{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261009-advance-20261013-r06-event-0001","batch_id":"batch-20261009-advance-20261013-r06-event-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7048a020ed4cc39c6aea2d810ae49eaf1d7eb07bc6b0edaaf2781994c36a43c1","family_id":"qf-density13-event-complement-partition","seed":202610136001,"semantic_template":"qf-density13-event-complement-partition/[1,[]]"},"fields":{"title":"Declared Event and Relative Complement Partition","stem":"S=Fin(1)={0,…,0}; the event family is explicitly F=P(S). E=[]⊆S. Give (increasing E, increasing S\\E, n).","answer":{"kind":"expression","value":"([], [0], 1)","explanation":"Take the complement inside the explicitly declared full event family. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is ([], [0], 1)."},"skills":["Declared Event and Relative Complement Partition"],"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-20261009-advance-20261013-r06-event-0002","batch_id":"batch-20261009-advance-20261013-r06-event-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cd5875c59e6caa5227285e7d5ad5a26488ae207973af16dc839f80290a726710","family_id":"qf-density13-event-complement-partition","seed":202610136002,"semantic_template":"qf-density13-event-complement-partition/[1,[0]]"},"fields":{"title":"Declared Event and Relative Complement Partition","stem":"S=Fin(1)={0,…,0}; the event family is explicitly F=P(S). E=[0]⊆S. Give (increasing E, increasing S\\E, n).","answer":{"kind":"expression","value":"([0], [], 1)","explanation":"Take the complement inside the explicitly declared full event family. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is ([0], [], 1)."},"skills":["Declared Event and Relative Complement Partition"],"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-20261009-advance-20261013-r06-event-0003","batch_id":"batch-20261009-advance-20261013-r06-event-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"048eabe9c8275a128bc88739190adc7b67f51852ff6c4049d2c63cb860ede284","family_id":"qf-density13-event-complement-partition","seed":202610136003,"semantic_template":"qf-density13-event-complement-partition/[5,[0]]"},"fields":{"title":"Declared Event and Relative Complement Partition","stem":"S=Fin(5)={0,…,4}; the event family is explicitly F=P(S). E=[0]⊆S. Give (increasing E, increasing S\\E, n).","answer":{"kind":"expression","value":"([0], [1, 2, 3, 4], 5)","explanation":"Take the complement inside the explicitly declared full event family. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is ([0], [1, 2, 3, 4], 5)."},"skills":["Declared Event and Relative Complement Partition"],"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-20261009-advance-20261013-r06-event-0004","batch_id":"batch-20261009-advance-20261013-r06-event-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"199c732b57a275f3885697521cf3834f418d1a185b8ecc15bab1c1dffa2fcfe8","family_id":"qf-density13-event-complement-partition","seed":202610136004,"semantic_template":"qf-density13-event-complement-partition/[5,[1]]"},"fields":{"title":"Declared Event and Relative Complement Partition","stem":"S=Fin(5)={0,…,4}; the event family is explicitly F=P(S). E=[1]⊆S. Give (increasing E, increasing S\\E, n).","answer":{"kind":"expression","value":"([1], [0, 2, 3, 4], 5)","explanation":"Take the complement inside the explicitly declared full event family. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is ([1], [0, 2, 3, 4], 5)."},"skills":["Declared Event and Relative Complement Partition"],"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-20261009-advance-20261013-r06-event-0005","batch_id":"batch-20261009-advance-20261013-r06-event-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"76603b340e4666fc7a50c7034fad91198af45f1516140c3a1b0104ae3b93799a","family_id":"qf-density13-event-complement-partition","seed":202610136005,"semantic_template":"qf-density13-event-complement-partition/[5,[0,1]]"},"fields":{"title":"Declared Event and Relative Complement Partition","stem":"S=Fin(5)={0,…,4}; the event family is explicitly F=P(S). E=[0, 1]⊆S. Give (increasing E, increasing S\\E, n).","answer":{"kind":"expression","value":"([0, 1], [2, 3, 4], 5)","explanation":"Take the complement inside the explicitly declared full event family. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is ([0, 1], [2, 3, 4], 5)."},"skills":["Declared Event and Relative Complement Partition"],"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-20261009-advance-20261013-r06-event-0006","batch_id":"batch-20261009-advance-20261013-r06-event-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a0c52251929e6758a041fcb28daa2871bcd2f23969ee20ba61923e9f8e3886f1","family_id":"qf-density13-event-complement-partition","seed":202610136006,"semantic_template":"qf-density13-event-complement-partition/[5,[2]]"},"fields":{"title":"Declared Event and Relative Complement Partition","stem":"S=Fin(5)={0,…,4}; the event family is explicitly F=P(S). E=[2]⊆S. Give (increasing E, increasing S\\E, n).","answer":{"kind":"expression","value":"([2], [0, 1, 3, 4], 5)","explanation":"Take the complement inside the explicitly declared full event family. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is ([2], [0, 1, 3, 4], 5)."},"skills":["Declared Event and Relative Complement Partition"],"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-20261009-advance-20261013-r06-event-0007","batch_id":"batch-20261009-advance-20261013-r06-event-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e3b3dcd12c138ee02946fc648a4b268429dbaf32eede90ed8fb91843953eacc3","family_id":"qf-density13-event-complement-partition","seed":202610136007,"semantic_template":"qf-density13-event-complement-partition/[5,[0,2]]"},"fields":{"title":"Declared Event and Relative Complement Partition","stem":"S=Fin(5)={0,…,4}; the event family is explicitly F=P(S). E=[0, 2]⊆S. Give (increasing E, increasing S\\E, n).","answer":{"kind":"expression","value":"([0, 2], [1, 3, 4], 5)","explanation":"Take the complement inside the explicitly declared full event family. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is ([0, 2], [1, 3, 4], 5)."},"skills":["Declared Event and Relative Complement Partition"],"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-20261009-advance-20261013-r06-event-0008","batch_id":"batch-20261009-advance-20261013-r06-event-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"45bf2516b8eb7e34f8c94232f9e77542ffd54dee2faf1922312d924e75dcb130","family_id":"qf-density13-event-complement-partition","seed":202610136008,"semantic_template":"qf-density13-event-complement-partition/[5,[1,2]]"},"fields":{"title":"Declared Event and Relative Complement Partition","stem":"S=Fin(5)={0,…,4}; the event family is explicitly F=P(S). E=[1, 2]⊆S. Give (increasing E, increasing S\\E, n).","answer":{"kind":"expression","value":"([1, 2], [0, 3, 4], 5)","explanation":"Take the complement inside the explicitly declared full event family. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is ([1, 2], [0, 3, 4], 5)."},"skills":["Declared Event and Relative Complement Partition"],"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-20261009-advance-20261013-r06-event-0009","batch_id":"batch-20261009-advance-20261013-r06-event-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d3f8498cfe4b6d994d9906041813e8697ebf3beb6248184c342f1078c7041d5c","family_id":"qf-density13-event-complement-partition","seed":202610136009,"semantic_template":"qf-density13-event-complement-partition/[5,[0,1,2]]"},"fields":{"title":"Declared Event and Relative Complement Partition","stem":"S=Fin(5)={0,…,4}; the event family is explicitly F=P(S). E=[0, 1, 2]⊆S. Give (increasing E, increasing S\\E, n).","answer":{"kind":"expression","value":"([0, 1, 2], [3, 4], 5)","explanation":"Take the complement inside the explicitly declared full event family. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is ([0, 1, 2], [3, 4], 5)."},"skills":["Declared Event and Relative Complement Partition"],"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-20261009-advance-20261013-r06-event-0010","batch_id":"batch-20261009-advance-20261013-r06-event-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4d76fd299bcf3d8ca3f82189ba593d8f719267562477c9359d3025ce233e990e","family_id":"qf-density13-event-complement-partition","seed":202610136010,"semantic_template":"qf-density13-event-complement-partition/[5,[3]]"},"fields":{"title":"Declared Event and Relative Complement Partition","stem":"S=Fin(5)={0,…,4}; the event family is explicitly F=P(S). E=[3]⊆S. Give (increasing E, increasing S\\E, n).","answer":{"kind":"expression","value":"([3], [0, 1, 2, 4], 5)","explanation":"Take the complement inside the explicitly declared full event family. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is ([3], [0, 1, 2, 4], 5)."},"skills":["Declared Event and Relative Complement Partition"],"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-20261009-advance-20261013-r06-event-0011","batch_id":"batch-20261009-advance-20261013-r06-event-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5745af628b7e2dbd2bba1d920066b07ad9b32767970bc3b10c3786f177e87a06","family_id":"qf-density13-event-complement-partition","seed":202610136011,"semantic_template":"qf-density13-event-complement-partition/[5,[0,3]]"},"fields":{"title":"Declared Event and Relative Complement Partition","stem":"S=Fin(5)={0,…,4}; the event family is explicitly F=P(S). E=[0, 3]⊆S. Give (increasing E, increasing S\\E, n).","answer":{"kind":"expression","value":"([0, 3], [1, 2, 4], 5)","explanation":"Take the complement inside the explicitly declared full event family. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is ([0, 3], [1, 2, 4], 5)."},"skills":["Declared Event and Relative Complement Partition"],"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-20261009-advance-20261013-r06-event-0012","batch_id":"batch-20261009-advance-20261013-r06-event-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f75917d59ad79e5b0999ea77101701a5eec627d19562e914936f19398d5e5fcd","family_id":"qf-density13-event-complement-partition","seed":202610136012,"semantic_template":"qf-density13-event-complement-partition/[5,[1,3]]"},"fields":{"title":"Declared Event and Relative Complement Partition","stem":"S=Fin(5)={0,…,4}; the event family is explicitly F=P(S). E=[1, 3]⊆S. Give (increasing E, increasing S\\E, n).","answer":{"kind":"expression","value":"([1, 3], [0, 2, 4], 5)","explanation":"Take the complement inside the explicitly declared full event family. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is ([1, 3], [0, 2, 4], 5)."},"skills":["Declared Event and Relative Complement Partition"],"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-20261009-advance-20261013-r06-event-0013","batch_id":"batch-20261009-advance-20261013-r06-event-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8e58e89eb240ad020648ab0fe43ad10a9d3c8eb436523ea4a228110c7ff95a0a","family_id":"qf-density13-event-complement-partition","seed":202610136013,"semantic_template":"qf-density13-event-complement-partition/[5,[0,1,3]]"},"fields":{"title":"Declared Event and Relative Complement Partition","stem":"S=Fin(5)={0,…,4}; the event family is explicitly F=P(S). E=[0, 1, 3]⊆S. Give (increasing E, increasing S\\E, n).","answer":{"kind":"expression","value":"([0, 1, 3], [2, 4], 5)","explanation":"Take the complement inside the explicitly declared full event family. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is ([0, 1, 3], [2, 4], 5)."},"skills":["Declared Event and Relative Complement Partition"],"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-20261009-advance-20261013-r06-event-0014","batch_id":"batch-20261009-advance-20261013-r06-event-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9208a270a3f05ff8a1fd35657e6843b161d1b4c8369a8bfc3bc1389edfb10806","family_id":"qf-density13-event-complement-partition","seed":202610136014,"semantic_template":"qf-density13-event-complement-partition/[5,[2,3]]"},"fields":{"title":"Declared Event and Relative Complement Partition","stem":"S=Fin(5)={0,…,4}; the event family is explicitly F=P(S). E=[2, 3]⊆S. Give (increasing E, increasing S\\E, n).","answer":{"kind":"expression","value":"([2, 3], [0, 1, 4], 5)","explanation":"Take the complement inside the explicitly declared full event family. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is ([2, 3], [0, 1, 4], 5)."},"skills":["Declared Event and Relative Complement Partition"],"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-20261009-advance-20261013-r06-event-0015","batch_id":"batch-20261009-advance-20261013-r06-event-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"09d9442bf699ec3b8ae8bea0f07454fd99f30c7518162b32e527b10e7472da6e","family_id":"qf-density13-event-complement-partition","seed":202610136015,"semantic_template":"qf-density13-event-complement-partition/[5,[0,2,3]]"},"fields":{"title":"Declared Event and Relative Complement Partition","stem":"S=Fin(5)={0,…,4}; the event family is explicitly F=P(S). E=[0, 2, 3]⊆S. Give (increasing E, increasing S\\E, n).","answer":{"kind":"expression","value":"([0, 2, 3], [1, 4], 5)","explanation":"Take the complement inside the explicitly declared full event family. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is ([0, 2, 3], [1, 4], 5)."},"skills":["Declared Event and Relative Complement Partition"],"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-20261009-advance-20261013-r06-event-0016","batch_id":"batch-20261009-advance-20261013-r06-event-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"051d6a865779f7d3f63d57bb9c60e6713fd4dd88f3c01556092f1680c02faa18","family_id":"qf-density13-event-complement-partition","seed":202610136016,"semantic_template":"qf-density13-event-complement-partition/[5,[1,2,3]]"},"fields":{"title":"Declared Event and Relative Complement Partition","stem":"S=Fin(5)={0,…,4}; the event family is explicitly F=P(S). E=[1, 2, 3]⊆S. Give (increasing E, increasing S\\E, n).","answer":{"kind":"expression","value":"([1, 2, 3], [0, 4], 5)","explanation":"Take the complement inside the explicitly declared full event family. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is ([1, 2, 3], [0, 4], 5)."},"skills":["Declared Event and Relative Complement Partition"],"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-20261009-advance-20261013-r06-event-0017","batch_id":"batch-20261009-advance-20261013-r06-event-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bff5367c2ca1e7be9ad8361d9fd27822267d530814febf31453c42aacb326de6","family_id":"qf-density13-event-complement-partition","seed":202610136017,"semantic_template":"qf-density13-event-complement-partition/[5,[0,1,2,3]]"},"fields":{"title":"Declared Event and Relative Complement Partition","stem":"S=Fin(5)={0,…,4}; the event family is explicitly F=P(S). E=[0, 1, 2, 3]⊆S. Give (increasing E, increasing S\\E, n).","answer":{"kind":"expression","value":"([0, 1, 2, 3], [4], 5)","explanation":"Take the complement inside the explicitly declared full event family. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is ([0, 1, 2, 3], [4], 5)."},"skills":["Declared Event and Relative Complement Partition"],"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-20261009-advance-20261013-r06-event-0018","batch_id":"batch-20261009-advance-20261013-r06-event-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"846226621ad5975b5c855131d9acba2106b69eaabf852853e64590902359d4bb","family_id":"qf-density13-event-complement-partition","seed":202610136018,"semantic_template":"qf-density13-event-complement-partition/[5,[4]]"},"fields":{"title":"Declared Event and Relative Complement Partition","stem":"S=Fin(5)={0,…,4}; the event family is explicitly F=P(S). E=[4]⊆S. Give (increasing E, increasing S\\E, n).","answer":{"kind":"expression","value":"([4], [0, 1, 2, 3], 5)","explanation":"Take the complement inside the explicitly declared full event family. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is ([4], [0, 1, 2, 3], 5)."},"skills":["Declared Event and Relative Complement Partition"],"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-20261009-advance-20261013-r06-event-0019","batch_id":"batch-20261009-advance-20261013-r06-event-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9626174fe95959c9ab58a51689cf069a939116fd6f457fe832c310e46b48e19a","family_id":"qf-density13-event-complement-partition","seed":202610136019,"semantic_template":"qf-density13-event-complement-partition/[5,[0,4]]"},"fields":{"title":"Declared Event and Relative Complement Partition","stem":"S=Fin(5)={0,…,4}; the event family is explicitly F=P(S). E=[0, 4]⊆S. Give (increasing E, increasing S\\E, n).","answer":{"kind":"expression","value":"([0, 4], [1, 2, 3], 5)","explanation":"Take the complement inside the explicitly declared full event family. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is ([0, 4], [1, 2, 3], 5)."},"skills":["Declared Event and Relative Complement Partition"],"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-20261009-advance-20261013-r06-event-0020","batch_id":"batch-20261009-advance-20261013-r06-event-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b52d971daaa950b6583279e670a90a817f738091fff4433ec9873f65e226421a","family_id":"qf-density13-event-complement-partition","seed":202610136020,"semantic_template":"qf-density13-event-complement-partition/[5,[1,4]]"},"fields":{"title":"Declared Event and Relative Complement Partition","stem":"S=Fin(5)={0,…,4}; the event family is explicitly F=P(S). E=[1, 4]⊆S. Give (increasing E, increasing S\\E, n).","answer":{"kind":"expression","value":"([1, 4], [0, 2, 3], 5)","explanation":"Take the complement inside the explicitly declared full event family. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is ([1, 4], [0, 2, 3], 5)."},"skills":["Declared Event and Relative Complement Partition"],"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-20261009-advance-20261013-r06-event-0021","batch_id":"batch-20261009-advance-20261013-r06-event-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e87994a708dccd28069e19f89dcb23159243b331c373c73fe53d69c02ccbf5c5","family_id":"qf-density13-event-complement-partition","seed":202610136021,"semantic_template":"qf-density13-event-complement-partition/[5,[0,1,4]]"},"fields":{"title":"Declared Event and Relative Complement Partition","stem":"S=Fin(5)={0,…,4}; the event family is explicitly F=P(S). E=[0, 1, 4]⊆S. Give (increasing E, increasing S\\E, n).","answer":{"kind":"expression","value":"([0, 1, 4], [2, 3], 5)","explanation":"Take the complement inside the explicitly declared full event family. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is ([0, 1, 4], [2, 3], 5)."},"skills":["Declared Event and Relative Complement Partition"],"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-20261009-advance-20261013-r06-event-0022","batch_id":"batch-20261009-advance-20261013-r06-event-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1f437fbd13bdca2a24b2905e1570863fb5c5420e66ff0861a8c5e23ce4228359","family_id":"qf-density13-event-complement-partition","seed":202610136022,"semantic_template":"qf-density13-event-complement-partition/[5,[2,4]]"},"fields":{"title":"Declared Event and Relative Complement Partition","stem":"S=Fin(5)={0,…,4}; the event family is explicitly F=P(S). E=[2, 4]⊆S. Give (increasing E, increasing S\\E, n).","answer":{"kind":"expression","value":"([2, 4], [0, 1, 3], 5)","explanation":"Take the complement inside the explicitly declared full event family. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is ([2, 4], [0, 1, 3], 5)."},"skills":["Declared Event and Relative Complement Partition"],"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-20261009-advance-20261013-r06-event-0023","batch_id":"batch-20261009-advance-20261013-r06-event-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"dadce1008877c074d1d58ad1821811ceaa37d9d05044b6b4ae0f93445e80f9cb","family_id":"qf-density13-event-complement-partition","seed":202610136023,"semantic_template":"qf-density13-event-complement-partition/[5,[0,2,4]]"},"fields":{"title":"Declared Event and Relative Complement Partition","stem":"S=Fin(5)={0,…,4}; the event family is explicitly F=P(S). E=[0, 2, 4]⊆S. Give (increasing E, increasing S\\E, n).","answer":{"kind":"expression","value":"([0, 2, 4], [1, 3], 5)","explanation":"Take the complement inside the explicitly declared full event family. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is ([0, 2, 4], [1, 3], 5)."},"skills":["Declared Event and Relative Complement Partition"],"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-20261009-advance-20261013-r06-event-0024","batch_id":"batch-20261009-advance-20261013-r06-event-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"67806c12d566166d3451193d6246bf0f7d5748b0697f462163001784d02a21cc","family_id":"qf-density13-event-complement-partition","seed":202610136024,"semantic_template":"qf-density13-event-complement-partition/[5,[1,2,4]]"},"fields":{"title":"Declared Event and Relative Complement Partition","stem":"S=Fin(5)={0,…,4}; the event family is explicitly F=P(S). E=[1, 2, 4]⊆S. Give (increasing E, increasing S\\E, n).","answer":{"kind":"expression","value":"([1, 2, 4], [0, 3], 5)","explanation":"Take the complement inside the explicitly declared full event family. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is ([1, 2, 4], [0, 3], 5)."},"skills":["Declared Event and Relative Complement Partition"],"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-20261009-advance-20261013-r06-event-0025","batch_id":"batch-20261009-advance-20261013-r06-event-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"313a4e5bf033af50dd95e5a5e4011474771c38dae07aa3038aa88fad901f2810","family_id":"qf-density13-event-complement-partition","seed":202610136025,"semantic_template":"qf-density13-event-complement-partition/[5,[0,1,2,4]]"},"fields":{"title":"Declared Event and Relative Complement Partition","stem":"S=Fin(5)={0,…,4}; the event family is explicitly F=P(S). E=[0, 1, 2, 4]⊆S. Give (increasing E, increasing S\\E, n).","answer":{"kind":"expression","value":"([0, 1, 2, 4], [3], 5)","explanation":"Take the complement inside the explicitly declared full event family. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is ([0, 1, 2, 4], [3], 5)."},"skills":["Declared Event and Relative Complement Partition"],"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-20261009-advance-20261013-r06-event-0026","batch_id":"batch-20261009-advance-20261013-r06-event-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a01ce1a588c54639b68d31c857514cd598e9494f28fb6c1773251de1c17929bc","family_id":"qf-density13-event-boolean-algebra-pair","seed":202610136026,"semantic_template":"qf-density13-event-boolean-algebra-pair/[1,[],[]]"},"fields":{"title":"Event Union, Intersection, and Outside Complement","stem":"S=Fin(1)={0,…,0}; the event family is explicitly F=P(S). A=[], B=[] are subsets of S. Give (A∪B, A∩B, S\\(A∪B)), each deduplicated and increasing.","answer":{"kind":"expression","value":"([], [], [0])","explanation":"Enumerate all three sets in the same sample space without confusing an outcome with an event. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is ([], [], [0])."},"skills":["Event Union, Intersection, and Outside Complement"],"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-20261009-advance-20261013-r06-event-0027","batch_id":"batch-20261009-advance-20261013-r06-event-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"04b78f564730a3c6f4b7d1584ca938b745bf61bf58283f090e0e1c009499547f","family_id":"qf-density13-event-boolean-algebra-pair","seed":202610136027,"semantic_template":"qf-density13-event-boolean-algebra-pair/[1,[0],[0]]"},"fields":{"title":"Event Union, Intersection, and Outside Complement","stem":"S=Fin(1)={0,…,0}; the event family is explicitly F=P(S). A=[0], B=[0] are subsets of S. Give (A∪B, A∩B, S\\(A∪B)), each deduplicated and increasing.","answer":{"kind":"expression","value":"([0], [0], [])","explanation":"Enumerate all three sets in the same sample space without confusing an outcome with an event. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is ([0], [0], [])."},"skills":["Event Union, Intersection, and Outside Complement"],"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-20261009-advance-20261013-r06-event-0028","batch_id":"batch-20261009-advance-20261013-r06-event-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b0cfa0e2301409df16494743bde5a64cda84645e22bc512570a9ec8e6d1e2385","family_id":"qf-density13-event-boolean-algebra-pair","seed":202610136028,"semantic_template":"qf-density13-event-boolean-algebra-pair/[5,[0],[]]"},"fields":{"title":"Event Union, Intersection, and Outside Complement","stem":"S=Fin(5)={0,…,4}; the event family is explicitly F=P(S). A=[0], B=[] are subsets of S. Give (A∪B, A∩B, S\\(A∪B)), each deduplicated and increasing.","answer":{"kind":"expression","value":"([0], [], [1, 2, 3, 4])","explanation":"Enumerate all three sets in the same sample space without confusing an outcome with an event. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is ([0], [], [1, 2, 3, 4])."},"skills":["Event Union, Intersection, and Outside Complement"],"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-20261009-advance-20261013-r06-event-0029","batch_id":"batch-20261009-advance-20261013-r06-event-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"867d3f711d912ce0878fc57bbd4a4ffe5d8f94e3d73ad0c17efe4d184009dc75","family_id":"qf-density13-event-boolean-algebra-pair","seed":202610136029,"semantic_template":"qf-density13-event-boolean-algebra-pair/[5,[1],[1]]"},"fields":{"title":"Event Union, Intersection, and Outside Complement","stem":"S=Fin(5)={0,…,4}; the event family is explicitly F=P(S). A=[1], B=[1] are subsets of S. Give (A∪B, A∩B, S\\(A∪B)), each deduplicated and increasing.","answer":{"kind":"expression","value":"([1], [1], [0, 2, 3, 4])","explanation":"Enumerate all three sets in the same sample space without confusing an outcome with an event. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is ([1], [1], [0, 2, 3, 4])."},"skills":["Event Union, Intersection, and Outside Complement"],"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-20261009-advance-20261013-r06-event-0030","batch_id":"batch-20261009-advance-20261013-r06-event-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"505563d05581076d535e41315cb6e08bf2caee01696e52f15bb0116a9084aa95","family_id":"qf-density13-event-boolean-algebra-pair","seed":202610136030,"semantic_template":"qf-density13-event-boolean-algebra-pair/[5,[0,1],[]]"},"fields":{"title":"Event Union, Intersection, and Outside Complement","stem":"S=Fin(5)={0,…,4}; the event family is explicitly F=P(S). A=[0, 1], B=[] are subsets of S. Give (A∪B, A∩B, S\\(A∪B)), each deduplicated and increasing.","answer":{"kind":"expression","value":"([0, 1], [], [2, 3, 4])","explanation":"Enumerate all three sets in the same sample space without confusing an outcome with an event. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is ([0, 1], [], [2, 3, 4])."},"skills":["Event Union, Intersection, and Outside Complement"],"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-20261009-advance-20261013-r06-event-0031","batch_id":"batch-20261009-advance-20261013-r06-event-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b8c0ba46c2dc337114d38a56a0ae207388fe3df85fabb634282740544d93c5e8","family_id":"qf-density13-event-boolean-algebra-pair","seed":202610136031,"semantic_template":"qf-density13-event-boolean-algebra-pair/[5,[2],[2]]"},"fields":{"title":"Event Union, Intersection, and Outside Complement","stem":"S=Fin(5)={0,…,4}; the event family is explicitly F=P(S). A=[2], B=[2] are subsets of S. Give (A∪B, A∩B, S\\(A∪B)), each deduplicated and increasing.","answer":{"kind":"expression","value":"([2], [2], [0, 1, 3, 4])","explanation":"Enumerate all three sets in the same sample space without confusing an outcome with an event. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is ([2], [2], [0, 1, 3, 4])."},"skills":["Event Union, Intersection, and Outside Complement"],"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-20261009-advance-20261013-r06-event-0032","batch_id":"batch-20261009-advance-20261013-r06-event-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8769de00380890bee14305155e235bdc7a21e96cd45b94dd2b36a179729a59fa","family_id":"qf-density13-event-boolean-algebra-pair","seed":202610136032,"semantic_template":"qf-density13-event-boolean-algebra-pair/[5,[0,2],[]]"},"fields":{"title":"Event Union, Intersection, and Outside Complement","stem":"S=Fin(5)={0,…,4}; the event family is explicitly F=P(S). A=[0, 2], B=[] are subsets of S. Give (A∪B, A∩B, S\\(A∪B)), each deduplicated and increasing.","answer":{"kind":"expression","value":"([0, 2], [], [1, 3, 4])","explanation":"Enumerate all three sets in the same sample space without confusing an outcome with an event. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is ([0, 2], [], [1, 3, 4])."},"skills":["Event Union, Intersection, and Outside Complement"],"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-20261009-advance-20261013-r06-event-0033","batch_id":"batch-20261009-advance-20261013-r06-event-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6bd6fd284677025f4239a67b28cdea8022e203631f157952ff1bb106a9aa2bef","family_id":"qf-density13-event-boolean-algebra-pair","seed":202610136033,"semantic_template":"qf-density13-event-boolean-algebra-pair/[5,[1,2],[1]]"},"fields":{"title":"Event Union, Intersection, and Outside Complement","stem":"S=Fin(5)={0,…,4}; the event family is explicitly F=P(S). A=[1, 2], B=[1] are subsets of S. Give (A∪B, A∩B, S\\(A∪B)), each deduplicated and increasing.","answer":{"kind":"expression","value":"([1, 2], [1], [0, 3, 4])","explanation":"Enumerate all three sets in the same sample space without confusing an outcome with an event. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is ([1, 2], [1], [0, 3, 4])."},"skills":["Event Union, Intersection, and Outside Complement"],"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-20261009-advance-20261013-r06-event-0034","batch_id":"batch-20261009-advance-20261013-r06-event-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9a3a5638a2383a1ec6e4fee4a9cd72a5a5e55fcdcd103e0cef82e24e4247073c","family_id":"qf-density13-event-boolean-algebra-pair","seed":202610136034,"semantic_template":"qf-density13-event-boolean-algebra-pair/[5,[0,1,2],[]]"},"fields":{"title":"Event Union, Intersection, and Outside Complement","stem":"S=Fin(5)={0,…,4}; the event family is explicitly F=P(S). A=[0, 1, 2], B=[] are subsets of S. Give (A∪B, A∩B, S\\(A∪B)), each deduplicated and increasing.","answer":{"kind":"expression","value":"([0, 1, 2], [], [3, 4])","explanation":"Enumerate all three sets in the same sample space without confusing an outcome with an event. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is ([0, 1, 2], [], [3, 4])."},"skills":["Event Union, Intersection, and Outside Complement"],"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-20261009-advance-20261013-r06-event-0035","batch_id":"batch-20261009-advance-20261013-r06-event-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"61529c9d91ce16180b943e6cdac7bed56d3a34a5f48c6de073aaf25a5c500e0c","family_id":"qf-density13-event-boolean-algebra-pair","seed":202610136035,"semantic_template":"qf-density13-event-boolean-algebra-pair/[5,[3],[3]]"},"fields":{"title":"Event Union, Intersection, and Outside Complement","stem":"S=Fin(5)={0,…,4}; the event family is explicitly F=P(S). A=[3], B=[3] are subsets of S. Give (A∪B, A∩B, S\\(A∪B)), each deduplicated and increasing.","answer":{"kind":"expression","value":"([3], [3], [0, 1, 2, 4])","explanation":"Enumerate all three sets in the same sample space without confusing an outcome with an event. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is ([3], [3], [0, 1, 2, 4])."},"skills":["Event Union, Intersection, and Outside Complement"],"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-20261009-advance-20261013-r06-event-0036","batch_id":"batch-20261009-advance-20261013-r06-event-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a8858a02b9eb96fe605d24a5349ac3793103b1a2eae868cd1c7f20117da13ea1","family_id":"qf-density13-event-boolean-algebra-pair","seed":202610136036,"semantic_template":"qf-density13-event-boolean-algebra-pair/[5,[0,3],[]]"},"fields":{"title":"Event Union, Intersection, and Outside Complement","stem":"S=Fin(5)={0,…,4}; the event family is explicitly F=P(S). A=[0, 3], B=[] are subsets of S. Give (A∪B, A∩B, S\\(A∪B)), each deduplicated and increasing.","answer":{"kind":"expression","value":"([0, 3], [], [1, 2, 4])","explanation":"Enumerate all three sets in the same sample space without confusing an outcome with an event. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is ([0, 3], [], [1, 2, 4])."},"skills":["Event Union, Intersection, and Outside Complement"],"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-20261009-advance-20261013-r06-event-0037","batch_id":"batch-20261009-advance-20261013-r06-event-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5a8ae43757c88677e2a1ef42d054a554188ea6a13534836905f996623c7609d3","family_id":"qf-density13-event-boolean-algebra-pair","seed":202610136037,"semantic_template":"qf-density13-event-boolean-algebra-pair/[5,[1,3],[1]]"},"fields":{"title":"Event Union, Intersection, and Outside Complement","stem":"S=Fin(5)={0,…,4}; the event family is explicitly F=P(S). A=[1, 3], B=[1] are subsets of S. Give (A∪B, A∩B, S\\(A∪B)), each deduplicated and increasing.","answer":{"kind":"expression","value":"([1, 3], [1], [0, 2, 4])","explanation":"Enumerate all three sets in the same sample space without confusing an outcome with an event. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is ([1, 3], [1], [0, 2, 4])."},"skills":["Event Union, Intersection, and Outside Complement"],"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-20261009-advance-20261013-r06-event-0038","batch_id":"batch-20261009-advance-20261013-r06-event-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"dcd92c15979f05311f1f21c9d853f71a98f7c0bd2fc425cdf0bf14912ae637fd","family_id":"qf-density13-event-boolean-algebra-pair","seed":202610136038,"semantic_template":"qf-density13-event-boolean-algebra-pair/[5,[0,1,3],[]]"},"fields":{"title":"Event Union, Intersection, and Outside Complement","stem":"S=Fin(5)={0,…,4}; the event family is explicitly F=P(S). A=[0, 1, 3], B=[] are subsets of S. Give (A∪B, A∩B, S\\(A∪B)), each deduplicated and increasing.","answer":{"kind":"expression","value":"([0, 1, 3], [], [2, 4])","explanation":"Enumerate all three sets in the same sample space without confusing an outcome with an event. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is ([0, 1, 3], [], [2, 4])."},"skills":["Event Union, Intersection, and Outside Complement"],"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-20261009-advance-20261013-r06-event-0039","batch_id":"batch-20261009-advance-20261013-r06-event-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9ca72e4a7473e0096f9cf71d3f66c559d18590b300fc3727cda4c292e6913b5b","family_id":"qf-density13-event-boolean-algebra-pair","seed":202610136039,"semantic_template":"qf-density13-event-boolean-algebra-pair/[5,[2,3],[2]]"},"fields":{"title":"Event Union, Intersection, and Outside Complement","stem":"S=Fin(5)={0,…,4}; the event family is explicitly F=P(S). A=[2, 3], B=[2] are subsets of S. Give (A∪B, A∩B, S\\(A∪B)), each deduplicated and increasing.","answer":{"kind":"expression","value":"([2, 3], [2], [0, 1, 4])","explanation":"Enumerate all three sets in the same sample space without confusing an outcome with an event. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is ([2, 3], [2], [0, 1, 4])."},"skills":["Event Union, Intersection, and Outside Complement"],"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-20261009-advance-20261013-r06-event-0040","batch_id":"batch-20261009-advance-20261013-r06-event-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ad5ee8770e45975871b0beb00396b375d84401edd53e73a0ac9f6a03f4311b27","family_id":"qf-density13-event-boolean-algebra-pair","seed":202610136040,"semantic_template":"qf-density13-event-boolean-algebra-pair/[5,[0,2,3],[]]"},"fields":{"title":"Event Union, Intersection, and Outside Complement","stem":"S=Fin(5)={0,…,4}; the event family is explicitly F=P(S). A=[0, 2, 3], B=[] are subsets of S. Give (A∪B, A∩B, S\\(A∪B)), each deduplicated and increasing.","answer":{"kind":"expression","value":"([0, 2, 3], [], [1, 4])","explanation":"Enumerate all three sets in the same sample space without confusing an outcome with an event. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is ([0, 2, 3], [], [1, 4])."},"skills":["Event Union, Intersection, and Outside Complement"],"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-20261009-advance-20261013-r06-event-0041","batch_id":"batch-20261009-advance-20261013-r06-event-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e507f237e2d7dc7306061ee689a8eca6fc25d818d412771cd927751cef948eda","family_id":"qf-density13-event-boolean-algebra-pair","seed":202610136041,"semantic_template":"qf-density13-event-boolean-algebra-pair/[5,[1,2,3],[1]]"},"fields":{"title":"Event Union, Intersection, and Outside Complement","stem":"S=Fin(5)={0,…,4}; the event family is explicitly F=P(S). A=[1, 2, 3], B=[1] are subsets of S. Give (A∪B, A∩B, S\\(A∪B)), each deduplicated and increasing.","answer":{"kind":"expression","value":"([1, 2, 3], [1], [0, 4])","explanation":"Enumerate all three sets in the same sample space without confusing an outcome with an event. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is ([1, 2, 3], [1], [0, 4])."},"skills":["Event Union, Intersection, and Outside Complement"],"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-20261009-advance-20261013-r06-event-0042","batch_id":"batch-20261009-advance-20261013-r06-event-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6c1916f5ea3c51f9a33ebf01aa0a82f7ac11a19a5da5e3e50ac8c89446ab43cb","family_id":"qf-density13-event-boolean-algebra-pair","seed":202610136042,"semantic_template":"qf-density13-event-boolean-algebra-pair/[5,[0,1,2,3],[]]"},"fields":{"title":"Event Union, Intersection, and Outside Complement","stem":"S=Fin(5)={0,…,4}; the event family is explicitly F=P(S). A=[0, 1, 2, 3], B=[] are subsets of S. Give (A∪B, A∩B, S\\(A∪B)), each deduplicated and increasing.","answer":{"kind":"expression","value":"([0, 1, 2, 3], [], [4])","explanation":"Enumerate all three sets in the same sample space without confusing an outcome with an event. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is ([0, 1, 2, 3], [], [4])."},"skills":["Event Union, Intersection, and Outside Complement"],"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-20261009-advance-20261013-r06-event-0043","batch_id":"batch-20261009-advance-20261013-r06-event-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"833bfbcfebb928ab6e615503c2b767fc2f5c3ff781343f054d2f1be021c1d165","family_id":"qf-density13-event-boolean-algebra-pair","seed":202610136043,"semantic_template":"qf-density13-event-boolean-algebra-pair/[5,[4],[4]]"},"fields":{"title":"Event Union, Intersection, and Outside Complement","stem":"S=Fin(5)={0,…,4}; the event family is explicitly F=P(S). A=[4], B=[4] are subsets of S. Give (A∪B, A∩B, S\\(A∪B)), each deduplicated and increasing.","answer":{"kind":"expression","value":"([4], [4], [0, 1, 2, 3])","explanation":"Enumerate all three sets in the same sample space without confusing an outcome with an event. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is ([4], [4], [0, 1, 2, 3])."},"skills":["Event Union, Intersection, and Outside Complement"],"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-20261009-advance-20261013-r06-event-0044","batch_id":"batch-20261009-advance-20261013-r06-event-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ecac73633ad66bef3d2c1a4321fe67e708ae2382ef56101133777b8243f16b53","family_id":"qf-density13-event-boolean-algebra-pair","seed":202610136044,"semantic_template":"qf-density13-event-boolean-algebra-pair/[5,[0,4],[]]"},"fields":{"title":"Event Union, Intersection, and Outside Complement","stem":"S=Fin(5)={0,…,4}; the event family is explicitly F=P(S). A=[0, 4], B=[] are subsets of S. Give (A∪B, A∩B, S\\(A∪B)), each deduplicated and increasing.","answer":{"kind":"expression","value":"([0, 4], [], [1, 2, 3])","explanation":"Enumerate all three sets in the same sample space without confusing an outcome with an event. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is ([0, 4], [], [1, 2, 3])."},"skills":["Event Union, Intersection, and Outside Complement"],"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-20261009-advance-20261013-r06-event-0045","batch_id":"batch-20261009-advance-20261013-r06-event-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0a3c212adca84afde49d1f541b1051c21f9d744de5ab19723110895e3980378f","family_id":"qf-density13-event-boolean-algebra-pair","seed":202610136045,"semantic_template":"qf-density13-event-boolean-algebra-pair/[5,[1,4],[1]]"},"fields":{"title":"Event Union, Intersection, and Outside Complement","stem":"S=Fin(5)={0,…,4}; the event family is explicitly F=P(S). A=[1, 4], B=[1] are subsets of S. Give (A∪B, A∩B, S\\(A∪B)), each deduplicated and increasing.","answer":{"kind":"expression","value":"([1, 4], [1], [0, 2, 3])","explanation":"Enumerate all three sets in the same sample space without confusing an outcome with an event. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is ([1, 4], [1], [0, 2, 3])."},"skills":["Event Union, Intersection, and Outside Complement"],"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-20261009-advance-20261013-r06-event-0046","batch_id":"batch-20261009-advance-20261013-r06-event-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"651e5344085f327e7d17b7b1abb35e3aec484a9c5647a7177bef911ab0dd05fd","family_id":"qf-density13-event-boolean-algebra-pair","seed":202610136046,"semantic_template":"qf-density13-event-boolean-algebra-pair/[5,[0,1,4],[]]"},"fields":{"title":"Event Union, Intersection, and Outside Complement","stem":"S=Fin(5)={0,…,4}; the event family is explicitly F=P(S). A=[0, 1, 4], B=[] are subsets of S. Give (A∪B, A∩B, S\\(A∪B)), each deduplicated and increasing.","answer":{"kind":"expression","value":"([0, 1, 4], [], [2, 3])","explanation":"Enumerate all three sets in the same sample space without confusing an outcome with an event. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is ([0, 1, 4], [], [2, 3])."},"skills":["Event Union, Intersection, and Outside Complement"],"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-20261009-advance-20261013-r06-event-0047","batch_id":"batch-20261009-advance-20261013-r06-event-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"191471be81806a3312c6655271101062f5cc95d2e3d95f73a5f488ee35899b5e","family_id":"qf-density13-event-boolean-algebra-pair","seed":202610136047,"semantic_template":"qf-density13-event-boolean-algebra-pair/[5,[2,4],[2]]"},"fields":{"title":"Event Union, Intersection, and Outside Complement","stem":"S=Fin(5)={0,…,4}; the event family is explicitly F=P(S). A=[2, 4], B=[2] are subsets of S. Give (A∪B, A∩B, S\\(A∪B)), each deduplicated and increasing.","answer":{"kind":"expression","value":"([2, 4], [2], [0, 1, 3])","explanation":"Enumerate all three sets in the same sample space without confusing an outcome with an event. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is ([2, 4], [2], [0, 1, 3])."},"skills":["Event Union, Intersection, and Outside Complement"],"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-20261009-advance-20261013-r06-event-0048","batch_id":"batch-20261009-advance-20261013-r06-event-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2027640d0ba13fc9a61c1b9fb7acc02c236b299dc54a8215bfb15627ed5f6f86","family_id":"qf-density13-event-boolean-algebra-pair","seed":202610136048,"semantic_template":"qf-density13-event-boolean-algebra-pair/[5,[0,2,4],[]]"},"fields":{"title":"Event Union, Intersection, and Outside Complement","stem":"S=Fin(5)={0,…,4}; the event family is explicitly F=P(S). A=[0, 2, 4], B=[] are subsets of S. Give (A∪B, A∩B, S\\(A∪B)), each deduplicated and increasing.","answer":{"kind":"expression","value":"([0, 2, 4], [], [1, 3])","explanation":"Enumerate all three sets in the same sample space without confusing an outcome with an event. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is ([0, 2, 4], [], [1, 3])."},"skills":["Event Union, Intersection, and Outside Complement"],"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-20261009-advance-20261013-r06-event-0049","batch_id":"batch-20261009-advance-20261013-r06-event-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f12b9529ffeac20244862dd808caf7c78c284ae4638c4e282cde9162b201ecec","family_id":"qf-density13-event-boolean-algebra-pair","seed":202610136049,"semantic_template":"qf-density13-event-boolean-algebra-pair/[5,[1,2,4],[1]]"},"fields":{"title":"Event Union, Intersection, and Outside Complement","stem":"S=Fin(5)={0,…,4}; the event family is explicitly F=P(S). A=[1, 2, 4], B=[1] are subsets of S. Give (A∪B, A∩B, S\\(A∪B)), each deduplicated and increasing.","answer":{"kind":"expression","value":"([1, 2, 4], [1], [0, 3])","explanation":"Enumerate all three sets in the same sample space without confusing an outcome with an event. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is ([1, 2, 4], [1], [0, 3])."},"skills":["Event Union, Intersection, and Outside Complement"],"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-20261009-advance-20261013-r06-event-0050","batch_id":"batch-20261009-advance-20261013-r06-event-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"49419257f523ecd40371dc7ccadecfc34d89c162381c950ada0b5bc8d876694b","family_id":"qf-density13-event-boolean-algebra-pair","seed":202610136050,"semantic_template":"qf-density13-event-boolean-algebra-pair/[5,[0,1,2,4],[]]"},"fields":{"title":"Event Union, Intersection, and Outside Complement","stem":"S=Fin(5)={0,…,4}; the event family is explicitly F=P(S). A=[0, 1, 2, 4], B=[] are subsets of S. Give (A∪B, A∩B, S\\(A∪B)), each deduplicated and increasing.","answer":{"kind":"expression","value":"([0, 1, 2, 4], [], [3])","explanation":"Enumerate all three sets in the same sample space without confusing an outcome with an event. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is ([0, 1, 2, 4], [], [3])."},"skills":["Event Union, Intersection, and Outside Complement"],"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-20261009-advance-20261013-r06-event-0051","batch_id":"batch-20261009-advance-20261013-r06-event-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ee2d07efaaeced6fe2685b9b6bdcabbf0cb7903de5140a9976f0c2ffb2a66af1","family_id":"qf-density13-event-partition-family-qualification","seed":202610136051,"semantic_template":"qf-density13-event-partition-family-qualification/[true,[],[]]"},"fields":{"title":"Partition-Union Event Eligibility and Least Containing Event","stem":"S=Fin(1)={0,…,0}; F consists of unions of complete disjoint nonempty blocks=[[0]]. Candidate E=[]⊆S. Give (is E a union of whole blocks?, partially hit block-member lists, smallest F set containing E). Sort inside each block and order blocks numerically lexicographically.","answer":{"kind":"expression","value":"(Yes, [], [])","explanation":"Each block is taken wholly or not at all; the least containing event takes every block hit by E, while fully hit blocks are not partial hits. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is (Yes, [], [])."},"skills":["Partition-Union Event Eligibility and Least Containing Event"],"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-20261009-advance-20261013-r06-event-0052","batch_id":"batch-20261009-advance-20261013-r06-event-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"666acfa66ab86b0df5561b2aaa93ff4d487cf54df55752f54e5daf2d79f920c2","family_id":"qf-density13-event-partition-family-qualification","seed":202610136052,"semantic_template":"qf-density13-event-partition-family-qualification/[true,[],[0,1]]"},"fields":{"title":"Partition-Union Event Eligibility and Least Containing Event","stem":"S=Fin(3)={0,…,2}; F consists of unions of complete disjoint nonempty blocks=[[0, 1], [2]]. Candidate E=[0, 1]⊆S. Give (is E a union of whole blocks?, partially hit block-member lists, smallest F set containing E). Sort inside each block and order blocks numerically lexicographically.","answer":{"kind":"expression","value":"(Yes, [], [0, 1])","explanation":"Each block is taken wholly or not at all; the least containing event takes every block hit by E, while fully hit blocks are not partial hits. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is (Yes, [], [0, 1])."},"skills":["Partition-Union Event Eligibility and Least Containing Event"],"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-20261009-advance-20261013-r06-event-0053","batch_id":"batch-20261009-advance-20261013-r06-event-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2bbf89a89fd57279774d012f00a1347374046da6374b1e4595dbfd108829d579","family_id":"qf-density13-event-partition-family-qualification","seed":202610136053,"semantic_template":"qf-density13-event-partition-family-qualification/[false,[[0,1]],[0,1]]"},"fields":{"title":"Partition-Union Event Eligibility and Least Containing Event","stem":"S=Fin(3)={0,…,2}; F consists of unions of complete disjoint nonempty blocks=[[0, 1], [2]]. Candidate E=[0]⊆S. Give (is E a union of whole blocks?, partially hit block-member lists, smallest F set containing E). Sort inside each block and order blocks numerically lexicographically.","answer":{"kind":"expression","value":"(No, [[0, 1]], [0, 1])","explanation":"Each block is taken wholly or not at all; the least containing event takes every block hit by E, while fully hit blocks are not partial hits. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is (No, [[0, 1]], [0, 1])."},"skills":["Partition-Union Event Eligibility and Least Containing Event"],"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-20261009-advance-20261013-r06-event-0054","batch_id":"batch-20261009-advance-20261013-r06-event-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a68131e47fce8a823e27a249cd27867f305d4489c1ed014fa7601e86b9e6f6e5","family_id":"qf-density13-event-partition-family-qualification","seed":202610136054,"semantic_template":"qf-density13-event-partition-family-qualification/[false,[[0,2]],[0,2]]"},"fields":{"title":"Partition-Union Event Eligibility and Least Containing Event","stem":"S=Fin(8)={0,…,7}; F consists of unions of complete disjoint nonempty blocks=[[0, 2], [1], [3], [4], [5], [6], [7]]. Candidate E=[0]⊆S. Give (is E a union of whole blocks?, partially hit block-member lists, smallest F set containing E). Sort inside each block and order blocks numerically lexicographically.","answer":{"kind":"expression","value":"(No, [[0, 2]], [0, 2])","explanation":"Each block is taken wholly or not at all; the least containing event takes every block hit by E, while fully hit blocks are not partial hits. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is (No, [[0, 2]], [0, 2])."},"skills":["Partition-Union Event Eligibility and Least Containing Event"],"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-20261009-advance-20261013-r06-event-0055","batch_id":"batch-20261009-advance-20261013-r06-event-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"06dba60251c3295195626198d7a80f27797a155d193020e1413433c608409b88","family_id":"qf-density13-event-partition-family-qualification","seed":202610136055,"semantic_template":"qf-density13-event-partition-family-qualification/[false,[[1,2]],[1,2]]"},"fields":{"title":"Partition-Union Event Eligibility and Least Containing Event","stem":"S=Fin(8)={0,…,7}; F consists of unions of complete disjoint nonempty blocks=[[1, 2], [0], [3], [4], [5], [6], [7]]. Candidate E=[1]⊆S. Give (is E a union of whole blocks?, partially hit block-member lists, smallest F set containing E). Sort inside each block and order blocks numerically lexicographically.","answer":{"kind":"expression","value":"(No, [[1, 2]], [1, 2])","explanation":"Each block is taken wholly or not at all; the least containing event takes every block hit by E, while fully hit blocks are not partial hits. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is (No, [[1, 2]], [1, 2])."},"skills":["Partition-Union Event Eligibility and Least Containing Event"],"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-20261009-advance-20261013-r06-event-0056","batch_id":"batch-20261009-advance-20261013-r06-event-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bf6748ed03ec131586eb51e667f7f34c236fd6b7dba85a9a693cf2aa6b0c625c","family_id":"qf-density13-event-partition-family-qualification","seed":202610136056,"semantic_template":"qf-density13-event-partition-family-qualification/[false,[[0,1,2]],[0,1,2]]"},"fields":{"title":"Partition-Union Event Eligibility and Least Containing Event","stem":"S=Fin(8)={0,…,7}; F consists of unions of complete disjoint nonempty blocks=[[0, 1, 2], [3], [4], [5], [6], [7]]. Candidate E=[0]⊆S. Give (is E a union of whole blocks?, partially hit block-member lists, smallest F set containing E). Sort inside each block and order blocks numerically lexicographically.","answer":{"kind":"expression","value":"(No, [[0, 1, 2]], [0, 1, 2])","explanation":"Each block is taken wholly or not at all; the least containing event takes every block hit by E, while fully hit blocks are not partial hits. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is (No, [[0, 1, 2]], [0, 1, 2])."},"skills":["Partition-Union Event Eligibility and Least Containing Event"],"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-20261009-advance-20261013-r06-event-0057","batch_id":"batch-20261009-advance-20261013-r06-event-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"efdfae3fb8c7229f1f2a542875027bdc351da5c583a4b4d780d9984548744b4b","family_id":"qf-density13-event-partition-family-qualification","seed":202610136057,"semantic_template":"qf-density13-event-partition-family-qualification/[false,[[0,3]],[0,3]]"},"fields":{"title":"Partition-Union Event Eligibility and Least Containing Event","stem":"S=Fin(8)={0,…,7}; F consists of unions of complete disjoint nonempty blocks=[[0, 3], [1], [2], [4], [5], [6], [7]]. Candidate E=[0]⊆S. Give (is E a union of whole blocks?, partially hit block-member lists, smallest F set containing E). Sort inside each block and order blocks numerically lexicographically.","answer":{"kind":"expression","value":"(No, [[0, 3]], [0, 3])","explanation":"Each block is taken wholly or not at all; the least containing event takes every block hit by E, while fully hit blocks are not partial hits. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is (No, [[0, 3]], [0, 3])."},"skills":["Partition-Union Event Eligibility and Least Containing Event"],"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-20261009-advance-20261013-r06-event-0058","batch_id":"batch-20261009-advance-20261013-r06-event-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7fe5a958dc4ec261245cc0c70f13c91de46c3c9ca927818e55505e3ea4a7625f","family_id":"qf-density13-event-partition-family-qualification","seed":202610136058,"semantic_template":"qf-density13-event-partition-family-qualification/[false,[[1,3]],[1,3]]"},"fields":{"title":"Partition-Union Event Eligibility and Least Containing Event","stem":"S=Fin(8)={0,…,7}; F consists of unions of complete disjoint nonempty blocks=[[1, 3], [0], [2], [4], [5], [6], [7]]. Candidate E=[1]⊆S. Give (is E a union of whole blocks?, partially hit block-member lists, smallest F set containing E). Sort inside each block and order blocks numerically lexicographically.","answer":{"kind":"expression","value":"(No, [[1, 3]], [1, 3])","explanation":"Each block is taken wholly or not at all; the least containing event takes every block hit by E, while fully hit blocks are not partial hits. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is (No, [[1, 3]], [1, 3])."},"skills":["Partition-Union Event Eligibility and Least Containing Event"],"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-20261009-advance-20261013-r06-event-0059","batch_id":"batch-20261009-advance-20261013-r06-event-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e7fff843a52fea727641f5a6f563c9a1d894704ebd03a8cdbce934658b96184e","family_id":"qf-density13-event-partition-family-qualification","seed":202610136059,"semantic_template":"qf-density13-event-partition-family-qualification/[false,[[0,1,3]],[0,1,3]]"},"fields":{"title":"Partition-Union Event Eligibility and Least Containing Event","stem":"S=Fin(8)={0,…,7}; F consists of unions of complete disjoint nonempty blocks=[[0, 1, 3], [2], [4], [5], [6], [7]]. Candidate E=[0]⊆S. Give (is E a union of whole blocks?, partially hit block-member lists, smallest F set containing E). Sort inside each block and order blocks numerically lexicographically.","answer":{"kind":"expression","value":"(No, [[0, 1, 3]], [0, 1, 3])","explanation":"Each block is taken wholly or not at all; the least containing event takes every block hit by E, while fully hit blocks are not partial hits. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is (No, [[0, 1, 3]], [0, 1, 3])."},"skills":["Partition-Union Event Eligibility and Least Containing Event"],"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-20261009-advance-20261013-r06-event-0060","batch_id":"batch-20261009-advance-20261013-r06-event-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a51aeaf6accf9b3c2e2d35e54fa195af74a3f82cc0a5d1907360453fa74d3712","family_id":"qf-density13-event-partition-family-qualification","seed":202610136060,"semantic_template":"qf-density13-event-partition-family-qualification/[false,[[2,3]],[2,3]]"},"fields":{"title":"Partition-Union Event Eligibility and Least Containing Event","stem":"S=Fin(8)={0,…,7}; F consists of unions of complete disjoint nonempty blocks=[[2, 3], [0], [1], [4], [5], [6], [7]]. Candidate E=[2]⊆S. Give (is E a union of whole blocks?, partially hit block-member lists, smallest F set containing E). Sort inside each block and order blocks numerically lexicographically.","answer":{"kind":"expression","value":"(No, [[2, 3]], [2, 3])","explanation":"Each block is taken wholly or not at all; the least containing event takes every block hit by E, while fully hit blocks are not partial hits. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is (No, [[2, 3]], [2, 3])."},"skills":["Partition-Union Event Eligibility and Least Containing Event"],"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-20261009-advance-20261013-r06-event-0061","batch_id":"batch-20261009-advance-20261013-r06-event-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e84f538a9c6022cefa22a26de88a1f212a762c88a538137ac628c7a6eaf4e70c","family_id":"qf-density13-event-partition-family-qualification","seed":202610136061,"semantic_template":"qf-density13-event-partition-family-qualification/[false,[[0,2,3]],[0,2,3]]"},"fields":{"title":"Partition-Union Event Eligibility and Least Containing Event","stem":"S=Fin(8)={0,…,7}; F consists of unions of complete disjoint nonempty blocks=[[0, 2, 3], [1], [4], [5], [6], [7]]. Candidate E=[0]⊆S. Give (is E a union of whole blocks?, partially hit block-member lists, smallest F set containing E). Sort inside each block and order blocks numerically lexicographically.","answer":{"kind":"expression","value":"(No, [[0, 2, 3]], [0, 2, 3])","explanation":"Each block is taken wholly or not at all; the least containing event takes every block hit by E, while fully hit blocks are not partial hits. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is (No, [[0, 2, 3]], [0, 2, 3])."},"skills":["Partition-Union Event Eligibility and Least Containing Event"],"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-20261009-advance-20261013-r06-event-0062","batch_id":"batch-20261009-advance-20261013-r06-event-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b4cd69847eab8733bc0049dbf8308deee696f3103003aeff480eee2c882f5c94","family_id":"qf-density13-event-partition-family-qualification","seed":202610136062,"semantic_template":"qf-density13-event-partition-family-qualification/[false,[[1,2,3]],[1,2,3]]"},"fields":{"title":"Partition-Union Event Eligibility and Least Containing Event","stem":"S=Fin(8)={0,…,7}; F consists of unions of complete disjoint nonempty blocks=[[1, 2, 3], [0], [4], [5], [6], [7]]. Candidate E=[1]⊆S. Give (is E a union of whole blocks?, partially hit block-member lists, smallest F set containing E). Sort inside each block and order blocks numerically lexicographically.","answer":{"kind":"expression","value":"(No, [[1, 2, 3]], [1, 2, 3])","explanation":"Each block is taken wholly or not at all; the least containing event takes every block hit by E, while fully hit blocks are not partial hits. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is (No, [[1, 2, 3]], [1, 2, 3])."},"skills":["Partition-Union Event Eligibility and Least Containing Event"],"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-20261009-advance-20261013-r06-event-0063","batch_id":"batch-20261009-advance-20261013-r06-event-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"41a07242c6731dbc33e03d0084d5493a3efd2c1d935380f596889a8684eb6a0c","family_id":"qf-density13-event-partition-family-qualification","seed":202610136063,"semantic_template":"qf-density13-event-partition-family-qualification/[false,[[0,1,2,3]],[0,1,2,3]]"},"fields":{"title":"Partition-Union Event Eligibility and Least Containing Event","stem":"S=Fin(8)={0,…,7}; F consists of unions of complete disjoint nonempty blocks=[[0, 1, 2, 3], [4], [5], [6], [7]]. Candidate E=[0]⊆S. Give (is E a union of whole blocks?, partially hit block-member lists, smallest F set containing E). Sort inside each block and order blocks numerically lexicographically.","answer":{"kind":"expression","value":"(No, [[0, 1, 2, 3]], [0, 1, 2, 3])","explanation":"Each block is taken wholly or not at all; the least containing event takes every block hit by E, while fully hit blocks are not partial hits. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is (No, [[0, 1, 2, 3]], [0, 1, 2, 3])."},"skills":["Partition-Union Event Eligibility and Least Containing Event"],"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-20261009-advance-20261013-r06-event-0064","batch_id":"batch-20261009-advance-20261013-r06-event-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a30b2ba9d46b665427d20147a221a05bc435f7c13b99f9b75710dee7dd9f4d86","family_id":"qf-density13-event-partition-family-qualification","seed":202610136064,"semantic_template":"qf-density13-event-partition-family-qualification/[false,[[0,4]],[0,4]]"},"fields":{"title":"Partition-Union Event Eligibility and Least Containing Event","stem":"S=Fin(8)={0,…,7}; F consists of unions of complete disjoint nonempty blocks=[[0, 4], [1], [2], [3], [5], [6], [7]]. Candidate E=[0]⊆S. Give (is E a union of whole blocks?, partially hit block-member lists, smallest F set containing E). Sort inside each block and order blocks numerically lexicographically.","answer":{"kind":"expression","value":"(No, [[0, 4]], [0, 4])","explanation":"Each block is taken wholly or not at all; the least containing event takes every block hit by E, while fully hit blocks are not partial hits. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is (No, [[0, 4]], [0, 4])."},"skills":["Partition-Union Event Eligibility and Least Containing Event"],"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-20261009-advance-20261013-r06-event-0065","batch_id":"batch-20261009-advance-20261013-r06-event-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"95124faadf3a3af229fed223ce75d338266cab9bc3609a0723cbfcd38eb02f88","family_id":"qf-density13-event-partition-family-qualification","seed":202610136065,"semantic_template":"qf-density13-event-partition-family-qualification/[false,[[1,4]],[1,4]]"},"fields":{"title":"Partition-Union Event Eligibility and Least Containing Event","stem":"S=Fin(8)={0,…,7}; F consists of unions of complete disjoint nonempty blocks=[[1, 4], [0], [2], [3], [5], [6], [7]]. Candidate E=[1]⊆S. Give (is E a union of whole blocks?, partially hit block-member lists, smallest F set containing E). Sort inside each block and order blocks numerically lexicographically.","answer":{"kind":"expression","value":"(No, [[1, 4]], [1, 4])","explanation":"Each block is taken wholly or not at all; the least containing event takes every block hit by E, while fully hit blocks are not partial hits. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is (No, [[1, 4]], [1, 4])."},"skills":["Partition-Union Event Eligibility and Least Containing Event"],"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-20261009-advance-20261013-r06-event-0066","batch_id":"batch-20261009-advance-20261013-r06-event-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"01f3c95dfdbf638eca0833cef27cc9bbbfd0dce53107dfd692a1754d4c5ed527","family_id":"qf-density13-event-partition-family-qualification","seed":202610136066,"semantic_template":"qf-density13-event-partition-family-qualification/[false,[[0,1,4]],[0,1,4]]"},"fields":{"title":"Partition-Union Event Eligibility and Least Containing Event","stem":"S=Fin(8)={0,…,7}; F consists of unions of complete disjoint nonempty blocks=[[0, 1, 4], [2], [3], [5], [6], [7]]. Candidate E=[0]⊆S. Give (is E a union of whole blocks?, partially hit block-member lists, smallest F set containing E). Sort inside each block and order blocks numerically lexicographically.","answer":{"kind":"expression","value":"(No, [[0, 1, 4]], [0, 1, 4])","explanation":"Each block is taken wholly or not at all; the least containing event takes every block hit by E, while fully hit blocks are not partial hits. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is (No, [[0, 1, 4]], [0, 1, 4])."},"skills":["Partition-Union Event Eligibility and Least Containing Event"],"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-20261009-advance-20261013-r06-event-0067","batch_id":"batch-20261009-advance-20261013-r06-event-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cc502170d438204c1c09c2ce6c90a9679baac6c0dc07f0c92d0cb48c2ed7340f","family_id":"qf-density13-event-partition-family-qualification","seed":202610136067,"semantic_template":"qf-density13-event-partition-family-qualification/[false,[[2,4]],[2,4]]"},"fields":{"title":"Partition-Union Event Eligibility and Least Containing Event","stem":"S=Fin(8)={0,…,7}; F consists of unions of complete disjoint nonempty blocks=[[2, 4], [0], [1], [3], [5], [6], [7]]. Candidate E=[2]⊆S. Give (is E a union of whole blocks?, partially hit block-member lists, smallest F set containing E). Sort inside each block and order blocks numerically lexicographically.","answer":{"kind":"expression","value":"(No, [[2, 4]], [2, 4])","explanation":"Each block is taken wholly or not at all; the least containing event takes every block hit by E, while fully hit blocks are not partial hits. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is (No, [[2, 4]], [2, 4])."},"skills":["Partition-Union Event Eligibility and Least Containing Event"],"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-20261009-advance-20261013-r06-event-0068","batch_id":"batch-20261009-advance-20261013-r06-event-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a76ee0794ce857174018252dbb56e03ee6f47c9c81451b627df8cc35cf6e1994","family_id":"qf-density13-event-partition-family-qualification","seed":202610136068,"semantic_template":"qf-density13-event-partition-family-qualification/[false,[[0,2,4]],[0,2,4]]"},"fields":{"title":"Partition-Union Event Eligibility and Least Containing Event","stem":"S=Fin(8)={0,…,7}; F consists of unions of complete disjoint nonempty blocks=[[0, 2, 4], [1], [3], [5], [6], [7]]. Candidate E=[0]⊆S. Give (is E a union of whole blocks?, partially hit block-member lists, smallest F set containing E). Sort inside each block and order blocks numerically lexicographically.","answer":{"kind":"expression","value":"(No, [[0, 2, 4]], [0, 2, 4])","explanation":"Each block is taken wholly or not at all; the least containing event takes every block hit by E, while fully hit blocks are not partial hits. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is (No, [[0, 2, 4]], [0, 2, 4])."},"skills":["Partition-Union Event Eligibility and Least Containing Event"],"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-20261009-advance-20261013-r06-event-0069","batch_id":"batch-20261009-advance-20261013-r06-event-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"63967b3af4c0b9ac751b20fbfbd181d6ddc44b2644dc38d861e2a4bffe30eb47","family_id":"qf-density13-event-partition-family-qualification","seed":202610136069,"semantic_template":"qf-density13-event-partition-family-qualification/[false,[[1,2,4]],[1,2,4]]"},"fields":{"title":"Partition-Union Event Eligibility and Least Containing Event","stem":"S=Fin(8)={0,…,7}; F consists of unions of complete disjoint nonempty blocks=[[1, 2, 4], [0], [3], [5], [6], [7]]. Candidate E=[1]⊆S. Give (is E a union of whole blocks?, partially hit block-member lists, smallest F set containing E). Sort inside each block and order blocks numerically lexicographically.","answer":{"kind":"expression","value":"(No, [[1, 2, 4]], [1, 2, 4])","explanation":"Each block is taken wholly or not at all; the least containing event takes every block hit by E, while fully hit blocks are not partial hits. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is (No, [[1, 2, 4]], [1, 2, 4])."},"skills":["Partition-Union Event Eligibility and Least Containing Event"],"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-20261009-advance-20261013-r06-event-0070","batch_id":"batch-20261009-advance-20261013-r06-event-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f485ba14eb16fd8e610a70b2a1ca594238838e315a8eeb35c7bda3b62a3d5ff9","family_id":"qf-density13-event-partition-family-qualification","seed":202610136070,"semantic_template":"qf-density13-event-partition-family-qualification/[false,[[0,1,2,4]],[0,1,2,4]]"},"fields":{"title":"Partition-Union Event Eligibility and Least Containing Event","stem":"S=Fin(8)={0,…,7}; F consists of unions of complete disjoint nonempty blocks=[[0, 1, 2, 4], [3], [5], [6], [7]]. Candidate E=[0]⊆S. Give (is E a union of whole blocks?, partially hit block-member lists, smallest F set containing E). Sort inside each block and order blocks numerically lexicographically.","answer":{"kind":"expression","value":"(No, [[0, 1, 2, 4]], [0, 1, 2, 4])","explanation":"Each block is taken wholly or not at all; the least containing event takes every block hit by E, while fully hit blocks are not partial hits. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is (No, [[0, 1, 2, 4]], [0, 1, 2, 4])."},"skills":["Partition-Union Event Eligibility and Least Containing Event"],"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-20261009-advance-20261013-r06-event-0071","batch_id":"batch-20261009-advance-20261013-r06-event-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b3386f4bcf8bda6d00d57dadf3a63c491772afda57666a5dc83f534c8f70814a","family_id":"qf-density13-event-partition-family-qualification","seed":202610136071,"semantic_template":"qf-density13-event-partition-family-qualification/[false,[[3,4]],[3,4]]"},"fields":{"title":"Partition-Union Event Eligibility and Least Containing Event","stem":"S=Fin(8)={0,…,7}; F consists of unions of complete disjoint nonempty blocks=[[3, 4], [0], [1], [2], [5], [6], [7]]. Candidate E=[3]⊆S. Give (is E a union of whole blocks?, partially hit block-member lists, smallest F set containing E). Sort inside each block and order blocks numerically lexicographically.","answer":{"kind":"expression","value":"(No, [[3, 4]], [3, 4])","explanation":"Each block is taken wholly or not at all; the least containing event takes every block hit by E, while fully hit blocks are not partial hits. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is (No, [[3, 4]], [3, 4])."},"skills":["Partition-Union Event Eligibility and Least Containing Event"],"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-20261009-advance-20261013-r06-event-0072","batch_id":"batch-20261009-advance-20261013-r06-event-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b6f65c9045b98fa2cc7c00070086881cff5ba2c61f701308f6b83b87e6df6f7b","family_id":"qf-density13-event-partition-family-qualification","seed":202610136072,"semantic_template":"qf-density13-event-partition-family-qualification/[false,[[0,3,4]],[0,3,4]]"},"fields":{"title":"Partition-Union Event Eligibility and Least Containing Event","stem":"S=Fin(8)={0,…,7}; F consists of unions of complete disjoint nonempty blocks=[[0, 3, 4], [1], [2], [5], [6], [7]]. Candidate E=[0]⊆S. Give (is E a union of whole blocks?, partially hit block-member lists, smallest F set containing E). Sort inside each block and order blocks numerically lexicographically.","answer":{"kind":"expression","value":"(No, [[0, 3, 4]], [0, 3, 4])","explanation":"Each block is taken wholly or not at all; the least containing event takes every block hit by E, while fully hit blocks are not partial hits. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is (No, [[0, 3, 4]], [0, 3, 4])."},"skills":["Partition-Union Event Eligibility and Least Containing Event"],"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-20261009-advance-20261013-r06-event-0073","batch_id":"batch-20261009-advance-20261013-r06-event-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7b6bfa46f196c94daf5300ac7bc32f613bbc0634e43d9d0483fadac2407d9f44","family_id":"qf-density13-event-partition-family-qualification","seed":202610136073,"semantic_template":"qf-density13-event-partition-family-qualification/[false,[[1,3,4]],[1,3,4]]"},"fields":{"title":"Partition-Union Event Eligibility and Least Containing Event","stem":"S=Fin(8)={0,…,7}; F consists of unions of complete disjoint nonempty blocks=[[1, 3, 4], [0], [2], [5], [6], [7]]. Candidate E=[1]⊆S. Give (is E a union of whole blocks?, partially hit block-member lists, smallest F set containing E). Sort inside each block and order blocks numerically lexicographically.","answer":{"kind":"expression","value":"(No, [[1, 3, 4]], [1, 3, 4])","explanation":"Each block is taken wholly or not at all; the least containing event takes every block hit by E, while fully hit blocks are not partial hits. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is (No, [[1, 3, 4]], [1, 3, 4])."},"skills":["Partition-Union Event Eligibility and Least Containing Event"],"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-20261009-advance-20261013-r06-event-0074","batch_id":"batch-20261009-advance-20261013-r06-event-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8683124c6e30493e7df804de95d057551bc790cda39716e38eb88ed5b47aec87","family_id":"qf-density13-event-partition-family-qualification","seed":202610136074,"semantic_template":"qf-density13-event-partition-family-qualification/[false,[[0,1,3,4]],[0,1,3,4]]"},"fields":{"title":"Partition-Union Event Eligibility and Least Containing Event","stem":"S=Fin(8)={0,…,7}; F consists of unions of complete disjoint nonempty blocks=[[0, 1, 3, 4], [2], [5], [6], [7]]. Candidate E=[0]⊆S. Give (is E a union of whole blocks?, partially hit block-member lists, smallest F set containing E). Sort inside each block and order blocks numerically lexicographically.","answer":{"kind":"expression","value":"(No, [[0, 1, 3, 4]], [0, 1, 3, 4])","explanation":"Each block is taken wholly or not at all; the least containing event takes every block hit by E, while fully hit blocks are not partial hits. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is (No, [[0, 1, 3, 4]], [0, 1, 3, 4])."},"skills":["Partition-Union Event Eligibility and Least Containing Event"],"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-20261009-advance-20261013-r06-event-0075","batch_id":"batch-20261009-advance-20261013-r06-event-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a2d41c96fb9458856270bae6d1853dfc25b89363d88fff5ab96b13e37c9023cd","family_id":"qf-density13-event-partition-family-qualification","seed":202610136075,"semantic_template":"qf-density13-event-partition-family-qualification/[false,[[2,3,4]],[2,3,4]]"},"fields":{"title":"Partition-Union Event Eligibility and Least Containing Event","stem":"S=Fin(8)={0,…,7}; F consists of unions of complete disjoint nonempty blocks=[[2, 3, 4], [0], [1], [5], [6], [7]]. Candidate E=[2]⊆S. Give (is E a union of whole blocks?, partially hit block-member lists, smallest F set containing E). Sort inside each block and order blocks numerically lexicographically.","answer":{"kind":"expression","value":"(No, [[2, 3, 4]], [2, 3, 4])","explanation":"Each block is taken wholly or not at all; the least containing event takes every block hit by E, while fully hit blocks are not partial hits. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is (No, [[2, 3, 4]], [2, 3, 4])."},"skills":["Partition-Union Event Eligibility and Least Containing Event"],"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-20261009-advance-20261013-r06-event-0076","batch_id":"batch-20261009-advance-20261013-r06-event-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"959a86db9f8289550684fdb79cab4225077ed88a0aeb0985982376725c3b87e0","family_id":"qf-density13-event-occurrence-first-hit","seed":202610136076,"semantic_template":"qf-density13-event-occurrence-first-hit/[]"},"fields":{"title":"Typed Observations, Event Occurrence, and First Hit","stem":"S=Fin(1)={0,…,0}; the event family is explicitly F=P(S). E=[], ordered outcomes=[]. Give (sample-eligibility mask, True/False/invalid-outcome occurrence list, first legal E hit’s zero-based index or none).","answer":{"kind":"expression","value":"([], [], none)","explanation":"Only a legal sample-space value belonging to E is an occurrence; an outside candidate is invalid-outcome, not False. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is ([], [], none)."},"skills":["Typed Observations, Event Occurrence, and First Hit"],"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-20261009-advance-20261013-r06-event-0077","batch_id":"batch-20261009-advance-20261013-r06-event-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5101b2c94493d72278af89545042a53433d7b4ec454849863d9447ccff461804","family_id":"qf-density13-event-occurrence-first-hit","seed":202610136077,"semantic_template":"qf-density13-event-occurrence-first-hit/[\"outside\",\"outside\",\"outside\"]"},"fields":{"title":"Typed Observations, Event Occurrence, and First Hit","stem":"S=Fin(3)={0,…,2}; the event family is explicitly F=P(S). E=[1], ordered outcomes=[100, 101, 102]. Give (sample-eligibility mask, True/False/invalid-outcome occurrence list, first legal E hit’s zero-based index or none).","answer":{"kind":"expression","value":"([No, No, No], [invalid-outcome, invalid-outcome, invalid-outcome], none)","explanation":"Only a legal sample-space value belonging to E is an occurrence; an outside candidate is invalid-outcome, not False. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is ([No, No, No], [invalid-outcome, invalid-outcome, invalid-outcome], none)."},"skills":["Typed Observations, Event Occurrence, and First Hit"],"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-20261009-advance-20261013-r06-event-0078","batch_id":"batch-20261009-advance-20261013-r06-event-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8f92a3d6ecc2a46dd9479217f4f984db8fd426b6987613fb1b87da6f1287844e","family_id":"qf-density13-event-occurrence-first-hit","seed":202610136078,"semantic_template":"qf-density13-event-occurrence-first-hit/[\"nonmember\",\"nonmember\",\"nonmember\"]"},"fields":{"title":"Typed Observations, Event Occurrence, and First Hit","stem":"S=Fin(3)={0,…,2}; the event family is explicitly F=P(S). E=[1], ordered outcomes=[0, 2, 0]. Give (sample-eligibility mask, True/False/invalid-outcome occurrence list, first legal E hit’s zero-based index or none).","answer":{"kind":"expression","value":"([Yes, Yes, Yes], [False, False, False], none)","explanation":"Only a legal sample-space value belonging to E is an occurrence; an outside candidate is invalid-outcome, not False. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is ([Yes, Yes, Yes], [False, False, False], none)."},"skills":["Typed Observations, Event Occurrence, and First Hit"],"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-20261009-advance-20261013-r06-event-0079","batch_id":"batch-20261009-advance-20261013-r06-event-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b824a62e48fbc1388bf7b883692f743ed70d434a553aa9b14b21c73ddd37f49f","family_id":"qf-density13-event-occurrence-first-hit","seed":202610136079,"semantic_template":"qf-density13-event-occurrence-first-hit/[\"member\",\"member\",\"member\"]"},"fields":{"title":"Typed Observations, Event Occurrence, and First Hit","stem":"S=Fin(3)={0,…,2}; the event family is explicitly F=P(S). E=[1], ordered outcomes=[1, 1, 1]. Give (sample-eligibility mask, True/False/invalid-outcome occurrence list, first legal E hit’s zero-based index or none).","answer":{"kind":"expression","value":"([Yes, Yes, Yes], [True, True, True], 0)","explanation":"Only a legal sample-space value belonging to E is an occurrence; an outside candidate is invalid-outcome, not False. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is ([Yes, Yes, Yes], [True, True, True], 0)."},"skills":["Typed Observations, Event Occurrence, and First Hit"],"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-20261009-advance-20261013-r06-event-0080","batch_id":"batch-20261009-advance-20261013-r06-event-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0c68aec56c37cc27fb5788f53697b233905042ccef7d44ee9e0bd9d638210957","family_id":"qf-density13-event-occurrence-first-hit","seed":202610136080,"semantic_template":"qf-density13-event-occurrence-first-hit/[\"nonmember\",\"outside\",\"outside\"]"},"fields":{"title":"Typed Observations, Event Occurrence, and First Hit","stem":"S=Fin(3)={0,…,2}; the event family is explicitly F=P(S). E=[1], ordered outcomes=[0, 101, 102]. Give (sample-eligibility mask, True/False/invalid-outcome occurrence list, first legal E hit’s zero-based index or none).","answer":{"kind":"expression","value":"([Yes, No, No], [False, invalid-outcome, invalid-outcome], none)","explanation":"Only a legal sample-space value belonging to E is an occurrence; an outside candidate is invalid-outcome, not False. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is ([Yes, No, No], [False, invalid-outcome, invalid-outcome], none)."},"skills":["Typed Observations, Event Occurrence, and First Hit"],"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-20261009-advance-20261013-r06-event-0081","batch_id":"batch-20261009-advance-20261013-r06-event-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5deb50f8469bf0d6771cba3ac928bf31887d884b45cc0d5388950c7c170226cf","family_id":"qf-density13-event-occurrence-first-hit","seed":202610136081,"semantic_template":"qf-density13-event-occurrence-first-hit/[\"member\",\"outside\",\"outside\"]"},"fields":{"title":"Typed Observations, Event Occurrence, and First Hit","stem":"S=Fin(3)={0,…,2}; the event family is explicitly F=P(S). E=[1], ordered outcomes=[1, 101, 102]. Give (sample-eligibility mask, True/False/invalid-outcome occurrence list, first legal E hit’s zero-based index or none).","answer":{"kind":"expression","value":"([Yes, No, No], [True, invalid-outcome, invalid-outcome], 0)","explanation":"Only a legal sample-space value belonging to E is an occurrence; an outside candidate is invalid-outcome, not False. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is ([Yes, No, No], [True, invalid-outcome, invalid-outcome], 0)."},"skills":["Typed Observations, Event Occurrence, and First Hit"],"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-20261009-advance-20261013-r06-event-0082","batch_id":"batch-20261009-advance-20261013-r06-event-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7ad77b1c5ed18dccb91911b44c0001a81b0993afb5c89140989172f8535a289f","family_id":"qf-density13-event-occurrence-first-hit","seed":202610136082,"semantic_template":"qf-density13-event-occurrence-first-hit/[\"outside\",\"nonmember\",\"outside\"]"},"fields":{"title":"Typed Observations, Event Occurrence, and First Hit","stem":"S=Fin(3)={0,…,2}; the event family is explicitly F=P(S). E=[1], ordered outcomes=[100, 2, 102]. Give (sample-eligibility mask, True/False/invalid-outcome occurrence list, first legal E hit’s zero-based index or none).","answer":{"kind":"expression","value":"([No, Yes, No], [invalid-outcome, False, invalid-outcome], none)","explanation":"Only a legal sample-space value belonging to E is an occurrence; an outside candidate is invalid-outcome, not False. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is ([No, Yes, No], [invalid-outcome, False, invalid-outcome], none)."},"skills":["Typed Observations, Event Occurrence, and First Hit"],"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-20261009-advance-20261013-r06-event-0083","batch_id":"batch-20261009-advance-20261013-r06-event-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4cb21088a359828b33ca07d99d5017efb8a57de3de3702fc828e580e01ff546e","family_id":"qf-density13-event-occurrence-first-hit","seed":202610136083,"semantic_template":"qf-density13-event-occurrence-first-hit/[\"nonmember\",\"nonmember\",\"outside\"]"},"fields":{"title":"Typed Observations, Event Occurrence, and First Hit","stem":"S=Fin(3)={0,…,2}; the event family is explicitly F=P(S). E=[1], ordered outcomes=[0, 2, 102]. Give (sample-eligibility mask, True/False/invalid-outcome occurrence list, first legal E hit’s zero-based index or none).","answer":{"kind":"expression","value":"([Yes, Yes, No], [False, False, invalid-outcome], none)","explanation":"Only a legal sample-space value belonging to E is an occurrence; an outside candidate is invalid-outcome, not False. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is ([Yes, Yes, No], [False, False, invalid-outcome], none)."},"skills":["Typed Observations, Event Occurrence, and First Hit"],"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-20261009-advance-20261013-r06-event-0084","batch_id":"batch-20261009-advance-20261013-r06-event-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0e6970605a4e96a4c285dd0036da9b4b239ae1102f6862033e01ac61ed576351","family_id":"qf-density13-event-occurrence-first-hit","seed":202610136084,"semantic_template":"qf-density13-event-occurrence-first-hit/[\"member\",\"nonmember\",\"outside\"]"},"fields":{"title":"Typed Observations, Event Occurrence, and First Hit","stem":"S=Fin(3)={0,…,2}; the event family is explicitly F=P(S). E=[1], ordered outcomes=[1, 2, 102]. Give (sample-eligibility mask, True/False/invalid-outcome occurrence list, first legal E hit’s zero-based index or none).","answer":{"kind":"expression","value":"([Yes, Yes, No], [True, False, invalid-outcome], 0)","explanation":"Only a legal sample-space value belonging to E is an occurrence; an outside candidate is invalid-outcome, not False. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is ([Yes, Yes, No], [True, False, invalid-outcome], 0)."},"skills":["Typed Observations, Event Occurrence, and First Hit"],"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-20261009-advance-20261013-r06-event-0085","batch_id":"batch-20261009-advance-20261013-r06-event-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"122c3507d1dbb208ce3af94adb5712f535564e5f16b409909e6e097cfeb00953","family_id":"qf-density13-event-occurrence-first-hit","seed":202610136085,"semantic_template":"qf-density13-event-occurrence-first-hit/[\"outside\",\"member\",\"outside\"]"},"fields":{"title":"Typed Observations, Event Occurrence, and First Hit","stem":"S=Fin(3)={0,…,2}; the event family is explicitly F=P(S). E=[1], ordered outcomes=[100, 1, 102]. Give (sample-eligibility mask, True/False/invalid-outcome occurrence list, first legal E hit’s zero-based index or none).","answer":{"kind":"expression","value":"([No, Yes, No], [invalid-outcome, True, invalid-outcome], 1)","explanation":"Only a legal sample-space value belonging to E is an occurrence; an outside candidate is invalid-outcome, not False. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is ([No, Yes, No], [invalid-outcome, True, invalid-outcome], 1)."},"skills":["Typed Observations, Event Occurrence, and First Hit"],"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-20261009-advance-20261013-r06-event-0086","batch_id":"batch-20261009-advance-20261013-r06-event-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"72b0bd5f1eb43b70498f7ddcbd92ac4acc8503cf09811f58a8eef7d7c4f6c9b6","family_id":"qf-density13-event-occurrence-first-hit","seed":202610136086,"semantic_template":"qf-density13-event-occurrence-first-hit/[\"nonmember\",\"member\",\"outside\"]"},"fields":{"title":"Typed Observations, Event Occurrence, and First Hit","stem":"S=Fin(3)={0,…,2}; the event family is explicitly F=P(S). E=[1], ordered outcomes=[0, 1, 102]. Give (sample-eligibility mask, True/False/invalid-outcome occurrence list, first legal E hit’s zero-based index or none).","answer":{"kind":"expression","value":"([Yes, Yes, No], [False, True, invalid-outcome], 1)","explanation":"Only a legal sample-space value belonging to E is an occurrence; an outside candidate is invalid-outcome, not False. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is ([Yes, Yes, No], [False, True, invalid-outcome], 1)."},"skills":["Typed Observations, Event Occurrence, and First Hit"],"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-20261009-advance-20261013-r06-event-0087","batch_id":"batch-20261009-advance-20261013-r06-event-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fe7db0edef950aa9a3295feb9d2bf16570c8ff4d2c677461e6c679027f931763","family_id":"qf-density13-event-occurrence-first-hit","seed":202610136087,"semantic_template":"qf-density13-event-occurrence-first-hit/[\"member\",\"member\",\"outside\"]"},"fields":{"title":"Typed Observations, Event Occurrence, and First Hit","stem":"S=Fin(3)={0,…,2}; the event family is explicitly F=P(S). E=[1], ordered outcomes=[1, 1, 102]. Give (sample-eligibility mask, True/False/invalid-outcome occurrence list, first legal E hit’s zero-based index or none).","answer":{"kind":"expression","value":"([Yes, Yes, No], [True, True, invalid-outcome], 0)","explanation":"Only a legal sample-space value belonging to E is an occurrence; an outside candidate is invalid-outcome, not False. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is ([Yes, Yes, No], [True, True, invalid-outcome], 0)."},"skills":["Typed Observations, Event Occurrence, and First Hit"],"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-20261009-advance-20261013-r06-event-0088","batch_id":"batch-20261009-advance-20261013-r06-event-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fc8f93ab2f354c11e3afe1f873154d068278c97c241ad4c8cf54276a3b28d681","family_id":"qf-density13-event-occurrence-first-hit","seed":202610136088,"semantic_template":"qf-density13-event-occurrence-first-hit/[\"outside\",\"outside\",\"nonmember\"]"},"fields":{"title":"Typed Observations, Event Occurrence, and First Hit","stem":"S=Fin(3)={0,…,2}; the event family is explicitly F=P(S). E=[1], ordered outcomes=[100, 101, 0]. Give (sample-eligibility mask, True/False/invalid-outcome occurrence list, first legal E hit’s zero-based index or none).","answer":{"kind":"expression","value":"([No, No, Yes], [invalid-outcome, invalid-outcome, False], none)","explanation":"Only a legal sample-space value belonging to E is an occurrence; an outside candidate is invalid-outcome, not False. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is ([No, No, Yes], [invalid-outcome, invalid-outcome, False], none)."},"skills":["Typed Observations, Event Occurrence, and First Hit"],"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-20261009-advance-20261013-r06-event-0089","batch_id":"batch-20261009-advance-20261013-r06-event-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4bd3ccc8fba100e74fedfc8f8cb95a95e850b3f5b1c7d079b79062853565f564","family_id":"qf-density13-event-occurrence-first-hit","seed":202610136089,"semantic_template":"qf-density13-event-occurrence-first-hit/[\"nonmember\",\"outside\",\"nonmember\"]"},"fields":{"title":"Typed Observations, Event Occurrence, and First Hit","stem":"S=Fin(3)={0,…,2}; the event family is explicitly F=P(S). E=[1], ordered outcomes=[0, 101, 0]. Give (sample-eligibility mask, True/False/invalid-outcome occurrence list, first legal E hit’s zero-based index or none).","answer":{"kind":"expression","value":"([Yes, No, Yes], [False, invalid-outcome, False], none)","explanation":"Only a legal sample-space value belonging to E is an occurrence; an outside candidate is invalid-outcome, not False. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is ([Yes, No, Yes], [False, invalid-outcome, False], none)."},"skills":["Typed Observations, Event Occurrence, and First Hit"],"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-20261009-advance-20261013-r06-event-0090","batch_id":"batch-20261009-advance-20261013-r06-event-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1a97ad481c67b96c4a1186ea06f79d29554957066e501b133a7180124ed85064","family_id":"qf-density13-event-occurrence-first-hit","seed":202610136090,"semantic_template":"qf-density13-event-occurrence-first-hit/[\"member\",\"outside\",\"nonmember\"]"},"fields":{"title":"Typed Observations, Event Occurrence, and First Hit","stem":"S=Fin(3)={0,…,2}; the event family is explicitly F=P(S). E=[1], ordered outcomes=[1, 101, 0]. Give (sample-eligibility mask, True/False/invalid-outcome occurrence list, first legal E hit’s zero-based index or none).","answer":{"kind":"expression","value":"([Yes, No, Yes], [True, invalid-outcome, False], 0)","explanation":"Only a legal sample-space value belonging to E is an occurrence; an outside candidate is invalid-outcome, not False. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is ([Yes, No, Yes], [True, invalid-outcome, False], 0)."},"skills":["Typed Observations, Event Occurrence, and First Hit"],"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-20261009-advance-20261013-r06-event-0091","batch_id":"batch-20261009-advance-20261013-r06-event-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f4c9848ab7b2ecd0bc760b6306f1a3818dc82a3556762188c01835001505c6a2","family_id":"qf-density13-event-occurrence-first-hit","seed":202610136091,"semantic_template":"qf-density13-event-occurrence-first-hit/[\"outside\",\"nonmember\",\"nonmember\"]"},"fields":{"title":"Typed Observations, Event Occurrence, and First Hit","stem":"S=Fin(3)={0,…,2}; the event family is explicitly F=P(S). E=[1], ordered outcomes=[100, 2, 0]. Give (sample-eligibility mask, True/False/invalid-outcome occurrence list, first legal E hit’s zero-based index or none).","answer":{"kind":"expression","value":"([No, Yes, Yes], [invalid-outcome, False, False], none)","explanation":"Only a legal sample-space value belonging to E is an occurrence; an outside candidate is invalid-outcome, not False. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is ([No, Yes, Yes], [invalid-outcome, False, False], none)."},"skills":["Typed Observations, Event Occurrence, and First Hit"],"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-20261009-advance-20261013-r06-event-0092","batch_id":"batch-20261009-advance-20261013-r06-event-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2001b48957cbcebd02dcb613055fedb667c439fd907acdecabec6353f58402a6","family_id":"qf-density13-event-occurrence-first-hit","seed":202610136092,"semantic_template":"qf-density13-event-occurrence-first-hit/[\"member\",\"nonmember\",\"nonmember\"]"},"fields":{"title":"Typed Observations, Event Occurrence, and First Hit","stem":"S=Fin(3)={0,…,2}; the event family is explicitly F=P(S). E=[1], ordered outcomes=[1, 2, 0]. Give (sample-eligibility mask, True/False/invalid-outcome occurrence list, first legal E hit’s zero-based index or none).","answer":{"kind":"expression","value":"([Yes, Yes, Yes], [True, False, False], 0)","explanation":"Only a legal sample-space value belonging to E is an occurrence; an outside candidate is invalid-outcome, not False. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is ([Yes, Yes, Yes], [True, False, False], 0)."},"skills":["Typed Observations, Event Occurrence, and First Hit"],"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-20261009-advance-20261013-r06-event-0093","batch_id":"batch-20261009-advance-20261013-r06-event-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a98812156039939cab559f120648f4fa87ee4afcd2edea5823c123898fdcf841","family_id":"qf-density13-event-occurrence-first-hit","seed":202610136093,"semantic_template":"qf-density13-event-occurrence-first-hit/[\"outside\",\"member\",\"nonmember\"]"},"fields":{"title":"Typed Observations, Event Occurrence, and First Hit","stem":"S=Fin(3)={0,…,2}; the event family is explicitly F=P(S). E=[1], ordered outcomes=[100, 1, 0]. Give (sample-eligibility mask, True/False/invalid-outcome occurrence list, first legal E hit’s zero-based index or none).","answer":{"kind":"expression","value":"([No, Yes, Yes], [invalid-outcome, True, False], 1)","explanation":"Only a legal sample-space value belonging to E is an occurrence; an outside candidate is invalid-outcome, not False. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is ([No, Yes, Yes], [invalid-outcome, True, False], 1)."},"skills":["Typed Observations, Event Occurrence, and First Hit"],"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-20261009-advance-20261013-r06-event-0094","batch_id":"batch-20261009-advance-20261013-r06-event-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ecbf7fbb80d94e3ffe50704df58539e19bf1b722e8d48810c2efe0385fc988b9","family_id":"qf-density13-event-occurrence-first-hit","seed":202610136094,"semantic_template":"qf-density13-event-occurrence-first-hit/[\"nonmember\",\"member\",\"nonmember\"]"},"fields":{"title":"Typed Observations, Event Occurrence, and First Hit","stem":"S=Fin(3)={0,…,2}; the event family is explicitly F=P(S). E=[1], ordered outcomes=[0, 1, 0]. Give (sample-eligibility mask, True/False/invalid-outcome occurrence list, first legal E hit’s zero-based index or none).","answer":{"kind":"expression","value":"([Yes, Yes, Yes], [False, True, False], 1)","explanation":"Only a legal sample-space value belonging to E is an occurrence; an outside candidate is invalid-outcome, not False. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is ([Yes, Yes, Yes], [False, True, False], 1)."},"skills":["Typed Observations, Event Occurrence, and First Hit"],"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-20261009-advance-20261013-r06-event-0095","batch_id":"batch-20261009-advance-20261013-r06-event-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cb1487ea5b39120bbc216fa1d20c14801e20cb43c164d6683602c9307dd7d382","family_id":"qf-density13-event-occurrence-first-hit","seed":202610136095,"semantic_template":"qf-density13-event-occurrence-first-hit/[\"member\",\"member\",\"nonmember\"]"},"fields":{"title":"Typed Observations, Event Occurrence, and First Hit","stem":"S=Fin(3)={0,…,2}; the event family is explicitly F=P(S). E=[1], ordered outcomes=[1, 1, 0]. Give (sample-eligibility mask, True/False/invalid-outcome occurrence list, first legal E hit’s zero-based index or none).","answer":{"kind":"expression","value":"([Yes, Yes, Yes], [True, True, False], 0)","explanation":"Only a legal sample-space value belonging to E is an occurrence; an outside candidate is invalid-outcome, not False. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is ([Yes, Yes, Yes], [True, True, False], 0)."},"skills":["Typed Observations, Event Occurrence, and First Hit"],"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-20261009-advance-20261013-r06-event-0096","batch_id":"batch-20261009-advance-20261013-r06-event-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bf89910576e8fcfb2ffd5cdcd36558849d0c3ab2ebb9d5645c093f89652515c2","family_id":"qf-density13-event-occurrence-first-hit","seed":202610136096,"semantic_template":"qf-density13-event-occurrence-first-hit/[\"outside\",\"outside\",\"member\"]"},"fields":{"title":"Typed Observations, Event Occurrence, and First Hit","stem":"S=Fin(3)={0,…,2}; the event family is explicitly F=P(S). E=[1], ordered outcomes=[100, 101, 1]. Give (sample-eligibility mask, True/False/invalid-outcome occurrence list, first legal E hit’s zero-based index or none).","answer":{"kind":"expression","value":"([No, No, Yes], [invalid-outcome, invalid-outcome, True], 2)","explanation":"Only a legal sample-space value belonging to E is an occurrence; an outside candidate is invalid-outcome, not False. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is ([No, No, Yes], [invalid-outcome, invalid-outcome, True], 2)."},"skills":["Typed Observations, Event Occurrence, and First Hit"],"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-20261009-advance-20261013-r06-event-0097","batch_id":"batch-20261009-advance-20261013-r06-event-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b1da755124d99d712e5c16d2f1fdb70f09ada7c449e5a7b959c0bf058d33253a","family_id":"qf-density13-event-occurrence-first-hit","seed":202610136097,"semantic_template":"qf-density13-event-occurrence-first-hit/[\"nonmember\",\"outside\",\"member\"]"},"fields":{"title":"Typed Observations, Event Occurrence, and First Hit","stem":"S=Fin(3)={0,…,2}; the event family is explicitly F=P(S). E=[1], ordered outcomes=[0, 101, 1]. Give (sample-eligibility mask, True/False/invalid-outcome occurrence list, first legal E hit’s zero-based index or none).","answer":{"kind":"expression","value":"([Yes, No, Yes], [False, invalid-outcome, True], 2)","explanation":"Only a legal sample-space value belonging to E is an occurrence; an outside candidate is invalid-outcome, not False. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is ([Yes, No, Yes], [False, invalid-outcome, True], 2)."},"skills":["Typed Observations, Event Occurrence, and First Hit"],"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-20261009-advance-20261013-r06-event-0098","batch_id":"batch-20261009-advance-20261013-r06-event-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9415e745990cbcce4c9297feaf7d2d0055c09d62edf4a52b48a767a37062a436","family_id":"qf-density13-event-occurrence-first-hit","seed":202610136098,"semantic_template":"qf-density13-event-occurrence-first-hit/[\"member\",\"outside\",\"member\"]"},"fields":{"title":"Typed Observations, Event Occurrence, and First Hit","stem":"S=Fin(3)={0,…,2}; the event family is explicitly F=P(S). E=[1], ordered outcomes=[1, 101, 1]. Give (sample-eligibility mask, True/False/invalid-outcome occurrence list, first legal E hit’s zero-based index or none).","answer":{"kind":"expression","value":"([Yes, No, Yes], [True, invalid-outcome, True], 0)","explanation":"Only a legal sample-space value belonging to E is an occurrence; an outside candidate is invalid-outcome, not False. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is ([Yes, No, Yes], [True, invalid-outcome, True], 0)."},"skills":["Typed Observations, Event Occurrence, and First Hit"],"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-20261009-advance-20261013-r06-event-0099","batch_id":"batch-20261009-advance-20261013-r06-event-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fed44c60d07eb546837af7690da3253dc44db7392f47cb9b9be4fe7fd82764df","family_id":"qf-density13-event-occurrence-first-hit","seed":202610136099,"semantic_template":"qf-density13-event-occurrence-first-hit/[\"outside\",\"nonmember\",\"member\"]"},"fields":{"title":"Typed Observations, Event Occurrence, and First Hit","stem":"S=Fin(3)={0,…,2}; the event family is explicitly F=P(S). E=[1], ordered outcomes=[100, 2, 1]. Give (sample-eligibility mask, True/False/invalid-outcome occurrence list, first legal E hit’s zero-based index or none).","answer":{"kind":"expression","value":"([No, Yes, Yes], [invalid-outcome, False, True], 2)","explanation":"Only a legal sample-space value belonging to E is an occurrence; an outside candidate is invalid-outcome, not False. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is ([No, Yes, Yes], [invalid-outcome, False, True], 2)."},"skills":["Typed Observations, Event Occurrence, and First Hit"],"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-20261009-advance-20261013-r06-event-0100","batch_id":"batch-20261009-advance-20261013-r06-event-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a3a7c6df94c6d2dfd9f1611e339a323480fc6e2a67cd803c224ac5a873c8e3ea","family_id":"qf-density13-event-occurrence-first-hit","seed":202610136100,"semantic_template":"qf-density13-event-occurrence-first-hit/[\"nonmember\",\"nonmember\",\"member\"]"},"fields":{"title":"Typed Observations, Event Occurrence, and First Hit","stem":"S=Fin(3)={0,…,2}; the event family is explicitly F=P(S). E=[1], ordered outcomes=[0, 2, 1]. Give (sample-eligibility mask, True/False/invalid-outcome occurrence list, first legal E hit’s zero-based index or none).","answer":{"kind":"expression","value":"([Yes, Yes, Yes], [False, False, True], 2)","explanation":"Only a legal sample-space value belonging to E is an occurrence; an outside candidate is invalid-outcome, not False. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is ([Yes, Yes, Yes], [False, False, True], 2)."},"skills":["Typed Observations, Event Occurrence, and First Hit"],"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."}}