b8dc29a8007edc3891ae0aa41ef0bc96dc5dc423df1838edc1cf","family_id":"qf-ext19-openclosed-status","seed":202610191024,"semantic_template":"qf-ext19-openclosed-status/{\"n\":4,\"opens\":[[],[3],[2,3],[1,2,3],[0,1,2,3]],\"subset\":[0,1,2]}"},"fields":{"title":"Open, Closed, Both, or Neither in One Fixed Topology","stem":"Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [3], [2, 3], [1, 2, 3], [0, 1, 2, 3]]. Normalize subsets and repeated members as sets; order families by binary subset masks. Complements are relative to this same X. Closed does not mean not open. Candidate=[0, 1, 2]. Give (n, normalized candidate, open?, closed?, clopen?, clopen/open/closed/neither).","answer":{"kind":"expression","value":"(4, [0, 1, 2], No, Yes, No, closed)","explanation":"Check membership in the open family and openness of the relative complement independently. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [0, 1, 2], No, Yes, No, closed)."},"skills":["Open, Closed, Both, or Neither in One Fixed Topology"],"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-20261013-advance-20261019-r01-open-closed-set-0025","batch_id":"batch-20261013-advance-20261019-r01-open-closed-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"02e27599dda337e2c8d3b2db85f2eeb995c9d5972b6ead5a1a57ecb590dfb807","family_id":"qf-ext19-openclosed-status","seed":202610191025,"semantic_template":"qf-ext19-openclosed-status/{\"n\":3,\"opens\":[[],[0],[0,1],[0,1,2]],\"subset\":[]}"},"fields":{"title":"Open, Closed, Both, or Neither in One Fixed Topology","stem":"Use the complete finite topology on X=Fin(3)={0,…,2} with opens=[[], [0], [0, 1], [0, 1, 2]]. Normalize subsets and repeated members as sets; order families by binary subset masks. Complements are relative to this same X. Closed does not mean not open. Candidate=[]. Give (n, normalized candidate, open?, closed?, clopen?, clopen/open/closed/neither).","answer":{"kind":"expression","value":"(3, [], Yes, Yes, Yes, clopen)","explanation":"Check membership in the open family and openness of the relative complement independently. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (3, [], Yes, Yes, Yes, clopen)."},"skills":["Open, Closed, Both, or Neither in One Fixed Topology"],"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-20261013-advance-20261019-r01-open-closed-set-0026","batch_id":"batch-20261013-advance-20261019-r01-open-closed-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"dde6944053d7d40b7c10e9f4a21fcc3de1939d837a532219c1e825decce937a2","family_id":"qf-ext19-openclosed-complement-duality","seed":202610191026,"semantic_template":"qf-ext19-openclosed-complement-duality/{\"n\":4,\"opens\":[[],[0],[0,1],[0,1,2],[0,1,2,3]],\"subset\":[]}"},"fields":{"title":"Relative Complement and Open/Closed Duality","stem":"Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3]]. Normalize subsets and repeated members as sets; order families by binary subset masks. Complements are relative to this same X. Closed does not mean not open. Candidate=[]. Give (n, candidate, relative complement, candidate open?, complement open?, candidate closed?, do the last two flags agree?).","answer":{"kind":"expression","value":"(4, [], [0, 1, 2, 3], Yes, Yes, Yes, Yes)","explanation":"Use the same underlying set throughout; closure is the complement-open condition, not negated openness. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [], [0, 1, 2, 3], Yes, Yes, Yes, Yes)."},"skills":["Relative Complement and Open/Closed Duality"],"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-20261013-advance-20261019-r01-open-closed-set-0027","batch_id":"batch-20261013-advance-20261019-r01-open-closed-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5fb7eb384974f6c84a03994008148d92e0943c60036361a5ddb9b15d054d7747","family_id":"qf-ext19-openclosed-complement-duality","seed":202610191027,"semantic_template":"qf-ext19-openclosed-complement-duality/{\"n\":4,\"opens\":[[],[1],[0,1],[0,1,2],[0,1,2,3]],\"subset\":[0]}"},"fields":{"title":"Relative Complement and Open/Closed Duality","stem":"Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [1], [0, 1], [0, 1, 2], [0, 1, 2, 3]]. Normalize subsets and repeated members as sets; order families by binary subset masks. Complements are relative to this same X. Closed does not mean not open. Candidate=[0]. Give (n, candidate, relative complement, candidate open?, complement open?, candidate closed?, do the last two flags agree?).","answer":{"kind":"expression","value":"(4, [0], [1, 2, 3], No, No, No, Yes)","explanation":"Use the same underlying set throughout; closure is the complement-open condition, not negated openness. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [0], [1, 2, 3], No, No, No, Yes)."},"skills":["Relative Complement and Open/Closed Duality"],"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-20261013-advance-20261019-r01-open-closed-set-0028","batch_id":"batch-20261013-advance-20261019-r01-open-closed-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b947a9daa6314215ef51ee52058cce348d03b28739fbe58496b5a14b29c45f42","family_id":"qf-ext19-openclosed-complement-duality","seed":202610191028,"semantic_template":"qf-ext19-openclosed-complement-duality/{\"n\":4,\"opens\":[[],[2],[0,2],[0,1,2],[0,1,2,3]],\"subset\":[1]}"},"fields":{"title":"Relative Complement and Open/Closed Duality","stem":"Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [2], [0, 2], [0, 1, 2], [0, 1, 2, 3]]. Normalize subsets and repeated members as sets; order families by binary subset masks. Complements are relative to this same X. Closed does not mean not open. Candidate=[1]. Give (n, candidate, relative complement, candidate open?, complement open?, candidate closed?, do the last two flags agree?).","answer":{"kind":"expression","value":"(4, [1], [0, 2, 3], No, No, No, Yes)","explanation":"Use the same underlying set throughout; closure is the complement-open condition, not negated openness. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [1], [0, 2, 3], No, No, No, Yes)."},"skills":["Relative Complement and Open/Closed Duality"],"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-20261013-advance-20261019-r01-open-closed-set-0029","batch_id":"batch-20261013-advance-20261019-r01-open-closed-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"25ce9bc91cb8d3528ebeb6315c8158ff253f6cd9dc678e1d01cad5ef9a41e10e","family_id":"qf-ext19-openclosed-complement-duality","seed":202610191029,"semantic_template":"qf-ext19-openclosed-complement-duality/{\"n\":4,\"opens\":[[],[3],[0,3],[0,1,3],[0,1,2,3]],\"subset\":[0,1]}"},"fields":{"title":"Relative Complement and Open/Closed Duality","stem":"Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [3], [0, 3], [0, 1, 3], [0, 1, 2, 3]]. Normalize subsets and repeated members as sets; order families by binary subset masks. Complements are relative to this same X. Closed does not mean not open. Candidate=[0, 1]. Give (n, candidate, relative complement, candidate open?, complement open?, candidate closed?, do the last two flags agree?).","answer":{"kind":"expression","value":"(4, [0, 1], [2, 3], No, No, No, Yes)","explanation":"Use the same underlying set throughout; closure is the complement-open condition, not negated openness. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [0, 1], [2, 3], No, No, No, Yes)."},"skills":["Relative Complement and Open/Closed Duality"],"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-20261013-advance-20261019-r01-open-closed-set-0030","batch_id":"batch-20261013-advance-20261019-r01-open-closed-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"01c4b414922de0224d9feab931725bbd13d3151ddc688ae0ceb072c25b4591b7","family_id":"qf-ext19-openclosed-complement-duality","seed":202610191030,"semantic_template":"qf-ext19-openclosed-complement-duality/{\"n\":4,\"opens\":[[],[0],[0,2],[0,1,2],[0,1,2,3]],\"subset\":[2]}"},"fields":{"title":"Relative Complement and Open/Closed Duality","stem":"Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [0], [0, 2], [0, 1, 2], [0, 1, 2, 3]]. Normalize subsets and repeated members as sets; order families by binary subset masks. Complements are relative to this same X. Closed does not mean not open. Candidate=[2]. Give (n, candidate, relative complement, candidate open?, complement open?, candidate closed?, do the last two flags agree?).","answer":{"kind":"expression","value":"(4, [2], [0, 1, 3], No, No, No, Yes)","explanation":"Use the same underlying set throughout; closure is the complement-open condition, not negated openness. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [2], [0, 1, 3], No, No, No, Yes)."},"skills":["Relative Complement and Open/Closed Duality"],"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-20261013-advance-20261019-r01-open-closed-set-0031","batch_id":"batch-20261013-advance-20261019-r01-open-closed-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"95c518a146ac5e2b7a49f6132087df5224665d0c1bfb3e03068b08ac254985cc","family_id":"qf-ext19-openclosed-complement-duality","seed":202610191031,"semantic_template":"qf-ext19-openclosed-complement-duality/{\"n\":4,\"opens\":[[],[1],[1,2],[0,1,2],[0,1,2,3]],\"subset\":[0,2]}"},"fields":{"title":"Relative Complement and Open/Closed Duality","stem":"Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [1], [1, 2], [0, 1, 2], [0, 1, 2, 3]]. Normalize subsets and repeated members as sets; order families by binary subset masks. Complements are relative to this same X. Closed does not mean not open. Candidate=[0, 2]. Give (n, candidate, relative complement, candidate open?, complement open?, candidate closed?, do the last two flags agree?).","answer":{"kind":"expression","value":"(4, [0, 2], [1, 3], No, No, No, Yes)","explanation":"Use the same underlying set throughout; closure is the complement-open condition, not negated openness. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [0, 2], [1, 3], No, No, No, Yes)."},"skills":["Relative Complement and Open/Closed Duality"],"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-20261013-advance-20261019-r01-open-closed-set-0032","batch_id":"batch-20261013-advance-20261019-r01-open-closed-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6cdf316aefd5e1d41b6d0074e31adda700fdcf093aea97af383bc28c8e8164be","family_id":"qf-ext19-openclosed-complement-duality","seed":202610191032,"semantic_template":"qf-ext19-openclosed-complement-duality/{\"n\":4,\"opens\":[[],[2],[1,2],[0,1,2],[0,1,2,3]],\"subset\":[1,2]}"},"fields":{"title":"Relative Complement and Open/Closed Duality","stem":"Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [2], [1, 2], [0, 1, 2], [0, 1, 2, 3]]. Normalize subsets and repeated members as sets; order families by binary subset masks. Complements are relative to this same X. Closed does not mean not open. Candidate=[1, 2]. Give (n, candidate, relative complement, candidate open?, complement open?, candidate closed?, do the last two flags agree?).","answer":{"kind":"expression","value":"(4, [1, 2], [0, 3], Yes, No, No, Yes)","explanation":"Use the same underlying set throughout; closure is the complement-open condition, not negated openness. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [1, 2], [0, 3], Yes, No, No, Yes)."},"skills":["Relative Complement and Open/Closed Duality"],"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-20261013-advance-20261019-r01-open-closed-set-0033","batch_id":"batch-20261013-advance-20261019-r01-open-closed-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"73185d1b3620aa41d8faa9b0b41921a7ef1c516e5a50516fd30b3b0469f3df68","family_id":"qf-ext19-openclosed-complement-duality","seed":202610191033,"semantic_template":"qf-ext19-openclosed-complement-duality/{\"n\":4,\"opens\":[[],[3],[1,3],[0,1,3],[0,1,2,3]],\"subset\":[0,1,2]}"},"fields":{"title":"Relative Complement and Open/Closed Duality","stem":"Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [3], [1, 3], [0, 1, 3], [0, 1, 2, 3]]. Normalize subsets and repeated members as sets; order families by binary subset masks. Complements are relative to this same X. Closed does not mean not open. Candidate=[0, 1, 2]. Give (n, candidate, relative complement, candidate open?, complement open?, candidate closed?, do the last two flags agree?).","answer":{"kind":"expression","value":"(4, [0, 1, 2], [3], No, Yes, Yes, Yes)","explanation":"Use the same underlying set throughout; closure is the complement-open condition, not negated openness. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [0, 1, 2], [3], No, Yes, Yes, Yes)."},"skills":["Relative Complement and Open/Closed Duality"],"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-20261013-advance-20261019-r01-open-closed-set-0034","batch_id":"batch-20261013-advance-20261019-r01-open-closed-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f15817f5168eea4a005846dd78f21da3ff36bb1c63820c0fe1c423272631788d","family_id":"qf-ext19-openclosed-complement-duality","seed":202610191034,"semantic_template":"qf-ext19-openclosed-complement-duality/{\"n\":4,\"opens\":[[],[0],[0,3],[0,1,3],[0,1,2,3]],\"subset\":[3]}"},"fields":{"title":"Relative Complement and Open/Closed Duality","stem":"Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [0], [0, 3], [0, 1, 3], [0, 1, 2, 3]]. Normalize subsets and repeated members as sets; order families by binary subset masks. Complements are relative to this same X. Closed does not mean not open. Candidate=[3]. Give (n, candidate, relative complement, candidate open?, complement open?, candidate closed?, do the last two flags agree?).","answer":{"kind":"expression","value":"(4, [3], [0, 1, 2], No, No, No, Yes)","explanation":"Use the same underlying set throughout; closure is the complement-open condition, not negated openness. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [3], [0, 1, 2], No, No, No, Yes)."},"skills":["Relative Complement and Open/Closed Duality"],"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-20261013-advance-20261019-r01-open-closed-set-0035","batch_id":"batch-20261013-advance-20261019-r01-open-closed-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a7819ee4e7c9c16d154e6c3cc9bfb7cc62af41b0d1f9ec21be025eab7e94125e","family_id":"qf-ext19-openclosed-complement-duality","seed":202610191035,"semantic_template":"qf-ext19-openclosed-complement-duality/{\"n\":4,\"opens\":[[],[1],[1,3],[0,1,3],[0,1,2,3]],\"subset\":[0,3]}"},"fields":{"title":"Relative Complement and Open/Closed Duality","stem":"Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [1], [1, 3], [0, 1, 3], [0, 1, 2, 3]]. Normalize subsets and repeated members as sets; order families by binary subset masks. Complements are relative to this same X. Closed does not mean not open. Candidate=[0, 3]. Give (n, candidate, relative complement, candidate open?, complement open?, candidate closed?, do the last two flags agree?).","answer":{"kind":"expression","value":"(4, [0, 3], [1, 2], No, No, No, Yes)","explanation":"Use the same underlying set throughout; closure is the complement-open condition, not negated openness. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [0, 3], [1, 2], No, No, No, Yes)."},"skills":["Relative Complement and Open/Closed Duality"],"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-20261013-advance-20261019-r01-open-closed-set-0036","batch_id":"batch-20261013-advance-20261019-r01-open-closed-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"629bb5661bbdba1a3c86859c5f9a188b8267144b9a6a35a9fcbfc9da7850c116","family_id":"qf-ext19-openclosed-complement-duality","seed":202610191036,"semantic_template":"qf-ext19-openclosed-complement-duality/{\"n\":4,\"opens\":[[],[2],[2,3],[0,2,3],[0,1,2,3]],\"subset\":[1,3]}"},"fields":{"title":"Relative Complement and Open/Closed Duality","stem":"Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [2], [2, 3], [0, 2, 3], [0, 1, 2, 3]]. Normalize subsets and repeated members as sets; order families by binary subset masks. Complements are relative to this same X. Closed does not mean not open. Candidate=[1, 3]. Give (n, candidate, relative complement, candidate open?, complement open?, candidate closed?, do the last two flags agree?).","answer":{"kind":"expression","value":"(4, [1, 3], [0, 2], No, No, No, Yes)","explanation":"Use the same underlying set throughout; closure is the complement-open condition, not negated openness. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [1, 3], [0, 2], No, No, No, Yes)."},"skills":["Relative Complement and Open/Closed Duality"],"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-20261013-advance-20261019-r01-open-closed-set-0037","batch_id":"batch-20261013-advance-20261019-r01-open-closed-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"99148d6fbfbd4166b541eebeba194b1b1dc89488c3226a07a59ebf0fe5b5db92","family_id":"qf-ext19-openclosed-complement-duality","seed":202610191037,"semantic_template":"qf-ext19-openclosed-complement-duality/{\"n\":4,\"opens\":[[],[3],[2,3],[0,2,3],[0,1,2,3]],\"subset\":[0,1,3]}"},"fields":{"title":"Relative Complement and Open/Closed Duality","stem":"Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [3], [2, 3], [0, 2, 3], [0, 1, 2, 3]]. Normalize subsets and repeated members as sets; order families by binary subset masks. Complements are relative to this same X. Closed does not mean not open. Candidate=[0, 1, 3]. Give (n, candidate, relative complement, candidate open?, complement open?, candidate closed?, do the last two flags agree?).","answer":{"kind":"expression","value":"(4, [0, 1, 3], [2], No, No, No, Yes)","explanation":"Use the same underlying set throughout; closure is the complement-open condition, not negated openness. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [0, 1, 3], [2], No, No, No, Yes)."},"skills":["Relative Complement and Open/Closed Duality"],"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-20261013-advance-20261019-r01-open-closed-set-0038","batch_id":"batch-20261013-advance-20261019-r01-open-closed-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d664eb677329230be1602d3ce7fc58c33cafe094bc16d0ecfb8d60807d9d3011","family_id":"qf-ext19-openclosed-complement-duality","seed":202610191038,"semantic_template":"qf-ext19-openclosed-complement-duality/{\"n\":4,\"opens\":[[],[0],[0,1],[0,1,3],[0,1,2,3]],\"subset\":[2,3]}"},"fields":{"title":"Relative Complement and Open/Closed Duality","stem":"Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [0], [0, 1], [0, 1, 3], [0, 1, 2, 3]]. Normalize subsets and repeated members as sets; order families by binary subset masks. Complements are relative to this same X. Closed does not mean not open. Candidate=[2, 3]. Give (n, candidate, relative complement, candidate open?, complement open?, candidate closed?, do the last two flags agree?).","answer":{"kind":"expression","value":"(4, [2, 3], [0, 1], No, Yes, Yes, Yes)","explanation":"Use the same underlying set throughout; closure is the complement-open condition, not negated openness. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [2, 3], [0, 1], No, Yes, Yes, Yes)."},"skills":["Relative Complement and Open/Closed Duality"],"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-20261013-advance-20261019-r01-open-closed-set-0039","batch_id":"batch-20261013-advance-20261019-r01-open-closed-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cd55a7531f7e349035d8b35775e62170ddbf3a26bdcc95cf9a08a180d0eb5e54","family_id":"qf-ext19-openclosed-complement-duality","seed":202610191039,"semantic_template":"qf-ext19-openclosed-complement-duality/{\"n\":4,\"opens\":[[],[1],[0,1],[0,1,3],[0,1,2,3]],\"subset\":[0,2,3]}"},"fields":{"title":"Relative Complement and Open/Closed Duality","stem":"Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [1], [0, 1], [0, 1, 3], [0, 1, 2, 3]]. Normalize subsets and repeated members as sets; order families by binary subset masks. Complements are relative to this same X. Closed does not mean not open. Candidate=[0, 2, 3]. Give (n, candidate, relative complement, candidate open?, complement open?, candidate closed?, do the last two flags agree?).","answer":{"kind":"expression","value":"(4, [0, 2, 3], [1], No, Yes, Yes, Yes)","explanation":"Use the same underlying set throughout; closure is the complement-open condition, not negated openness. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [0, 2, 3], [1], No, Yes, Yes, Yes)."},"skills":["Relative Complement and Open/Closed Duality"],"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-20261013-advance-20261019-r01-open-closed-set-0040","batch_id":"batch-20261013-advance-20261019-r01-open-closed-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"faf80c09826daea4e5059c658cbf53375e86c56c493b5390deb0ca3fa5290239","family_id":"qf-ext19-openclosed-complement-duality","seed":202610191040,"semantic_template":"qf-ext19-openclosed-complement-duality/{\"n\":4,\"opens\":[[],[2],[0,2],[0,2,3],[0,1,2,3]],\"subset\":[1,2,3]}"},"fields":{"title":"Relative Complement and Open/Closed Duality","stem":"Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [2], [0, 2], [0, 2, 3], [0, 1, 2, 3]]. Normalize subsets and repeated members as sets; order families by binary subset masks. Complements are relative to this same X. Closed does not mean not open. Candidate=[1, 2, 3]. Give (n, candidate, relative complement, candidate open?, complement open?, candidate closed?, do the last two flags agree?).","answer":{"kind":"expression","value":"(4, [1, 2, 3], [0], No, No, No, Yes)","explanation":"Use the same underlying set throughout; closure is the complement-open condition, not negated openness. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [1, 2, 3], [0], No, No, No, Yes)."},"skills":["Relative Complement and Open/Closed Duality"],"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-20261013-advance-20261019-r01-open-closed-set-0041","batch_id":"batch-20261013-advance-20261019-r01-open-closed-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"34eddb0962a22cf0ea76509485ad9a180aebdf4e591a9e4b06810c9ef316d3cb","family_id":"qf-ext19-openclosed-complement-duality","seed":202610191041,"semantic_template":"qf-ext19-openclosed-complement-duality/{\"n\":4,\"opens\":[[],[3],[0,3],[0,2,3],[0,1,2,3]],\"subset\":[0,1,2,3]}"},"fields":{"title":"Relative Complement and Open/Closed Duality","stem":"Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [3], [0, 3], [0, 2, 3], [0, 1, 2, 3]]. Normalize subsets and repeated members as sets; order families by binary subset masks. Complements are relative to this same X. Closed does not mean not open. Candidate=[0, 1, 2, 3]. Give (n, candidate, relative complement, candidate open?, complement open?, candidate closed?, do the last two flags agree?).","answer":{"kind":"expression","value":"(4, [0, 1, 2, 3], [], Yes, Yes, Yes, Yes)","explanation":"Use the same underlying set throughout; closure is the complement-open condition, not negated openness. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [0, 1, 2, 3], [], Yes, Yes, Yes, Yes)."},"skills":["Relative Complement and Open/Closed Duality"],"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-20261013-advance-20261019-r01-open-closed-set-0042","batch_id":"batch-20261013-advance-20261019-r01-open-closed-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f7b4a1bfa9bcea5df6a488db24bfe2ea947418a9ccac7b289019e884ea6b1e55","family_id":"qf-ext19-openclosed-complement-duality","seed":202610191042,"semantic_template":"qf-ext19-openclosed-complement-duality/{\"n\":4,\"opens\":[[],[0],[0,2],[0,2,3],[0,1,2,3]],\"subset\":[]}"},"fields":{"title":"Relative Complement and Open/Closed Duality","stem":"Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [0], [0, 2], [0, 2, 3], [0, 1, 2, 3]]. Normalize subsets and repeated members as sets; order families by binary subset masks. Complements are relative to this same X. Closed does not mean not open. Candidate=[]. Give (n, candidate, relative complement, candidate open?, complement open?, candidate closed?, do the last two flags agree?).","answer":{"kind":"expression","value":"(4, [], [0, 1, 2, 3], Yes, Yes, Yes, Yes)","explanation":"Use the same underlying set throughout; closure is the complement-open condition, not negated openness. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [], [0, 1, 2, 3], Yes, Yes, Yes, Yes)."},"skills":["Relative Complement and Open/Closed Duality"],"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-20261013-advance-20261019-r01-open-closed-set-0043","batch_id":"batch-20261013-advance-20261019-r01-open-closed-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"19a233e9d0f403d6a0cf42f204f031bd270668fb8ba664b21b80cf554fbabd01","family_id":"qf-ext19-openclosed-complement-duality","seed":202610191043,"semantic_template":"qf-ext19-openclosed-complement-duality/{\"n\":4,\"opens\":[[],[1],[1,2],[1,2,3],[0,1,2,3]],\"subset\":[0]}"},"fields":{"title":"Relative Complement and Open/Closed Duality","stem":"Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [1], [1, 2], [1, 2, 3], [0, 1, 2, 3]]. Normalize subsets and repeated members as sets; order families by binary subset masks. Complements are relative to this same X. Closed does not mean not open. Candidate=[0]. Give (n, candidate, relative complement, candidate open?, complement open?, candidate closed?, do the last two flags agree?).","answer":{"kind":"expression","value":"(4, [0], [1, 2, 3], No, Yes, Yes, Yes)","explanation":"Use the same underlying set throughout; closure is the complement-open condition, not negated openness. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [0], [1, 2, 3], No, Yes, Yes, Yes)."},"skills":["Relative Complement and Open/Closed Duality"],"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-20261013-advance-20261019-r01-open-closed-set-0044","batch_id":"batch-20261013-advance-20261019-r01-open-closed-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7e832e1e7dc917a95680b0ab825b0fb6e7c10f742d6b5d5a4c00760fac4ccdff","family_id":"qf-ext19-openclosed-complement-duality","seed":202610191044,"semantic_template":"qf-ext19-openclosed-complement-duality/{\"n\":4,\"opens\":[[],[2],[1,2],[1,2,3],[0,1,2,3]],\"subset\":[1]}"},"fields":{"title":"Relative Complement and Open/Closed Duality","stem":"Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [2], [1, 2], [1, 2, 3], [0, 1, 2, 3]]. Normalize subsets and repeated members as sets; order families by binary subset masks. Complements are relative to this same X. Closed does not mean not open. Candidate=[1]. Give (n, candidate, relative complement, candidate open?, complement open?, candidate closed?, do the last two flags agree?).","answer":{"kind":"expression","value":"(4, [1], [0, 2, 3], No, No, No, Yes)","explanation":"Use the same underlying set throughout; closure is the complement-open condition, not negated openness. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [1], [0, 2, 3], No, No, No, Yes)."},"skills":["Relative Complement and Open/Closed Duality"],"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-20261013-advance-20261019-r01-open-closed-set-0045","batch_id":"batch-20261013-advance-20261019-r01-open-closed-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"566989b5aeb63dec38cfb340a01e380444e77e13dd41f15dff1c1f6ce58f2e43","family_id":"qf-ext19-openclosed-complement-duality","seed":202610191045,"semantic_template":"qf-ext19-openclosed-complement-duality/{\"n\":4,\"opens\":[[],[3],[1,3],[1,2,3],[0,1,2,3]],\"subset\":[0,1]}"},"fields":{"title":"Relative Complement and Open/Closed Duality","stem":"Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [3], [1, 3], [1, 2, 3], [0, 1, 2, 3]]. Normalize subsets and repeated members as sets; order families by binary subset masks. Complements are relative to this same X. Closed does not mean not open. Candidate=[0, 1]. Give (n, candidate, relative complement, candidate open?, complement open?, candidate closed?, do the last two flags agree?).","answer":{"kind":"expression","value":"(4, [0, 1], [2, 3], No, No, No, Yes)","explanation":"Use the same underlying set throughout; closure is the complement-open condition, not negated openness. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [0, 1], [2, 3], No, No, No, Yes)."},"skills":["Relative Complement and Open/Closed Duality"],"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-20261013-advance-20261019-r01-open-closed-set-0046","batch_id":"batch-20261013-advance-20261019-r01-open-closed-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"eda22c666e936d93798baed3b7e4e0af7a67cd777b5e6760b49cc3654e3f8b9e","family_id":"qf-ext19-openclosed-complement-duality","seed":202610191046,"semantic_template":"qf-ext19-openclosed-complement-duality/{\"n\":4,\"opens\":[[],[0],[0,3],[0,2,3],[0,1,2,3]],\"subset\":[2]}"},"fields":{"title":"Relative Complement and Open/Closed Duality","stem":"Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [0], [0, 3], [0, 2, 3], [0, 1, 2, 3]]. Normalize subsets and repeated members as sets; order families by binary subset masks. Complements are relative to this same X. Closed does not mean not open. Candidate=[2]. Give (n, candidate, relative complement, candidate open?, complement open?, candidate closed?, do the last two flags agree?).","answer":{"kind":"expression","value":"(4, [2], [0, 1, 3], No, No, No, Yes)","explanation":"Use the same underlying set throughout; closure is the complement-open condition, not negated openness. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [2], [0, 1, 3], No, No, No, Yes)."},"skills":["Relative Complement and Open/Closed Duality"],"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-20261013-advance-20261019-r01-open-closed-set-0047","batch_id":"batch-20261013-advance-20261019-r01-open-closed-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"10e179aecb570f019a34fa73ad9851c961f77966bd697dbad96424a22dd21f81","family_id":"qf-ext19-openclosed-complement-duality","seed":202610191047,"semantic_template":"qf-ext19-openclosed-complement-duality/{\"n\":4,\"opens\":[[],[1],[1,3],[1,2,3],[0,1,2,3]],\"subset\":[0,2]}"},"fields":{"title":"Relative Complement and Open/Closed Duality","stem":"Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [1], [1, 3], [1, 2, 3], [0, 1, 2, 3]]. Normalize subsets and repeated members as sets; order families by binary subset masks. Complements are relative to this same X. Closed does not mean not open. Candidate=[0, 2]. Give (n, candidate, relative complement, candidate open?, complement open?, candidate closed?, do the last two flags agree?).","answer":{"kind":"expression","value":"(4, [0, 2], [1, 3], No, Yes, Yes, Yes)","explanation":"Use the same underlying set throughout; closure is the complement-open condition, not negated openness. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [0, 2], [1, 3], No, Yes, Yes, Yes)."},"skills":["Relative Complement and Open/Closed Duality"],"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-20261013-advance-20261019-r01-open-closed-set-0048","batch_id":"batch-20261013-advance-20261019-r01-open-closed-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c3ac9e0850040b88690a159dd6248711b6db15b37c5e76a22efca1d5124b4cee","family_id":"qf-ext19-openclosed-complement-duality","seed":202610191048,"semantic_template":"qf-ext19-openclosed-complement-duality/{\"n\":4,\"opens\":[[],[2],[2,3],[1,2,3],[0,1,2,3]],\"subset\":[1,2]}"},"fields":{"title":"Relative Complement and Open/Closed Duality","stem":"Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [2], [2, 3], [1, 2, 3], [0, 1, 2, 3]]. Normalize subsets and repeated members as sets; order families by binary subset masks. Complements are relative to this same X. Closed does not mean not open. Candidate=[1, 2]. Give (n, candidate, relative complement, candidate open?, complement open?, candidate closed?, do the last two flags agree?).","answer":{"kind":"expression","value":"(4, [1, 2], [0, 3], No, No, No, Yes)","explanation":"Use the same underlying set throughout; closure is the complement-open condition, not negated openness. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [1, 2], [0, 3], No, No, No, Yes)."},"skills":["Relative Complement and Open/Closed Duality"],"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-20261013-advance-20261019-r01-open-closed-set-0049","batch_id":"batch-20261013-advance-20261019-r01-open-closed-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0c5c240b94728a94449a135720edcb83121a00c5f8fb78470b58f42fc71e8b73","family_id":"qf-ext19-openclosed-complement-duality","seed":202610191049,"semantic_template":"qf-ext19-openclosed-complement-duality/{\"n\":4,\"opens\":[[],[3],[2,3],[1,2,3],[0,1,2,3]],\"subset\":[0,1,2]}"},"fields":{"title":"Relative Complement and Open/Closed Duality","stem":"Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [3], [2, 3], [1, 2, 3], [0, 1, 2, 3]]. Normalize subsets and repeated members as sets; order families by binary subset masks. Complements are relative to this same X. Closed does not mean not open. Candidate=[0, 1, 2]. Give (n, candidate, relative complement, candidate open?, complement open?, candidate closed?, do the last two flags agree?).","answer":{"kind":"expression","value":"(4, [0, 1, 2], [3], No, Yes, Yes, Yes)","explanation":"Use the same underlying set throughout; closure is the complement-open condition, not negated openness. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [0, 1, 2], [3], No, Yes, Yes, Yes)."},"skills":["Relative Complement and Open/Closed Duality"],"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-20261013-advance-20261019-r01-open-closed-set-0050","batch_id":"batch-20261013-advance-20261019-r01-open-closed-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"266934c16349346b5b5661316968ced0440e4800fbdf72aa74daf8b66b46543c","family_id":"qf-ext19-openclosed-complement-duality","seed":202610191050,"semantic_template":"qf-ext19-openclosed-complement-duality/{\"n\":3,\"opens\":[[],[0],[0,1],[0,1,2]],\"subset\":[]}"},"fields":{"title":"Relative Complement and Open/Closed Duality","stem":"Use the complete finite topology on X=Fin(3)={0,…,2} with opens=[[], [0], [0, 1], [0, 1, 2]]. Normalize subsets and repeated members as sets; order families by binary subset masks. Complements are relative to this same X. Closed does not mean not open. Candidate=[]. Give (n, candidate, relative complement, candidate open?, complement open?, candidate closed?, do the last two flags agree?).","answer":{"kind":"expression","value":"(3, [], [0, 1, 2], Yes, Yes, Yes, Yes)","explanation":"Use the same underlying set throughout; closure is the complement-open condition, not negated openness. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (3, [], [0, 1, 2], Yes, Yes, Yes, Yes)."},"skills":["Relative Complement and Open/Closed Duality"],"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-20261013-advance-20261019-r01-open-closed-set-0051","batch_id":"batch-20261013-advance-20261019-r01-open-closed-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2505eaa894700d8b1d837bc967071d278bca16baeb36286ae01a0b3bdce79d2d","family_id":"qf-ext19-openclosed-clopen-inventory","seed":202610191051,"semantic_template":"qf-ext19-openclosed-clopen-inventory/{\"n\":4,\"opens\":[[],[0],[0,1],[0,1,2],[0,1,2,3]]}"},"fields":{"title":"Complete Open, Closed, and Clopen Families","stem":"Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3]]. Normalize subsets and repeated members as sets; order families by binary subset masks. Complements are relative to this same X. Closed does not mean not open. Give (n, all opens, all closed sets, all clopen sets), in subset-mask order.","answer":{"kind":"expression","value":"(4, [[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3]], [[], [3], [2, 3], [1, 2, 3], [0, 1, 2, 3]], [[], [0, 1, 2, 3]])","explanation":"Complement every open set and compare both full families; the empty set and whole space are clopen. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3]], [[], [3], [2, 3], [1, 2, 3], [0, 1, 2, 3]], [[], [0, 1, 2, 3]])."},"skills":["Complete Open, Closed, and Clopen Families"],"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-20261013-advance-20261019-r01-open-closed-set-0052","batch_id":"batch-20261013-advance-20261019-r01-open-closed-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"58fb6c7a1fc14661d6dab9cafe1a3c212b1c5556fdf1300c3e4122db644c2cf5","family_id":"qf-ext19-openclosed-clopen-inventory","seed":202610191052,"semantic_template":"qf-ext19-openclosed-clopen-inventory/{\"n\":4,\"opens\":[[],[1],[0,1],[0,1,2],[0,1,2,3]]}"},"fields":{"title":"Complete Open, Closed, and Clopen Families","stem":"Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [1], [0, 1], [0, 1, 2], [0, 1, 2, 3]]. Normalize subsets and repeated members as sets; order families by binary subset masks. Complements are relative to this same X. Closed does not mean not open. Give (n, all opens, all closed sets, all clopen sets), in subset-mask order.","answer":{"kind":"expression","value":"(4, [[], [1], [0, 1], [0, 1, 2], [0, 1, 2, 3]], [[], [3], [2, 3], [0, 2, 3], [0, 1, 2, 3]], [[], [0, 1, 2, 3]])","explanation":"Complement every open set and compare both full families; the empty set and whole space are clopen. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [[], [1], [0, 1], [0, 1, 2], [0, 1, 2, 3]], [[], [3], [2, 3], [0, 2, 3], [0, 1, 2, 3]], [[], [0, 1, 2, 3]])."},"skills":["Complete Open, Closed, and Clopen Families"],"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-20261013-advance-20261019-r01-open-closed-set-0053","batch_id":"batch-20261013-advance-20261019-r01-open-closed-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"23ef0eb5a77bbc0c8d8af4753c11a4b693af3710e4a0e42bace24ddc38dde4e7","family_id":"qf-ext19-openclosed-clopen-inventory","seed":202610191053,"semantic_template":"qf-ext19-openclosed-clopen-inventory/{\"n\":4,\"opens\":[[],[2],[0,2],[0,1,2],[0,1,2,3]]}"},"fields":{"title":"Complete Open, Closed, and Clopen Families","stem":"Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [2], [0, 2], [0, 1, 2], [0, 1, 2, 3]]. Normalize subsets and repeated members as sets; order families by binary subset masks. Complements are relative to this same X. Closed does not mean not open. Give (n, all opens, all closed sets, all clopen sets), in subset-mask order.","answer":{"kind":"expression","value":"(4, [[], [2], [0, 2], [0, 1, 2], [0, 1, 2, 3]], [[], [3], [1, 3], [0, 1, 3], [0, 1, 2, 3]], [[], [0, 1, 2, 3]])","explanation":"Complement every open set and compare both full families; the empty set and whole space are clopen. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [[], [2], [0, 2], [0, 1, 2], [0, 1, 2, 3]], [[], [3], [1, 3], [0, 1, 3], [0, 1, 2, 3]], [[], [0, 1, 2, 3]])."},"skills":["Complete Open, Closed, and Clopen Families"],"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-20261013-advance-20261019-r01-open-closed-set-0054","batch_id":"batch-20261013-advance-20261019-r01-open-closed-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"942c6b46591d4af8bb136f60e9503682888ca6b650ef9486c1f478fe6d1709ab","family_id":"qf-ext19-openclosed-clopen-inventory","seed":202610191054,"semantic_template":"qf-ext19-openclosed-clopen-inventory/{\"n\":4,\"opens\":[[],[3],[0,3],[0,1,3],[0,1,2,3]]}"},"fields":{"title":"Complete Open, Closed, and Clopen Families","stem":"Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [3], [0, 3], [0, 1, 3], [0, 1, 2, 3]]. Normalize subsets and repeated members as sets; order families by binary subset masks. Complements are relative to this same X. Closed does not mean not open. Give (n, all opens, all closed sets, all clopen sets), in subset-mask order.","answer":{"kind":"expression","value":"(4, [[], [3], [0, 3], [0, 1, 3], [0, 1, 2, 3]], [[], [2], [1, 2], [0, 1, 2], [0, 1, 2, 3]], [[], [0, 1, 2, 3]])","explanation":"Complement every open set and compare both full families; the empty set and whole space are clopen. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [[], [3], [0, 3], [0, 1, 3], [0, 1, 2, 3]], [[], [2], [1, 2], [0, 1, 2], [0, 1, 2, 3]], [[], [0, 1, 2, 3]])."},"skills":["Complete Open, Closed, and Clopen Families"],"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-20261013-advance-20261019-r01-open-closed-set-0055","batch_id":"batch-20261013-advance-20261019-r01-open-closed-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9842f06ae52d576fd4f248169a276df295a4f4615f2b365b99e0fbb7e98ce1cd","family_id":"qf-ext19-openclosed-clopen-inventory","seed":202610191055,"semantic_template":"qf-ext19-openclosed-clopen-inventory/{\"n\":4,\"opens\":[[],[0],[0,2],[0,1,2],[0,1,2,3]]}"},"fields":{"title":"Complete Open, Closed, and Clopen Families","stem":"Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [0], [0, 2], [0, 1, 2], [0, 1, 2, 3]]. Normalize subsets and repeated members as sets; order families by binary subset masks. Complements are relative to this same X. Closed does not mean not open. Give (n, all opens, all closed sets, all clopen sets), in subset-mask order.","answer":{"kind":"expression","value":"(4, [[], [0], [0, 2], [0, 1, 2], [0, 1, 2, 3]], [[], [3], [1, 3], [1, 2, 3], [0, 1, 2, 3]], [[], [0, 1, 2, 3]])","explanation":"Complement every open set and compare both full families; the empty set and whole space are clopen. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [[], [0], [0, 2], [0, 1, 2], [0, 1, 2, 3]], [[], [3], [1, 3], [1, 2, 3], [0, 1, 2, 3]], [[], [0, 1, 2, 3]])."},"skills":["Complete Open, Closed, and Clopen Families"],"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-20261013-advance-20261019-r01-open-closed-set-0056","batch_id":"batch-20261013-advance-20261019-r01-open-closed-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a94912c87613984d71b50eae514c189c0ee26e57a8beca588f55198e36d7ad15","family_id":"qf-ext19-openclosed-clopen-inventory","seed":202610191056,"semantic_template":"qf-ext19-openclosed-clopen-inventory/{\"n\":4,\"opens\":[[],[1],[1,2],[0,1,2],[0,1,2,3]]}"},"fields":{"title":"Complete Open, Closed, and Clopen Families","stem":"Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [1], [1, 2], [0, 1, 2], [0, 1, 2, 3]]. Normalize subsets and repeated members as sets; order families by binary subset masks. Complements are relative to this same X. Closed does not mean not open. Give (n, all opens, all closed sets, all clopen sets), in subset-mask order.","answer":{"kind":"expression","value":"(4, [[], [1], [1, 2], [0, 1, 2], [0, 1, 2, 3]], [[], [3], [0, 3], [0, 2, 3], [0, 1, 2, 3]], [[], [0, 1, 2, 3]])","explanation":"Complement every open set and compare both full families; the empty set and whole space are clopen. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [[], [1], [1, 2], [0, 1, 2], [0, 1, 2, 3]], [[], [3], [0, 3], [0, 2, 3], [0, 1, 2, 3]], [[], [0, 1, 2, 3]])."},"skills":["Complete Open, Closed, and Clopen Families"],"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-20261013-advance-20261019-r01-open-closed-set-0057","batch_id":"batch-20261013-advance-20261019-r01-open-closed-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bd846dea591d148c735d71783a2021c5de56469a8fb5ddd9e95742ead8abb66a","family_id":"qf-ext19-openclosed-clopen-inventory","seed":202610191057,"semantic_template":"qf-ext19-openclosed-clopen-inventory/{\"n\":4,\"opens\":[[],[2],[1,2],[0,1,2],[0,1,2,3]]}"},"fields":{"title":"Complete Open, Closed, and Clopen Families","stem":"Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [2], [1, 2], [0, 1, 2], [0, 1, 2, 3]]. Normalize subsets and repeated members as sets; order families by binary subset masks. Complements are relative to this same X. Closed does not mean not open. Give (n, all opens, all closed sets, all clopen sets), in subset-mask order.","answer":{"kind":"expression","value":"(4, [[], [2], [1, 2], [0, 1, 2], [0, 1, 2, 3]], [[], [3], [0, 3], [0, 1, 3], [0, 1, 2, 3]], [[], [0, 1, 2, 3]])","explanation":"Complement every open set and compare both full families; the empty set and whole space are clopen. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [[], [2], [1, 2], [0, 1, 2], [0, 1, 2, 3]], [[], [3], [0, 3], [0, 1, 3], [0, 1, 2, 3]], [[], [0, 1, 2, 3]])."},"skills":["Complete Open, Closed, and Clopen Families"],"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-20261013-advance-20261019-r01-open-closed-set-0058","batch_id":"batch-20261013-advance-20261019-r01-open-closed-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"27da8d60094913af78874f4ba6805766e96eb4e8f64ba7a6583be505a0450ba1","family_id":"qf-ext19-openclosed-clopen-inventory","seed":202610191058,"semantic_template":"qf-ext19-openclosed-clopen-inventory/{\"n\":4,\"opens\":[[],[3],[1,3],[0,1,3],[0,1,2,3]]}"},"fields":{"title":"Complete Open, Closed, and Clopen Families","stem":"Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [3], [1, 3], [0, 1, 3], [0, 1, 2, 3]]. Normalize subsets and repeated members as sets; order families by binary subset masks. Complements are relative to this same X. Closed does not mean not open. Give (n, all opens, all closed sets, all clopen sets), in subset-mask order.","answer":{"kind":"expression","value":"(4, [[], [3], [1, 3], [0, 1, 3], [0, 1, 2, 3]], [[], [2], [0, 2], [0, 1, 2], [0, 1, 2, 3]], [[], [0, 1, 2, 3]])","explanation":"Complement every open set and compare both full families; the empty set and whole space are clopen. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [[], [3], [1, 3], [0, 1, 3], [0, 1, 2, 3]], [[], [2], [0, 2], [0, 1, 2], [0, 1, 2, 3]], [[], [0, 1, 2, 3]])."},"skills":["Complete Open, Closed, and Clopen Families"],"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-20261013-advance-20261019-r01-open-closed-set-0059","batch_id":"batch-20261013-advance-20261019-r01-open-closed-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d26844da95960226c7b4d0cc64682dcccef9e4b30213ccf9df1e74b73e33b6a0","family_id":"qf-ext19-openclosed-clopen-inventory","seed":202610191059,"semantic_template":"qf-ext19-openclosed-clopen-inventory/{\"n\":4,\"opens\":[[],[0],[0,3],[0,1,3],[0,1,2,3]]}"},"fields":{"title":"Complete Open, Closed, and Clopen Families","stem":"Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [0], [0, 3], [0, 1, 3], [0, 1, 2, 3]]. Normalize subsets and repeated members as sets; order families by binary subset masks. Complements are relative to this same X. Closed does not mean not open. Give (n, all opens, all closed sets, all clopen sets), in subset-mask order.","answer":{"kind":"expression","value":"(4, [[], [0], [0, 3], [0, 1, 3], [0, 1, 2, 3]], [[], [2], [1, 2], [1, 2, 3], [0, 1, 2, 3]], [[], [0, 1, 2, 3]])","explanation":"Complement every open set and compare both full families; the empty set and whole space are clopen. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [[], [0], [0, 3], [0, 1, 3], [0, 1, 2, 3]], [[], [2], [1, 2], [1, 2, 3], [0, 1, 2, 3]], [[], [0, 1, 2, 3]])."},"skills":["Complete Open, Closed, and Clopen Families"],"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-20261013-advance-20261019-r01-open-closed-set-0060","batch_id":"batch-20261013-advance-20261019-r01-open-closed-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"63f46898555705d8df64ce3f7a528ada9efa0333c404b9750cd51bcb81bca983","family_id":"qf-ext19-openclosed-clopen-inventory","seed":202610191060,"semantic_template":"qf-ext19-openclosed-clopen-inventory/{\"n\":4,\"opens\":[[],[1],[1,3],[0,1,3],[0,1,2,3]]}"},"fields":{"title":"Complete Open, Closed, and Clopen Families","stem":"Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [1], [1, 3], [0, 1, 3], [0, 1, 2, 3]]. Normalize subsets and repeated members as sets; order families by binary subset masks. Complements are relative to this same X. Closed does not mean not open. Give (n, all opens, all closed sets, all clopen sets), in subset-mask order.","answer":{"kind":"expression","value":"(4, [[], [1], [1, 3], [0, 1, 3], [0, 1, 2, 3]], [[], [2], [0, 2], [0, 2, 3], [0, 1, 2, 3]], [[], [0, 1, 2, 3]])","explanation":"Complement every open set and compare both full families; the empty set and whole space are clopen. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [[], [1], [1, 3], [0, 1, 3], [0, 1, 2, 3]], [[], [2], [0, 2], [0, 2, 3], [0, 1, 2, 3]], [[], [0, 1, 2, 3]])."},"skills":["Complete Open, Closed, and Clopen Families"],"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-20261013-advance-20261019-r01-open-closed-set-0061","batch_id":"batch-20261013-advance-20261019-r01-open-closed-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"793cd660ed8eaa29cd87fef18ab9f158411381c79aecba932186bc7f57267eef","family_id":"qf-ext19-openclosed-clopen-inventory","seed":202610191061,"semantic_template":"qf-ext19-openclosed-clopen-inventory/{\"n\":4,\"opens\":[[],[2],[2,3],[0,2,3],[0,1,2,3]]}"},"fields":{"title":"Complete Open, Closed, and Clopen Families","stem":"Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [2], [2, 3], [0, 2, 3], [0, 1, 2, 3]]. Normalize subsets and repeated members as sets; order families by binary subset masks. Complements are relative to this same X. Closed does not mean not open. Give (n, all opens, all closed sets, all clopen sets), in subset-mask order.","answer":{"kind":"expression","value":"(4, [[], [2], [2, 3], [0, 2, 3], [0, 1, 2, 3]], [[], [1], [0, 1], [0, 1, 3], [0, 1, 2, 3]], [[], [0, 1, 2, 3]])","explanation":"Complement every open set and compare both full families; the empty set and whole space are clopen. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [[], [2], [2, 3], [0, 2, 3], [0, 1, 2, 3]], [[], [1], [0, 1], [0, 1, 3], [0, 1, 2, 3]], [[], [0, 1, 2, 3]])."},"skills":["Complete Open, Closed, and Clopen Families"],"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-20261013-advance-20261019-r01-open-closed-set-0062","batch_id":"batch-20261013-advance-20261019-r01-open-closed-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"84bd802a36109c211b779a480b4335951c860f874f77ac89d4fceab359faa7e3","family_id":"qf-ext19-openclosed-clopen-inventory","seed":202610191062,"semantic_template":"qf-ext19-openclosed-clopen-inventory/{\"n\":4,\"opens\":[[],[3],[2,3],[0,2,3],[0,1,2,3]]}"},"fields":{"title":"Complete Open, Closed, and Clopen Families","stem":"Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [3], [2, 3], [0, 2, 3], [0, 1, 2, 3]]. Normalize subsets and repeated members as sets; order families by binary subset masks. Complements are relative to this same X. Closed does not mean not open. Give (n, all opens, all closed sets, all clopen sets), in subset-mask order.","answer":{"kind":"expression","value":"(4, [[], [3], [2, 3], [0, 2, 3], [0, 1, 2, 3]], [[], [1], [0, 1], [0, 1, 2], [0, 1, 2, 3]], [[], [0, 1, 2, 3]])","explanation":"Complement every open set and compare both full families; the empty set and whole space are clopen. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [[], [3], [2, 3], [0, 2, 3], [0, 1, 2, 3]], [[], [1], [0, 1], [0, 1, 2], [0, 1, 2, 3]], [[], [0, 1, 2, 3]])."},"skills":["Complete Open, Closed, and Clopen Families"],"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-20261013-advance-20261019-r01-open-closed-set-0063","batch_id":"batch-20261013-advance-20261019-r01-open-closed-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"601d16d84005924f858e9ace3398378bb15426013c59d815c00ae1d252f7c0de","family_id":"qf-ext19-openclosed-clopen-inventory","seed":202610191063,"semantic_template":"qf-ext19-openclosed-clopen-inventory/{\"n\":4,\"opens\":[[],[0],[0,1],[0,1,3],[0,1,2,3]]}"},"fields":{"title":"Complete Open, Closed, and Clopen Families","stem":"Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [0], [0, 1], [0, 1, 3], [0, 1, 2, 3]]. Normalize subsets and repeated members as sets; order families by binary subset masks. Complements are relative to this same X. Closed does not mean not open. Give (n, all opens, all closed sets, all clopen sets), in subset-mask order.","answer":{"kind":"expression","value":"(4, [[], [0], [0, 1], [0, 1, 3], [0, 1, 2, 3]], [[], [2], [2, 3], [1, 2, 3], [0, 1, 2, 3]], [[], [0, 1, 2, 3]])","explanation":"Complement every open set and compare both full families; the empty set and whole space are clopen. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [[], [0], [0, 1], [0, 1, 3], [0, 1, 2, 3]], [[], [2], [2, 3], [1, 2, 3], [0, 1, 2, 3]], [[], [0, 1, 2, 3]])."},"skills":["Complete Open, Closed, and Clopen Families"],"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-20261013-advance-20261019-r01-open-closed-set-0064","batch_id":"batch-20261013-advance-20261019-r01-open-closed-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6e1b1e60068dfb037fed02caa9f2e4b2bc47b6670471bca9e24a89141751430c","family_id":"qf-ext19-openclosed-clopen-inventory","seed":202610191064,"semantic_template":"qf-ext19-openclosed-clopen-inventory/{\"n\":4,\"opens\":[[],[1],[0,1],[0,1,3],[0,1,2,3]]}"},"fields":{"title":"Complete Open, Closed, and Clopen Families","stem":"Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [1], [0, 1], [0, 1, 3], [0, 1, 2, 3]]. Normalize subsets and repeated members as sets; order families by binary subset masks. Complements are relative to this same X. Closed does not mean not open. Give (n, all opens, all closed sets, all clopen sets), in subset-mask order.","answer":{"kind":"expression","value":"(4, [[], [1], [0, 1], [0, 1, 3], [0, 1, 2, 3]], [[], [2], [2, 3], [0, 2, 3], [0, 1, 2, 3]], [[], [0, 1, 2, 3]])","explanation":"Complement every open set and compare both full families; the empty set and whole space are clopen. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [[], [1], [0, 1], [0, 1, 3], [0, 1, 2, 3]], [[], [2], [2, 3], [0, 2, 3], [0, 1, 2, 3]], [[], [0, 1, 2, 3]])."},"skills":["Complete Open, Closed, and Clopen Families"],"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-20261013-advance-20261019-r01-open-closed-set-0065","batch_id":"batch-20261013-advance-20261019-r01-open-closed-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"324030fd3ff025b0bfc94c26fd04de9aa23019e438bbc0f94979bfab78e0b887","family_id":"qf-ext19-openclosed-clopen-inventory","seed":202610191065,"semantic_template":"qf-ext19-openclosed-clopen-inventory/{\"n\":4,\"opens\":[[],[2],[0,2],[0,2,3],[0,1,2,3]]}"},"fields":{"title":"Complete Open, Closed, and Clopen Families","stem":"Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [2], [0, 2], [0, 2, 3], [0, 1, 2, 3]]. Normalize subsets and repeated members as sets; order families by binary subset masks. Complements are relative to this same X. Closed does not mean not open. Give (n, all opens, all closed sets, all clopen sets), in subset-mask order.","answer":{"kind":"expression","value":"(4, [[], [2], [0, 2], [0, 2, 3], [0, 1, 2, 3]], [[], [1], [1, 3], [0, 1, 3], [0, 1, 2, 3]], [[], [0, 1, 2, 3]])","explanation":"Complement every open set and compare both full families; the empty set and whole space are clopen. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [[], [2], [0, 2], [0, 2, 3], [0, 1, 2, 3]], [[], [1], [1, 3], [0, 1, 3], [0, 1, 2, 3]], [[], [0, 1, 2, 3]])."},"skills":["Complete Open, Closed, and Clopen Families"],"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-20261013-advance-20261019-r01-open-closed-set-0066","batch_id":"batch-20261013-advance-20261019-r01-open-closed-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"97dc5ec3903a630a0ba217121656c3753313135bb20164650895b738e33a293a","family_id":"qf-ext19-openclosed-clopen-inventory","seed":202610191066,"semantic_template":"qf-ext19-openclosed-clopen-inventory/{\"n\":4,\"opens\":[[],[3],[0,3],[0,2,3],[0,1,2,3]]}"},"fields":{"title":"Complete Open, Closed, and Clopen Families","stem":"Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [3], [0, 3], [0, 2, 3], [0, 1, 2, 3]]. Normalize subsets and repeated members as sets; order families by binary subset masks. Complements are relative to this same X. Closed does not mean not open. Give (n, all opens, all closed sets, all clopen sets), in subset-mask order.","answer":{"kind":"expression","value":"(4, [[], [3], [0, 3], [0, 2, 3], [0, 1, 2, 3]], [[], [1], [1, 2], [0, 1, 2], [0, 1, 2, 3]], [[], [0, 1, 2, 3]])","explanation":"Complement every open set and compare both full families; the empty set and whole space are clopen. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [[], [3], [0, 3], [0, 2, 3], [0, 1, 2, 3]], [[], [1], [1, 2], [0, 1, 2], [0, 1, 2, 3]], [[], [0, 1, 2, 3]])."},"skills":["Complete Open, Closed, and Clopen Families"],"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-20261013-advance-20261019-r01-open-closed-set-0067","batch_id":"batch-20261013-advance-20261019-r01-open-closed-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"826e1b8190a90736cdd89711ea1e343ebee96148674965bfbc6ae056dabf68e6","family_id":"qf-ext19-openclosed-clopen-inventory","seed":202610191067,"semantic_template":"qf-ext19-openclosed-clopen-inventory/{\"n\":4,\"opens\":[[],[0],[0,2],[0,2,3],[0,1,2,3]]}"},"fields":{"title":"Complete Open, Closed, and Clopen Families","stem":"Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [0], [0, 2], [0, 2, 3], [0, 1, 2, 3]]. Normalize subsets and repeated members as sets; order families by binary subset masks. Complements are relative to this same X. Closed does not mean not open. Give (n, all opens, all closed sets, all clopen sets), in subset-mask order.","answer":{"kind":"expression","value":"(4, [[], [0], [0, 2], [0, 2, 3], [0, 1, 2, 3]], [[], [1], [1, 3], [1, 2, 3], [0, 1, 2, 3]], [[], [0, 1, 2, 3]])","explanation":"Complement every open set and compare both full families; the empty set and whole space are clopen. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [[], [0], [0, 2], [0, 2, 3], [0, 1, 2, 3]], [[], [1], [1, 3], [1, 2, 3], [0, 1, 2, 3]], [[], [0, 1, 2, 3]])."},"skills":["Complete Open, Closed, and Clopen Families"],"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-20261013-advance-20261019-r01-open-closed-set-0068","batch_id":"batch-20261013-advance-20261019-r01-open-closed-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2999d957f0a27de003c1f5f84b8d0f0aace355ebe4885f58c4becacd0abb2b5b","family_id":"qf-ext19-openclosed-clopen-inventory","seed":202610191068,"semantic_template":"qf-ext19-openclosed-clopen-inventory/{\"n\":4,\"opens\":[[],[1],[1,2],[1,2,3],[0,1,2,3]]}"},"fields":{"title":"Complete Open, Closed, and Clopen Families","stem":"Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [1], [1, 2], [1, 2, 3], [0, 1, 2, 3]]. Normalize subsets and repeated members as sets; order families by binary subset masks. Complements are relative to this same X. Closed does not mean not open. Give (n, all opens, all closed sets, all clopen sets), in subset-mask order.","answer":{"kind":"expression","value":"(4, [[], [1], [1, 2], [1, 2, 3], [0, 1, 2, 3]], [[], [0], [0, 3], [0, 2, 3], [0, 1, 2, 3]], [[], [0, 1, 2, 3]])","explanation":"Complement every open set and compare both full families; the empty set and whole space are clopen. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [[], [1], [1, 2], [1, 2, 3], [0, 1, 2, 3]], [[], [0], [0, 3], [0, 2, 3], [0, 1, 2, 3]], [[], [0, 1, 2, 3]])."},"skills":["Complete Open, Closed, and Clopen Families"],"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-20261013-advance-20261019-r01-open-closed-set-0069","batch_id":"batch-20261013-advance-20261019-r01-open-closed-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bfd7f8b80d6d746518b3246ce89bdc87a889946808f72883b693ac01a7d51d9e","family_id":"qf-ext19-openclosed-clopen-inventory","seed":202610191069,"semantic_template":"qf-ext19-openclosed-clopen-inventory/{\"n\":4,\"opens\":[[],[2],[1,2],[1,2,3],[0,1,2,3]]}"},"fields":{"title":"Complete Open, Closed, and Clopen Families","stem":"Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [2], [1, 2], [1, 2, 3], [0, 1, 2, 3]]. Normalize subsets and repeated members as sets; order families by binary subset masks. Complements are relative to this same X. Closed does not mean not open. Give (n, all opens, all closed sets, all clopen sets), in subset-mask order.","answer":{"kind":"expression","value":"(4, [[], [2], [1, 2], [1, 2, 3], [0, 1, 2, 3]], [[], [0], [0, 3], [0, 1, 3], [0, 1, 2, 3]], [[], [0, 1, 2, 3]])","explanation":"Complement every open set and compare both full families; the empty set and whole space are clopen. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [[], [2], [1, 2], [1, 2, 3], [0, 1, 2, 3]], [[], [0], [0, 3], [0, 1, 3], [0, 1, 2, 3]], [[], [0, 1, 2, 3]])."},"skills":["Complete Open, Closed, and Clopen Families"],"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-20261013-advance-20261019-r01-open-closed-set-0070","batch_id":"batch-20261013-advance-20261019-r01-open-closed-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"431f0167109913ad7d4841b8fc68cffd3183c301a332e7e3a2b01602d1e5c62a","family_id":"qf-ext19-openclosed-clopen-inventory","seed":202610191070,"semantic_template":"qf-ext19-openclosed-clopen-inventory/{\"n\":4,\"opens\":[[],[3],[1,3],[1,2,3],[0,1,2,3]]}"},"fields":{"title":"Complete Open, Closed, and Clopen Families","stem":"Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [3], [1, 3], [1, 2, 3], [0, 1, 2, 3]]. Normalize subsets and repeated members as sets; order families by binary subset masks. Complements are relative to this same X. Closed does not mean not open. Give (n, all opens, all closed sets, all clopen sets), in subset-mask order.","answer":{"kind":"expression","value":"(4, [[], [3], [1, 3], [1, 2, 3], [0, 1, 2, 3]], [[], [0], [0, 2], [0, 1, 2], [0, 1, 2, 3]], [[], [0, 1, 2, 3]])","explanation":"Complement every open set and compare both full families; the empty set and whole space are clopen. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [[], [3], [1, 3], [1, 2, 3], [0, 1, 2, 3]], [[], [0], [0, 2], [0, 1, 2], [0, 1, 2, 3]], [[], [0, 1, 2, 3]])."},"skills":["Complete Open, Closed, and Clopen Families"],"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-20261013-advance-20261019-r01-open-closed-set-0071","batch_id":"batch-20261013-advance-20261019-r01-open-closed-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"87be80b0c1603140878da396c99183305f282ef7b4d12c53c4b322d35a30a99f","family_id":"qf-ext19-openclosed-clopen-inventory","seed":202610191071,"semantic_template":"qf-ext19-openclosed-clopen-inventory/{\"n\":4,\"opens\":[[],[0],[0,3],[0,2,3],[0,1,2,3]]}"},"fields":{"title":"Complete Open, Closed, and Clopen Families","stem":"Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [0], [0, 3], [0, 2, 3], [0, 1, 2, 3]]. Normalize subsets and repeated members as sets; order families by binary subset masks. Complements are relative to this same X. Closed does not mean not open. Give (n, all opens, all closed sets, all clopen sets), in subset-mask order.","answer":{"kind":"expression","value":"(4, [[], [0], [0, 3], [0, 2, 3], [0, 1, 2, 3]], [[], [1], [1, 2], [1, 2, 3], [0, 1, 2, 3]], [[], [0, 1, 2, 3]])","explanation":"Complement every open set and compare both full families; the empty set and whole space are clopen. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [[], [0], [0, 3], [0, 2, 3], [0, 1, 2, 3]], [[], [1], [1, 2], [1, 2, 3], [0, 1, 2, 3]], [[], [0, 1, 2, 3]])."},"skills":["Complete Open, Closed, and Clopen Families"],"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-20261013-advance-20261019-r01-open-closed-set-0072","batch_id":"batch-20261013-advance-20261019-r01-open-closed-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b518b29a2e4c71f6dac13e2f8cd7bba72d3dba4bdd14e23cacf5c729b9d4d0d6","family_id":"qf-ext19-openclosed-clopen-inventory","seed":202610191072,"semantic_template":"qf-ext19-openclosed-clopen-inventory/{\"n\":4,\"opens\":[[],[1],[1,3],[1,2,3],[0,1,2,3]]}"},"fields":{"title":"Complete Open, Closed, and Clopen Families","stem":"Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [1], [1, 3], [1, 2, 3], [0, 1, 2, 3]]. Normalize subsets and repeated members as sets; order families by binary subset masks. Complements are relative to this same X. Closed does not mean not open. Give (n, all opens, all closed sets, all clopen sets), in subset-mask order.","answer":{"kind":"expression","value":"(4, [[], [1], [1, 3], [1, 2, 3], [0, 1, 2, 3]], [[], [0], [0, 2], [0, 2, 3], [0, 1, 2, 3]], [[], [0, 1, 2, 3]])","explanation":"Complement every open set and compare both full families; the empty set and whole space are clopen. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [[], [1], [1, 3], [1, 2, 3], [0, 1, 2, 3]], [[], [0], [0, 2], [0, 2, 3], [0, 1, 2, 3]], [[], [0, 1, 2, 3]])."},"skills":["Complete Open, Closed, and Clopen Families"],"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-20261013-advance-20261019-r01-open-closed-set-0073","batch_id":"batch-20261013-advance-20261019-r01-open-closed-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"222cc0f05bd9b20ebd0eabbfc33f945f9b1a9e85e0464fe27a3654c627417692","family_id":"qf-ext19-openclosed-clopen-inventory","seed":202610191073,"semantic_template":"qf-ext19-openclosed-clopen-inventory/{\"n\":4,\"opens\":[[],[2],[2,3],[1,2,3],[0,1,2,3]]}"},"fields":{"title":"Complete Open, Closed, and Clopen Families","stem":"Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [2], [2, 3], [1, 2, 3], [0, 1, 2, 3]]. Normalize subsets and repeated members as sets; order families by binary subset masks. Complements are relative to this same X. Closed does not mean not open. Give (n, all opens, all closed sets, all clopen sets), in subset-mask order.","answer":{"kind":"expression","value":"(4, [[], [2], [2, 3], [1, 2, 3], [0, 1, 2, 3]], [[], [0], [0, 1], [0, 1, 3], [0, 1, 2, 3]], [[], [0, 1, 2, 3]])","explanation":"Complement every open set and compare both full families; the empty set and whole space are clopen. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [[], [2], [2, 3], [1, 2, 3], [0, 1, 2, 3]], [[], [0], [0, 1], [0, 1, 3], [0, 1, 2, 3]], [[], [0, 1, 2, 3]])."},"skills":["Complete Open, Closed, and Clopen Families"],"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-20261013-advance-20261019-r01-open-closed-set-0074","batch_id":"batch-20261013-advance-20261019-r01-open-closed-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f487fd5968ed996823fa8c346e9fd1d5f26003871853b22f2c53256b516305f4","family_id":"qf-ext19-openclosed-clopen-inventory","seed":202610191074,"semantic_template":"qf-ext19-openclosed-clopen-inventory/{\"n\":4,\"opens\":[[],[3],[2,3],[1,2,3],[0,1,2,3]]}"},"fields":{"title":"Complete Open, Closed, and Clopen Families","stem":"Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [3], [2, 3], [1, 2, 3], [0, 1, 2, 3]]. Normalize subsets and repeated members as sets; order families by binary subset masks. Complements are relative to this same X. Closed does not mean not open. Give (n, all opens, all closed sets, all clopen sets), in subset-mask order.","answer":{"kind":"expression","value":"(4, [[], [3], [2, 3], [1, 2, 3], [0, 1, 2, 3]], [[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3]], [[], [0, 1, 2, 3]])","explanation":"Complement every open set and compare both full families; the empty set and whole space are clopen. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, [[], [3], [2, 3], [1, 2, 3], [0, 1, 2, 3]], [[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3]], [[], [0, 1, 2, 3]])."},"skills":["Complete Open, Closed, and Clopen Families"],"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-20261013-advance-20261019-r01-open-closed-set-0075","batch_id":"batch-20261013-advance-20261019-r01-open-closed-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6a9643a5108688184e7edda7f3d5db7c8973ade0733ff4ae543d221c0a28b54d","family_id":"qf-ext19-openclosed-clopen-inventory","seed":202610191075,"semantic_template":"qf-ext19-openclosed-clopen-inventory/{\"n\":3,\"opens\":[[],[0],[0,1],[0,1,2]]}"},"fields":{"title":"Complete Open, Closed, and Clopen Families","stem":"Use the complete finite topology on X=Fin(3)={0,…,2} with opens=[[], [0], [0, 1], [0, 1, 2]]. Normalize subsets and repeated members as sets; order families by binary subset masks. Complements are relative to this same X. Closed does not mean not open. Give (n, all opens, all closed sets, all clopen sets), in subset-mask order.","answer":{"kind":"expression","value":"(3, [[], [0], [0, 1], [0, 1, 2]], [[], [2], [1, 2], [0, 1, 2]], [[], [0, 1, 2]])","explanation":"Complement every open set and compare both full families; the empty set and whole space are clopen. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (3, [[], [0], [0, 1], [0, 1, 2]], [[], [2], [1, 2], [0, 1, 2]], [[], [0, 1, 2]])."},"skills":["Complete Open, Closed, and Clopen Families"],"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-20261013-advance-20261019-r01-open-closed-set-0076","batch_id":"batch-20261013-advance-20261019-r01-open-closed-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fc88ebe1fb2a81bdb76ebaa4fee15155b692d99fdb80061334fc2a123696aab0","family_id":"qf-ext19-openclosed-family-closure","seed":202610191076,"semantic_template":"qf-ext19-openclosed-family-closure/{\"mode\":\"open_union\",\"n\":4,\"opens\":[[],[0],[0,1],[0,1,2],[0,1,2,3]],\"subsets\":[[],[0,1],[0,1,2,3]]}"},"fields":{"title":"Empty Families and Qualified Closure Premises","stem":"Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [0], [0, 1], [0, 1, 2], [0, 1, 2, 3]]. Normalize subsets and repeated members as sets; order families by binary subset masks. Complements are relative to this same X. Closed does not mean not open. mode=open_union, selected=[[], [0, 1], [0, 1, 2, 3]]. Modes are open_union, open_finite_intersection, closed_intersection, closed_finite_union. Give (n, mode, normalized selected family, all selected sets have the required open/closed property?, union/intersection result, result has that property?). Empty union is ∅ and empty intersection is X. Compute the result even if a premise fails.","answer":{"kind":"expression","value":"(4, open_union, [[], [0, 1], [0, 1, 2, 3]], Yes, [0, 1, 2, 3], Yes)","explanation":"Keep premise qualification separate from the resulting set’s property; a false premise does not prevent an accidentally qualified result. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, open_union, [[], [0, 1], [0, 1, 2, 3]], Yes, [0, 1, 2, 3], Yes)."},"skills":["Empty Families and Qualified Closure Premises"],"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-20261013-advance-20261019-r01-open-closed-set-0077","batch_id":"batch-20261013-advance-20261019-r01-open-closed-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"74a833cb7d314acd29e8e1760c9f74c2ad0d940a2220f46d7ac4ddc069307529","family_id":"qf-ext19-openclosed-family-closure","seed":202610191077,"semantic_template":"qf-ext19-openclosed-family-closure/{\"mode\":\"open_finite_intersection\",\"n\":4,\"opens\":[[],[1],[0,1],[0,1,2],[0,1,2,3]],\"subsets\":[[1],[0,1,2]]}"},"fields":{"title":"Empty Families and Qualified Closure Premises","stem":"Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [1], [0, 1], [0, 1, 2], [0, 1, 2, 3]]. Normalize subsets and repeated members as sets; order families by binary subset masks. Complements are relative to this same X. Closed does not mean not open. mode=open_finite_intersection, selected=[[1], [0, 1, 2]]. Modes are open_union, open_finite_intersection, closed_intersection, closed_finite_union. Give (n, mode, normalized selected family, all selected sets have the required open/closed property?, union/intersection result, result has that property?). Empty union is ∅ and empty intersection is X. Compute the result even if a premise fails.","answer":{"kind":"expression","value":"(4, open_finite_intersection, [[1], [0, 1, 2]], Yes, [1], Yes)","explanation":"Keep premise qualification separate from the resulting set’s property; a false premise does not prevent an accidentally qualified result. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, open_finite_intersection, [[1], [0, 1, 2]], Yes, [1], Yes)."},"skills":["Empty Families and Qualified Closure Premises"],"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-20261013-advance-20261019-r01-open-closed-set-0078","batch_id":"batch-20261013-advance-20261019-r01-open-closed-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"11352841e7a954ad2f35189ba6c2973d5f4ff49c05ec12803ff331658c9a3115","family_id":"qf-ext19-openclosed-family-closure","seed":202610191078,"semantic_template":"qf-ext19-openclosed-family-closure/{\"mode\":\"closed_intersection\",\"n\":4,\"opens\":[[],[2],[0,2],[0,1,2],[0,1,2,3]],\"subsets\":[[],[1,3],[0,1,2,3]]}"},"fields":{"title":"Empty Families and Qualified Closure Premises","stem":"Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [2], [0, 2], [0, 1, 2], [0, 1, 2, 3]]. Normalize subsets and repeated members as sets; order families by binary subset masks. Complements are relative to this same X. Closed does not mean not open. mode=closed_intersection, selected=[[], [1, 3], [0, 1, 2, 3]]. Modes are open_union, open_finite_intersection, closed_intersection, closed_finite_union. Give (n, mode, normalized selected family, all selected sets have the required open/closed property?, union/intersection result, result has that property?). Empty union is ∅ and empty intersection is X. Compute the result even if a premise fails.","answer":{"kind":"expression","value":"(4, closed_intersection, [[], [1, 3], [0, 1, 2, 3]], Yes, [], Yes)","explanation":"Keep premise qualification separate from the resulting set’s property; a false premise does not prevent an accidentally qualified result. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, closed_intersection, [[], [1, 3], [0, 1, 2, 3]], Yes, [], Yes)."},"skills":["Empty Families and Qualified Closure Premises"],"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-20261013-advance-20261019-r01-open-closed-set-0079","batch_id":"batch-20261013-advance-20261019-r01-open-closed-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9da15366695cadcdcae98f4407c81a28b51a89b455ed6dc14694cbdaa81c4c55","family_id":"qf-ext19-openclosed-family-closure","seed":202610191079,"semantic_template":"qf-ext19-openclosed-family-closure/{\"mode\":\"closed_finite_union\",\"n\":4,\"opens\":[[],[3],[0,3],[0,1,3],[0,1,2,3]],\"subsets\":[[2],[0,1,2]]}"},"fields":{"title":"Empty Families and Qualified Closure Premises","stem":"Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [3], [0, 3], [0, 1, 3], [0, 1, 2, 3]]. Normalize subsets and repeated members as sets; order families by binary subset masks. Complements are relative to this same X. Closed does not mean not open. mode=closed_finite_union, selected=[[2], [0, 1, 2]]. Modes are open_union, open_finite_intersection, closed_intersection, closed_finite_union. Give (n, mode, normalized selected family, all selected sets have the required open/closed property?, union/intersection result, result has that property?). Empty union is ∅ and empty intersection is X. Compute the result even if a premise fails.","answer":{"kind":"expression","value":"(4, closed_finite_union, [[2], [0, 1, 2]], Yes, [0, 1, 2], Yes)","explanation":"Keep premise qualification separate from the resulting set’s property; a false premise does not prevent an accidentally qualified result. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, closed_finite_union, [[2], [0, 1, 2]], Yes, [0, 1, 2], Yes)."},"skills":["Empty Families and Qualified Closure Premises"],"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-20261013-advance-20261019-r01-open-closed-set-0080","batch_id":"batch-20261013-advance-20261019-r01-open-closed-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"eab403c3304ee12b17fabba04ab1bb47b3b0d371b832ef62ab613ec792157200","family_id":"qf-ext19-openclosed-family-closure","seed":202610191080,"semantic_template":"qf-ext19-openclosed-family-closure/{\"mode\":\"open_union\",\"n\":4,\"opens\":[[],[0],[0,2],[0,1,2],[0,1,2,3]],\"subsets\":[[],[0,2],[0,1,2,3]]}"},"fields":{"title":"Empty Families and Qualified Closure Premises","stem":"Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [0], [0, 2], [0, 1, 2], [0, 1, 2, 3]]. Normalize subsets and repeated members as sets; order families by binary subset masks. Complements are relative to this same X. Closed does not mean not open. mode=open_union, selected=[[], [0, 2], [0, 1, 2, 3]]. Modes are open_union, open_finite_intersection, closed_intersection, closed_finite_union. Give (n, mode, normalized selected family, all selected sets have the required open/closed property?, union/intersection result, result has that property?). Empty union is ∅ and empty intersection is X. Compute the result even if a premise fails.","answer":{"kind":"expression","value":"(4, open_union, [[], [0, 2], [0, 1, 2, 3]], Yes, [0, 1, 2, 3], Yes)","explanation":"Keep premise qualification separate from the resulting set’s property; a false premise does not prevent an accidentally qualified result. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, open_union, [[], [0, 2], [0, 1, 2, 3]], Yes, [0, 1, 2, 3], Yes)."},"skills":["Empty Families and Qualified Closure Premises"],"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-20261013-advance-20261019-r01-open-closed-set-0081","batch_id":"batch-20261013-advance-20261019-r01-open-closed-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ed3c2ccc5b3eb4b239c2ac77f97277d4160f187978cf67cdce71947c333ae2ad","family_id":"qf-ext19-openclosed-family-closure","seed":202610191081,"semantic_template":"qf-ext19-openclosed-family-closure/{\"mode\":\"open_finite_intersection\",\"n\":4,\"opens\":[[],[1],[1,2],[0,1,2],[0,1,2,3]],\"subsets\":[[1],[0,1,2]]}"},"fields":{"title":"Empty Families and Qualified Closure Premises","stem":"Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [1], [1, 2], [0, 1, 2], [0, 1, 2, 3]]. Normalize subsets and repeated members as sets; order families by binary subset masks. Complements are relative to this same X. Closed does not mean not open. mode=open_finite_intersection, selected=[[1], [0, 1, 2]]. Modes are open_union, open_finite_intersection, closed_intersection, closed_finite_union. Give (n, mode, normalized selected family, all selected sets have the required open/closed property?, union/intersection result, result has that property?). Empty union is ∅ and empty intersection is X. Compute the result even if a premise fails.","answer":{"kind":"expression","value":"(4, open_finite_intersection, [[1], [0, 1, 2]], Yes, [1], Yes)","explanation":"Keep premise qualification separate from the resulting set’s property; a false premise does not prevent an accidentally qualified result. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, open_finite_intersection, [[1], [0, 1, 2]], Yes, [1], Yes)."},"skills":["Empty Families and Qualified Closure Premises"],"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-20261013-advance-20261019-r01-open-closed-set-0082","batch_id":"batch-20261013-advance-20261019-r01-open-closed-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fe42620b03f9ce59b602c526c1b47cf77e9d6f972d08b9e98fc512c0583aa2c9","family_id":"qf-ext19-openclosed-family-closure","seed":202610191082,"semantic_template":"qf-ext19-openclosed-family-closure/{\"mode\":\"closed_intersection\",\"n\":4,\"opens\":[[],[2],[1,2],[0,1,2],[0,1,2,3]],\"subsets\":[[],[0,3],[0,1,2,3]]}"},"fields":{"title":"Empty Families and Qualified Closure Premises","stem":"Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [2], [1, 2], [0, 1, 2], [0, 1, 2, 3]]. Normalize subsets and repeated members as sets; order families by binary subset masks. Complements are relative to this same X. Closed does not mean not open. mode=closed_intersection, selected=[[], [0, 3], [0, 1, 2, 3]]. Modes are open_union, open_finite_intersection, closed_intersection, closed_finite_union. Give (n, mode, normalized selected family, all selected sets have the required open/closed property?, union/intersection result, result has that property?). Empty union is ∅ and empty intersection is X. Compute the result even if a premise fails.","answer":{"kind":"expression","value":"(4, closed_intersection, [[], [0, 3], [0, 1, 2, 3]], Yes, [], Yes)","explanation":"Keep premise qualification separate from the resulting set’s property; a false premise does not prevent an accidentally qualified result. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, closed_intersection, [[], [0, 3], [0, 1, 2, 3]], Yes, [], Yes)."},"skills":["Empty Families and Qualified Closure Premises"],"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-20261013-advance-20261019-r01-open-closed-set-0083","batch_id":"batch-20261013-advance-20261019-r01-open-closed-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f15f7242f6043e6ae2853eafb5c380abd68dfeb1651b77c31697caca19388eb0","family_id":"qf-ext19-openclosed-family-closure","seed":202610191083,"semantic_template":"qf-ext19-openclosed-family-closure/{\"mode\":\"closed_finite_union\",\"n\":4,\"opens\":[[],[3],[1,3],[0,1,3],[0,1,2,3]],\"subsets\":[[2],[0,1,2]]}"},"fields":{"title":"Empty Families and Qualified Closure Premises","stem":"Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [3], [1, 3], [0, 1, 3], [0, 1, 2, 3]]. Normalize subsets and repeated members as sets; order families by binary subset masks. Complements are relative to this same X. Closed does not mean not open. mode=closed_finite_union, selected=[[2], [0, 1, 2]]. Modes are open_union, open_finite_intersection, closed_intersection, closed_finite_union. Give (n, mode, normalized selected family, all selected sets have the required open/closed property?, union/intersection result, result has that property?). Empty union is ∅ and empty intersection is X. Compute the result even if a premise fails.","answer":{"kind":"expression","value":"(4, closed_finite_union, [[2], [0, 1, 2]], Yes, [0, 1, 2], Yes)","explanation":"Keep premise qualification separate from the resulting set’s property; a false premise does not prevent an accidentally qualified result. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, closed_finite_union, [[2], [0, 1, 2]], Yes, [0, 1, 2], Yes)."},"skills":["Empty Families and Qualified Closure Premises"],"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-20261013-advance-20261019-r01-open-closed-set-0084","batch_id":"batch-20261013-advance-20261019-r01-open-closed-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1eedf81a40110a297bb8afdd34bddc358bccff7e796a10e22786ee889bcc2136","family_id":"qf-ext19-openclosed-family-closure","seed":202610191084,"semantic_template":"qf-ext19-openclosed-family-closure/{\"mode\":\"open_union\",\"n\":4,\"opens\":[[],[0],[0,3],[0,1,3],[0,1,2,3]],\"subsets\":[[],[0,3],[0,1,2,3]]}"},"fields":{"title":"Empty Families and Qualified Closure Premises","stem":"Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [0], [0, 3], [0, 1, 3], [0, 1, 2, 3]]. Normalize subsets and repeated members as sets; order families by binary subset masks. Complements are relative to this same X. Closed does not mean not open. mode=open_union, selected=[[], [0, 3], [0, 1, 2, 3]]. Modes are open_union, open_finite_intersection, closed_intersection, closed_finite_union. Give (n, mode, normalized selected family, all selected sets have the required open/closed property?, union/intersection result, result has that property?). Empty union is ∅ and empty intersection is X. Compute the result even if a premise fails.","answer":{"kind":"expression","value":"(4, open_union, [[], [0, 3], [0, 1, 2, 3]], Yes, [0, 1, 2, 3], Yes)","explanation":"Keep premise qualification separate from the resulting set’s property; a false premise does not prevent an accidentally qualified result. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, open_union, [[], [0, 3], [0, 1, 2, 3]], Yes, [0, 1, 2, 3], Yes)."},"skills":["Empty Families and Qualified Closure Premises"],"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-20261013-advance-20261019-r01-open-closed-set-0085","batch_id":"batch-20261013-advance-20261019-r01-open-closed-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4b17bf95cf4502ca50d7618bb994cb743324845f806a04e6d4fcae87adac7081","family_id":"qf-ext19-openclosed-family-closure","seed":202610191085,"semantic_template":"qf-ext19-openclosed-family-closure/{\"mode\":\"open_finite_intersection\",\"n\":4,\"opens\":[[],[1],[1,3],[0,1,3],[0,1,2,3]],\"subsets\":[[1],[0,1,3]]}"},"fields":{"title":"Empty Families and Qualified Closure Premises","stem":"Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [1], [1, 3], [0, 1, 3], [0, 1, 2, 3]]. Normalize subsets and repeated members as sets; order families by binary subset masks. Complements are relative to this same X. Closed does not mean not open. mode=open_finite_intersection, selected=[[1], [0, 1, 3]]. Modes are open_union, open_finite_intersection, closed_intersection, closed_finite_union. Give (n, mode, normalized selected family, all selected sets have the required open/closed property?, union/intersection result, result has that property?). Empty union is ∅ and empty intersection is X. Compute the result even if a premise fails.","answer":{"kind":"expression","value":"(4, open_finite_intersection, [[1], [0, 1, 3]], Yes, [1], Yes)","explanation":"Keep premise qualification separate from the resulting set’s property; a false premise does not prevent an accidentally qualified result. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, open_finite_intersection, [[1], [0, 1, 3]], Yes, [1], Yes)."},"skills":["Empty Families and Qualified Closure Premises"],"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-20261013-advance-20261019-r01-open-closed-set-0086","batch_id":"batch-20261013-advance-20261019-r01-open-closed-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5f73ad5dce62942d4e298235ae3af49c58e9220db4eeeb8a8c2707fc538735fa","family_id":"qf-ext19-openclosed-family-closure","seed":202610191086,"semantic_template":"qf-ext19-openclosed-family-closure/{\"mode\":\"closed_intersection\",\"n\":4,\"opens\":[[],[2],[2,3],[0,2,3],[0,1,2,3]],\"subsets\":[[],[0,1],[0,1,2,3]]}"},"fields":{"title":"Empty Families and Qualified Closure Premises","stem":"Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [2], [2, 3], [0, 2, 3], [0, 1, 2, 3]]. Normalize subsets and repeated members as sets; order families by binary subset masks. Complements are relative to this same X. Closed does not mean not open. mode=closed_intersection, selected=[[], [0, 1], [0, 1, 2, 3]]. Modes are open_union, open_finite_intersection, closed_intersection, closed_finite_union. Give (n, mode, normalized selected family, all selected sets have the required open/closed property?, union/intersection result, result has that property?). Empty union is ∅ and empty intersection is X. Compute the result even if a premise fails.","answer":{"kind":"expression","value":"(4, closed_intersection, [[], [0, 1], [0, 1, 2, 3]], Yes, [], Yes)","explanation":"Keep premise qualification separate from the resulting set’s property; a false premise does not prevent an accidentally qualified result. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, closed_intersection, [[], [0, 1], [0, 1, 2, 3]], Yes, [], Yes)."},"skills":["Empty Families and Qualified Closure Premises"],"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-20261013-advance-20261019-r01-open-closed-set-0087","batch_id":"batch-20261013-advance-20261019-r01-open-closed-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"72078317ee7ef8d821c3665511ed74ec28b0a4d644c46c4db6b9a6884a8c2f6f","family_id":"qf-ext19-openclosed-family-closure","seed":202610191087,"semantic_template":"qf-ext19-openclosed-family-closure/{\"mode\":\"closed_finite_union\",\"n\":4,\"opens\":[[],[3],[2,3],[0,2,3],[0,1,2,3]],\"subsets\":[[1],[0,1,2]]}"},"fields":{"title":"Empty Families and Qualified Closure Premises","stem":"Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [3], [2, 3], [0, 2, 3], [0, 1, 2, 3]]. Normalize subsets and repeated members as sets; order families by binary subset masks. Complements are relative to this same X. Closed does not mean not open. mode=closed_finite_union, selected=[[1], [0, 1, 2]]. Modes are open_union, open_finite_intersection, closed_intersection, closed_finite_union. Give (n, mode, normalized selected family, all selected sets have the required open/closed property?, union/intersection result, result has that property?). Empty union is ∅ and empty intersection is X. Compute the result even if a premise fails.","answer":{"kind":"expression","value":"(4, closed_finite_union, [[1], [0, 1, 2]], Yes, [0, 1, 2], Yes)","explanation":"Keep premise qualification separate from the resulting set’s property; a false premise does not prevent an accidentally qualified result. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, closed_finite_union, [[1], [0, 1, 2]], Yes, [0, 1, 2], Yes)."},"skills":["Empty Families and Qualified Closure Premises"],"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-20261013-advance-20261019-r01-open-closed-set-0088","batch_id":"batch-20261013-advance-20261019-r01-open-closed-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cf7cf95d5c13e31dd5476f4bee54d29e95f80ecf4e0626bd0a27efe30d2c85db","family_id":"qf-ext19-openclosed-family-closure","seed":202610191088,"semantic_template":"qf-ext19-openclosed-family-closure/{\"mode\":\"open_union\",\"n\":4,\"opens\":[[],[0],[0,1],[0,1,3],[0,1,2,3]],\"subsets\":[[],[0,1],[0,1,2,3]]}"},"fields":{"title":"Empty Families and Qualified Closure Premises","stem":"Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [0], [0, 1], [0, 1, 3], [0, 1, 2, 3]]. Normalize subsets and repeated members as sets; order families by binary subset masks. Complements are relative to this same X. Closed does not mean not open. mode=open_union, selected=[[], [0, 1], [0, 1, 2, 3]]. Modes are open_union, open_finite_intersection, closed_intersection, closed_finite_union. Give (n, mode, normalized selected family, all selected sets have the required open/closed property?, union/intersection result, result has that property?). Empty union is ∅ and empty intersection is X. Compute the result even if a premise fails.","answer":{"kind":"expression","value":"(4, open_union, [[], [0, 1], [0, 1, 2, 3]], Yes, [0, 1, 2, 3], Yes)","explanation":"Keep premise qualification separate from the resulting set’s property; a false premise does not prevent an accidentally qualified result. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, open_union, [[], [0, 1], [0, 1, 2, 3]], Yes, [0, 1, 2, 3], Yes)."},"skills":["Empty Families and Qualified Closure Premises"],"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-20261013-advance-20261019-r01-open-closed-set-0089","batch_id":"batch-20261013-advance-20261019-r01-open-closed-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6dd2780fc717c919b745219230ceece6295d7394758799ae3a908c74167e3bd5","family_id":"qf-ext19-openclosed-family-closure","seed":202610191089,"semantic_template":"qf-ext19-openclosed-family-closure/{\"mode\":\"open_finite_intersection\",\"n\":4,\"opens\":[[],[1],[0,1],[0,1,3],[0,1,2,3]],\"subsets\":[[1],[0,1,3]]}"},"fields":{"title":"Empty Families and Qualified Closure Premises","stem":"Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [1], [0, 1], [0, 1, 3], [0, 1, 2, 3]]. Normalize subsets and repeated members as sets; order families by binary subset masks. Complements are relative to this same X. Closed does not mean not open. mode=open_finite_intersection, selected=[[1], [0, 1, 3]]. Modes are open_union, open_finite_intersection, closed_intersection, closed_finite_union. Give (n, mode, normalized selected family, all selected sets have the required open/closed property?, union/intersection result, result has that property?). Empty union is ∅ and empty intersection is X. Compute the result even if a premise fails.","answer":{"kind":"expression","value":"(4, open_finite_intersection, [[1], [0, 1, 3]], Yes, [1], Yes)","explanation":"Keep premise qualification separate from the resulting set’s property; a false premise does not prevent an accidentally qualified result. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, open_finite_intersection, [[1], [0, 1, 3]], Yes, [1], Yes)."},"skills":["Empty Families and Qualified Closure Premises"],"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-20261013-advance-20261019-r01-open-closed-set-0090","batch_id":"batch-20261013-advance-20261019-r01-open-closed-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"affb09996f00f7bd75387ed7135a96cee04393105276c3554a51678cde29feab","family_id":"qf-ext19-openclosed-family-closure","seed":202610191090,"semantic_template":"qf-ext19-openclosed-family-closure/{\"mode\":\"closed_intersection\",\"n\":4,\"opens\":[[],[2],[0,2],[0,2,3],[0,1,2,3]],\"subsets\":[[],[1,3],[0,1,2,3]]}"},"fields":{"title":"Empty Families and Qualified Closure Premises","stem":"Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [2], [0, 2], [0, 2, 3], [0, 1, 2, 3]]. Normalize subsets and repeated members as sets; order families by binary subset masks. Complements are relative to this same X. Closed does not mean not open. mode=closed_intersection, selected=[[], [1, 3], [0, 1, 2, 3]]. Modes are open_union, open_finite_intersection, closed_intersection, closed_finite_union. Give (n, mode, normalized selected family, all selected sets have the required open/closed property?, union/intersection result, result has that property?). Empty union is ∅ and empty intersection is X. Compute the result even if a premise fails.","answer":{"kind":"expression","value":"(4, closed_intersection, [[], [1, 3], [0, 1, 2, 3]], Yes, [], Yes)","explanation":"Keep premise qualification separate from the resulting set’s property; a false premise does not prevent an accidentally qualified result. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, closed_intersection, [[], [1, 3], [0, 1, 2, 3]], Yes, [], Yes)."},"skills":["Empty Families and Qualified Closure Premises"],"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-20261013-advance-20261019-r01-open-closed-set-0091","batch_id":"batch-20261013-advance-20261019-r01-open-closed-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b816c5d77eeb3c95d5cc57ee1858e662b16981f0e389436e6d390e155c3b2499","family_id":"qf-ext19-openclosed-family-closure","seed":202610191091,"semantic_template":"qf-ext19-openclosed-family-closure/{\"mode\":\"closed_finite_union\",\"n\":4,\"opens\":[[],[3],[0,3],[0,2,3],[0,1,2,3]],\"subsets\":[[1],[0,1,2]]}"},"fields":{"title":"Empty Families and Qualified Closure Premises","stem":"Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [3], [0, 3], [0, 2, 3], [0, 1, 2, 3]]. Normalize subsets and repeated members as sets; order families by binary subset masks. Complements are relative to this same X. Closed does not mean not open. mode=closed_finite_union, selected=[[1], [0, 1, 2]]. Modes are open_union, open_finite_intersection, closed_intersection, closed_finite_union. Give (n, mode, normalized selected family, all selected sets have the required open/closed property?, union/intersection result, result has that property?). Empty union is ∅ and empty intersection is X. Compute the result even if a premise fails.","answer":{"kind":"expression","value":"(4, closed_finite_union, [[1], [0, 1, 2]], Yes, [0, 1, 2], Yes)","explanation":"Keep premise qualification separate from the resulting set’s property; a false premise does not prevent an accidentally qualified result. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, closed_finite_union, [[1], [0, 1, 2]], Yes, [0, 1, 2], Yes)."},"skills":["Empty Families and Qualified Closure Premises"],"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-20261013-advance-20261019-r01-open-closed-set-0092","batch_id":"batch-20261013-advance-20261019-r01-open-closed-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3585b4362980a027bb03fc5966b645809af9dbeec152fee678fa05210217d60d","family_id":"qf-ext19-openclosed-family-closure","seed":202610191092,"semantic_template":"qf-ext19-openclosed-family-closure/{\"mode\":\"open_union\",\"n\":4,\"opens\":[[],[0],[0,2],[0,2,3],[0,1,2,3]],\"subsets\":[[],[0,2],[0,1,2,3]]}"},"fields":{"title":"Empty Families and Qualified Closure Premises","stem":"Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [0], [0, 2], [0, 2, 3], [0, 1, 2, 3]]. Normalize subsets and repeated members as sets; order families by binary subset masks. Complements are relative to this same X. Closed does not mean not open. mode=open_union, selected=[[], [0, 2], [0, 1, 2, 3]]. Modes are open_union, open_finite_intersection, closed_intersection, closed_finite_union. Give (n, mode, normalized selected family, all selected sets have the required open/closed property?, union/intersection result, result has that property?). Empty union is ∅ and empty intersection is X. Compute the result even if a premise fails.","answer":{"kind":"expression","value":"(4, open_union, [[], [0, 2], [0, 1, 2, 3]], Yes, [0, 1, 2, 3], Yes)","explanation":"Keep premise qualification separate from the resulting set’s property; a false premise does not prevent an accidentally qualified result. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, open_union, [[], [0, 2], [0, 1, 2, 3]], Yes, [0, 1, 2, 3], Yes)."},"skills":["Empty Families and Qualified Closure Premises"],"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-20261013-advance-20261019-r01-open-closed-set-0093","batch_id":"batch-20261013-advance-20261019-r01-open-closed-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fc8ee5d04094be8d8cd8b57a0c5730969fa7b2b230ffc9f8da597eea4df72970","family_id":"qf-ext19-openclosed-family-closure","seed":202610191093,"semantic_template":"qf-ext19-openclosed-family-closure/{\"mode\":\"open_finite_intersection\",\"n\":4,\"opens\":[[],[1],[1,2],[1,2,3],[0,1,2,3]],\"subsets\":[[1],[1,2,3]]}"},"fields":{"title":"Empty Families and Qualified Closure Premises","stem":"Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [1], [1, 2], [1, 2, 3], [0, 1, 2, 3]]. Normalize subsets and repeated members as sets; order families by binary subset masks. Complements are relative to this same X. Closed does not mean not open. mode=open_finite_intersection, selected=[[1], [1, 2, 3]]. Modes are open_union, open_finite_intersection, closed_intersection, closed_finite_union. Give (n, mode, normalized selected family, all selected sets have the required open/closed property?, union/intersection result, result has that property?). Empty union is ∅ and empty intersection is X. Compute the result even if a premise fails.","answer":{"kind":"expression","value":"(4, open_finite_intersection, [[1], [1, 2, 3]], Yes, [1], Yes)","explanation":"Keep premise qualification separate from the resulting set’s property; a false premise does not prevent an accidentally qualified result. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, open_finite_intersection, [[1], [1, 2, 3]], Yes, [1], Yes)."},"skills":["Empty Families and Qualified Closure Premises"],"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-20261013-advance-20261019-r01-open-closed-set-0094","batch_id":"batch-20261013-advance-20261019-r01-open-closed-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"68ac7628b1cd9df0312c0a658efdcbf7a5118b921c289bcddde35c8bd96c6f7c","family_id":"qf-ext19-openclosed-family-closure","seed":202610191094,"semantic_template":"qf-ext19-openclosed-family-closure/{\"mode\":\"closed_intersection\",\"n\":4,\"opens\":[[],[2],[1,2],[1,2,3],[0,1,2,3]],\"subsets\":[[],[0,3],[0,1,2,3]]}"},"fields":{"title":"Empty Families and Qualified Closure Premises","stem":"Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [2], [1, 2], [1, 2, 3], [0, 1, 2, 3]]. Normalize subsets and repeated members as sets; order families by binary subset masks. Complements are relative to this same X. Closed does not mean not open. mode=closed_intersection, selected=[[], [0, 3], [0, 1, 2, 3]]. Modes are open_union, open_finite_intersection, closed_intersection, closed_finite_union. Give (n, mode, normalized selected family, all selected sets have the required open/closed property?, union/intersection result, result has that property?). Empty union is ∅ and empty intersection is X. Compute the result even if a premise fails.","answer":{"kind":"expression","value":"(4, closed_intersection, [[], [0, 3], [0, 1, 2, 3]], Yes, [], Yes)","explanation":"Keep premise qualification separate from the resulting set’s property; a false premise does not prevent an accidentally qualified result. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, closed_intersection, [[], [0, 3], [0, 1, 2, 3]], Yes, [], Yes)."},"skills":["Empty Families and Qualified Closure Premises"],"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-20261013-advance-20261019-r01-open-closed-set-0095","batch_id":"batch-20261013-advance-20261019-r01-open-closed-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d255cc50c49bb58b42e429ef94456d56d64e11d1ea3c91323e95afb88fc1af9d","family_id":"qf-ext19-openclosed-family-closure","seed":202610191095,"semantic_template":"qf-ext19-openclosed-family-closure/{\"mode\":\"closed_finite_union\",\"n\":4,\"opens\":[[],[3],[1,3],[1,2,3],[0,1,2,3]],\"subsets\":[[0],[0,1,2]]}"},"fields":{"title":"Empty Families and Qualified Closure Premises","stem":"Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [3], [1, 3], [1, 2, 3], [0, 1, 2, 3]]. Normalize subsets and repeated members as sets; order families by binary subset masks. Complements are relative to this same X. Closed does not mean not open. mode=closed_finite_union, selected=[[0], [0, 1, 2]]. Modes are open_union, open_finite_intersection, closed_intersection, closed_finite_union. Give (n, mode, normalized selected family, all selected sets have the required open/closed property?, union/intersection result, result has that property?). Empty union is ∅ and empty intersection is X. Compute the result even if a premise fails.","answer":{"kind":"expression","value":"(4, closed_finite_union, [[0], [0, 1, 2]], Yes, [0, 1, 2], Yes)","explanation":"Keep premise qualification separate from the resulting set’s property; a false premise does not prevent an accidentally qualified result. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, closed_finite_union, [[0], [0, 1, 2]], Yes, [0, 1, 2], Yes)."},"skills":["Empty Families and Qualified Closure Premises"],"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-20261013-advance-20261019-r01-open-closed-set-0096","batch_id":"batch-20261013-advance-20261019-r01-open-closed-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"04db17f6dd776c2bca1734ab729e28bd591725e8edd96d267f8a17974a02c6d8","family_id":"qf-ext19-openclosed-family-closure","seed":202610191096,"semantic_template":"qf-ext19-openclosed-family-closure/{\"mode\":\"open_union\",\"n\":4,\"opens\":[[],[0],[0,3],[0,2,3],[0,1,2,3]],\"subsets\":[[],[0,3],[0,1,2,3]]}"},"fields":{"title":"Empty Families and Qualified Closure Premises","stem":"Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [0], [0, 3], [0, 2, 3], [0, 1, 2, 3]]. Normalize subsets and repeated members as sets; order families by binary subset masks. Complements are relative to this same X. Closed does not mean not open. mode=open_union, selected=[[], [0, 3], [0, 1, 2, 3]]. Modes are open_union, open_finite_intersection, closed_intersection, closed_finite_union. Give (n, mode, normalized selected family, all selected sets have the required open/closed property?, union/intersection result, result has that property?). Empty union is ∅ and empty intersection is X. Compute the result even if a premise fails.","answer":{"kind":"expression","value":"(4, open_union, [[], [0, 3], [0, 1, 2, 3]], Yes, [0, 1, 2, 3], Yes)","explanation":"Keep premise qualification separate from the resulting set’s property; a false premise does not prevent an accidentally qualified result. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, open_union, [[], [0, 3], [0, 1, 2, 3]], Yes, [0, 1, 2, 3], Yes)."},"skills":["Empty Families and Qualified Closure Premises"],"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-20261013-advance-20261019-r01-open-closed-set-0097","batch_id":"batch-20261013-advance-20261019-r01-open-closed-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b56e9c100329dc85949279cfb3075f987696d1180ef8bed4a1b38f469acc2b61","family_id":"qf-ext19-openclosed-family-closure","seed":202610191097,"semantic_template":"qf-ext19-openclosed-family-closure/{\"mode\":\"open_finite_intersection\",\"n\":4,\"opens\":[[],[1],[1,3],[1,2,3],[0,1,2,3]],\"subsets\":[[1],[1,2,3]]}"},"fields":{"title":"Empty Families and Qualified Closure Premises","stem":"Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [1], [1, 3], [1, 2, 3], [0, 1, 2, 3]]. Normalize subsets and repeated members as sets; order families by binary subset masks. Complements are relative to this same X. Closed does not mean not open. mode=open_finite_intersection, selected=[[1], [1, 2, 3]]. Modes are open_union, open_finite_intersection, closed_intersection, closed_finite_union. Give (n, mode, normalized selected family, all selected sets have the required open/closed property?, union/intersection result, result has that property?). Empty union is ∅ and empty intersection is X. Compute the result even if a premise fails.","answer":{"kind":"expression","value":"(4, open_finite_intersection, [[1], [1, 2, 3]], Yes, [1], Yes)","explanation":"Keep premise qualification separate from the resulting set’s property; a false premise does not prevent an accidentally qualified result. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, open_finite_intersection, [[1], [1, 2, 3]], Yes, [1], Yes)."},"skills":["Empty Families and Qualified Closure Premises"],"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-20261013-advance-20261019-r01-open-closed-set-0098","batch_id":"batch-20261013-advance-20261019-r01-open-closed-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"15e435d64422e23310f85dd1ce4e4f825f8129c1ba055adb22f58a1a9120d252","family_id":"qf-ext19-openclosed-family-closure","seed":202610191098,"semantic_template":"qf-ext19-openclosed-family-closure/{\"mode\":\"closed_intersection\",\"n\":4,\"opens\":[[],[2],[2,3],[1,2,3],[0,1,2,3]],\"subsets\":[[],[0,1],[0,1,2,3]]}"},"fields":{"title":"Empty Families and Qualified Closure Premises","stem":"Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [2], [2, 3], [1, 2, 3], [0, 1, 2, 3]]. Normalize subsets and repeated members as sets; order families by binary subset masks. Complements are relative to this same X. Closed does not mean not open. mode=closed_intersection, selected=[[], [0, 1], [0, 1, 2, 3]]. Modes are open_union, open_finite_intersection, closed_intersection, closed_finite_union. Give (n, mode, normalized selected family, all selected sets have the required open/closed property?, union/intersection result, result has that property?). Empty union is ∅ and empty intersection is X. Compute the result even if a premise fails.","answer":{"kind":"expression","value":"(4, closed_intersection, [[], [0, 1], [0, 1, 2, 3]], Yes, [], Yes)","explanation":"Keep premise qualification separate from the resulting set’s property; a false premise does not prevent an accidentally qualified result. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, closed_intersection, [[], [0, 1], [0, 1, 2, 3]], Yes, [], Yes)."},"skills":["Empty Families and Qualified Closure Premises"],"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-20261013-advance-20261019-r01-open-closed-set-0099","batch_id":"batch-20261013-advance-20261019-r01-open-closed-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3e130c9d57bc35f3bf7d4499e2af7841b8c106d1c0125f7c9c7e69220b8e5395","family_id":"qf-ext19-openclosed-family-closure","seed":202610191099,"semantic_template":"qf-ext19-openclosed-family-closure/{\"mode\":\"closed_finite_union\",\"n\":4,\"opens\":[[],[3],[2,3],[1,2,3],[0,1,2,3]],\"subsets\":[[0],[0,1,2]]}"},"fields":{"title":"Empty Families and Qualified Closure Premises","stem":"Use the complete finite topology on X=Fin(4)={0,…,3} with opens=[[], [3], [2, 3], [1, 2, 3], [0, 1, 2, 3]]. Normalize subsets and repeated members as sets; order families by binary subset masks. Complements are relative to this same X. Closed does not mean not open. mode=closed_finite_union, selected=[[0], [0, 1, 2]]. Modes are open_union, open_finite_intersection, closed_intersection, closed_finite_union. Give (n, mode, normalized selected family, all selected sets have the required open/closed property?, union/intersection result, result has that property?). Empty union is ∅ and empty intersection is X. Compute the result even if a premise fails.","answer":{"kind":"expression","value":"(4, closed_finite_union, [[0], [0, 1, 2]], Yes, [0, 1, 2], Yes)","explanation":"Keep premise qualification separate from the resulting set’s property; a false premise does not prevent an accidentally qualified result. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (4, closed_finite_union, [[0], [0, 1, 2]], Yes, [0, 1, 2], Yes)."},"skills":["Empty Families and Qualified Closure Premises"],"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-20261013-advance-20261019-r01-open-closed-set-0100","batch_id":"batch-20261013-advance-20261019-r01-open-closed-set-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ba123faa4336c06848938f011df7e10f026779189793171894c562056ff9806a","family_id":"qf-ext19-openclosed-family-closure","seed":202610191100,"semantic_template":"qf-ext19-openclosed-family-closure/{\"mode\":\"open_union\",\"n\":3,\"opens\":[[],[0],[0,1],[0,1,2]],\"subsets\":[[],[0,1]]}"},"fields":{"title":"Empty Families and Qualified Closure Premises","stem":"Use the complete finite topology on X=Fin(3)={0,…,2} with opens=[[], [0], [0, 1], [0, 1, 2]]. Normalize subsets and repeated members as sets; order families by binary subset masks. Complements are relative to this same X. Closed does not mean not open. mode=open_union, selected=[[], [0, 1]]. Modes are open_union, open_finite_intersection, closed_intersection, closed_finite_union. Give (n, mode, normalized selected family, all selected sets have the required open/closed property?, union/intersection result, result has that property?). Empty union is ∅ and empty intersection is X. Compute the result even if a premise fails.","answer":{"kind":"expression","value":"(3, open_union, [[], [0, 1]], Yes, [0, 1], Yes)","explanation":"Keep premise qualification separate from the resulting set’s property; a false premise does not prevent an accidentally qualified result. Finite/synthetic checks are not new general formal Evidence. The exact result, retaining the requested order, is (3, open_union, [[], [0, 1]], Yes, [0, 1], Yes)."},"skills":["Empty Families and Qualified Closure Premises"],"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."}}