{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r03-relation-0001","batch_id":"batch-20261003-advance-20261004-r03-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-relation"],"title_zh":"二元關係的有序對成員","stem_zh":"二元關係的來源 A=∅、目標 B=∅，R=∅⊆A×B。有序對 (0, 1) 是否屬於 R？","answer":{"kind":"short_text","value":"否","explanation_zh":"(0, 1)∈R=否。逐一核對完整有序對及方向，不能自行交換兩分量。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["二元關係的有序對成員"]},"variant":{"family_id":"qf-relation-pair-membership","seed":202610043001,"parameters":{"A":[],"B":[],"R":[],"operation":"pair_membership","x":0,"y":1,"semantic_template":"qf-relation-pair-membership/{\"A\":[],\"B\":[],\"R\":[],\"operation\":\"pair_membership\",\"x\":0,\"y\":1}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r03-relation-0002","batch_id":"batch-20261003-advance-20261004-r03-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-relation"],"title_zh":"二元關係的有序對成員","stem_zh":"二元關係的來源 A={0}、目標 B={1}，R=∅⊆A×B。有序對 (1, 0) 是否屬於 R？","answer":{"kind":"short_text","value":"否","explanation_zh":"(1, 0)∈R=否。逐一核對完整有序對及方向，不能自行交換兩分量。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["二元關係的有序對成員"]},"variant":{"family_id":"qf-relation-pair-membership","seed":202610043002,"parameters":{"A":[0],"B":[1],"R":[],"operation":"pair_membership","x":1,"y":0,"semantic_template":"qf-relation-pair-membership/{\"A\":[0],\"B\":[1],\"R\":[],\"operation\":\"pair_membership\",\"x\":1,\"y\":0}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r03-relation-0003","batch_id":"batch-20261003-advance-20261004-r03-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-relation"],"title_zh":"二元關係的有序對成員","stem_zh":"二元關係的來源 A={-10, -9, -8}、目標 B={-6, -5, -4}，R={(-10, -6), (-10, -5)}⊆A×B。有序對 (-8, -6) 是否屬於 R？","answer":{"kind":"short_text","value":"否","explanation_zh":"(-8, -6)∈R=否。逐一核對完整有序對及方向，不能自行交換兩分量。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["二元關係的有序對成員"]},"variant":{"family_id":"qf-relation-pair-membership","seed":202610043003,"parameters":{"A":[-10,-9,-8],"B":[-6,-5,-4],"R":[[-10,-6],[-10,-5]],"operation":"pair_membership","x":-8,"y":-6,"semantic_template":"qf-relation-pair-membership/{\"A\":[-10,-9,-8],\"B\":[-6,-5,-4],\"R\":[[-10,-6],[-10,-5]],\"operation\":\"pair_membership\",\"x\":-8,\"y\":-6}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r03-relation-0004","batch_id":"batch-20261003-advance-20261004-r03-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-relation"],"title_zh":"二元關係的有序對成員","stem_zh":"二元關係的來源 A={-9, -8, -7}、目標 B={-5, -4, -3}，R={(-9, -4), (-8, -4)}⊆A×B。有序對 (-9, -4) 是否屬於 R？","answer":{"kind":"short_text","value":"是","explanation_zh":"(-9, -4)∈R=是。逐一核對完整有序對及方向，不能自行交換兩分量。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["二元關係的有序對成員"]},"variant":{"family_id":"qf-relation-pair-membership","seed":202610043004,"parameters":{"A":[-9,-8,-7],"B":[-5,-4,-3],"R":[[-9,-4],[-8,-4]],"operation":"pair_membership","x":-9,"y":-4,"semantic_template":"qf-relation-pair-membership/{\"A\":[-9,-8,-7],\"B\":[-5,-4,-3],\"R\":[[-9,-4],[-8,-4]],\"operation\":\"pair_membership\",\"x\":-9,\"y\":-4}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r03-relation-0005","batch_id":"batch-20261003-advance-20261004-r03-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-relation"],"title_zh":"二元關係的有序對成員","stem_zh":"二元關係的來源 A={-8, -7, -6}、目標 B={-4, -3, -2}，R={(-8, -4), (-7, -3), (-6, -2)}⊆A×B。有序對 (-4, -8) 是否屬於 R？","answer":{"kind":"short_text","value":"否","explanation_zh":"(-4, -8)∈R=否。逐一核對完整有序對及方向，不能自行交換兩分量。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["二元關係的有序對成員"]},"variant":{"family_id":"qf-relation-pair-membership","seed":202610043005,"parameters":{"A":[-8,-7,-6],"B":[-4,-3,-2],"R":[[-8,-4],[-7,-3],[-6,-2]],"operation":"pair_membership","x":-4,"y":-8,"semantic_template":"qf-relation-pair-membership/{\"A\":[-8,-7,-6],\"B\":[-4,-3,-2],\"R\":[[-8,-4],[-7,-3],[-6,-2]],\"operation\":\"pair_membership\",\"x\":-4,\"y\":-8}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r03-relation-0006","batch_id":"batch-20261003-advance-20261004-r03-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-relation"],"title_zh":"二元關係的有序對成員","stem_zh":"二元關係的來源 A={-7, -6, -5}、目標 B={-3, -2, -1}，R={(-6, -3), (-5, -2)}⊆A×B。有序對 (-5, -3) 是否屬於 R？","answer":{"kind":"short_text","value":"否","explanation_zh":"(-5, -3)∈R=否。逐一核對完整有序對及方向，不能自行交換兩分量。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["二元關係的有序對成員"]},"variant":{"family_id":"qf-relation-pair-membership","seed":202610043006,"parameters":{"A":[-7,-6,-5],"B":[-3,-2,-1],"R":[[-6,-3],[-5,-2]],"operation":"pair_membership","x":-5,"y":-3,"semantic_template":"qf-relation-pair-membership/{\"A\":[-7,-6,-5],\"B\":[-3,-2,-1],\"R\":[[-6,-3],[-5,-2]],\"operation\":\"pair_membership\",\"x\":-5,\"y\":-3}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r03-relation-0007","batch_id":"batch-20261003-advance-20261004-r03-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-relation"],"title_zh":"二元關係的有序對成員","stem_zh":"二元關係的來源 A={-6, -5, -4}、目標 B={-2, -1, 0}，R=∅⊆A×B。有序對 (-6, -2) 是否屬於 R？","answer":{"kind":"short_text","value":"否","explanation_zh":"(-6, -2)∈R=否。逐一核對完整有序對及方向，不能自行交換兩分量。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["二元關係的有序對成員"]},"variant":{"family_id":"qf-relation-pair-membership","seed":202610043007,"parameters":{"A":[-6,-5,-4],"B":[-2,-1,0],"R":[],"operation":"pair_membership","x":-6,"y":-2,"semantic_template":"qf-relation-pair-membership/{\"A\":[-6,-5,-4],\"B\":[-2,-1,0],\"R\":[],\"operation\":\"pair_membership\",\"x\":-6,\"y\":-2}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r03-relation-0008","batch_id":"batch-20261003-advance-20261004-r03-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-relation"],"title_zh":"二元關係的有序對成員","stem_zh":"二元關係的來源 A={-5, -4, -3}、目標 B={-1, 0, 1}，R={(-5, -1)}⊆A×B。有序對 (-1, -5) 是否屬於 R？","answer":{"kind":"short_text","value":"否","explanation_zh":"(-1, -5)∈R=否。逐一核對完整有序對及方向，不能自行交換兩分量。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["二元關係的有序對成員"]},"variant":{"family_id":"qf-relation-pair-membership","seed":202610043008,"parameters":{"A":[-5,-4,-3],"B":[-1,0,1],"R":[[-5,-1]],"operation":"pair_membership","x":-1,"y":-5,"semantic_template":"qf-relation-pair-membership/{\"A\":[-5,-4,-3],\"B\":[-1,0,1],\"R\":[[-5,-1]],\"operation\":\"pair_membership\",\"x\":-1,\"y\":-5}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r03-relation-0009","batch_id":"batch-20261003-advance-20261004-r03-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-relation"],"title_zh":"二元關係的有序對成員","stem_zh":"二元關係的來源 A={-4, -3, -2}、目標 B={0, 1, 2}，R={(-4, 0), (-4, 1)}⊆A×B。有序對 (-2, 0) 是否屬於 R？","answer":{"kind":"short_text","value":"否","explanation_zh":"(-2, 0)∈R=否。逐一核對完整有序對及方向，不能自行交換兩分量。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["二元關係的有序對成員"]},"variant":{"family_id":"qf-relation-pair-membership","seed":202610043009,"parameters":{"A":[-4,-3,-2],"B":[0,1,2],"R":[[-4,0],[-4,1]],"operation":"pair_membership","x":-2,"y":0,"semantic_template":"qf-relation-pair-membership/{\"A\":[-4,-3,-2],\"B\":[0,1,2],\"R\":[[-4,0],[-4,1]],\"operation\":\"pair_membership\",\"x\":-2,\"y\":0}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r03-relation-0010","batch_id":"batch-20261003-advance-20261004-r03-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-relation"],"title_zh":"二元關係的有序對成員","stem_zh":"二元關係的來源 A={-3, -2, -1}、目標 B={1, 2, 3}，R={(-3, 2), (-2, 2)}⊆A×B。有序對 (-3, 2) 是否屬於 R？","answer":{"kind":"short_text","value":"是","explanation_zh":"(-3, 2)∈R=是。逐一核對完整有序對及方向，不能自行交換兩分量。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["二元關係的有序對成員"]},"variant":{"family_id":"qf-relation-pair-membership","seed":202610043010,"parameters":{"A":[-3,-2,-1],"B":[1,2,3],"R":[[-3,2],[-2,2]],"operation":"pair_membership","x":-3,"y":2,"semantic_template":"qf-relation-pair-membership/{\"A\":[-3,-2,-1],\"B\":[1,2,3],\"R\":[[-3,2],[-2,2]],\"operation\":\"pair_membership\",\"x\":-3,\"y\":2}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r03-relation-0011","batch_id":"batch-20261003-advance-20261004-r03-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-relation"],"title_zh":"二元關係的有序對成員","stem_zh":"二元關係的來源 A={-2, -1, 0}、目標 B={2, 3, 4}，R={(-2, 2), (-1, 3), (0, 4)}⊆A×B。有序對 (2, -2) 是否屬於 R？","answer":{"kind":"short_text","value":"否","explanation_zh":"(2, -2)∈R=否。逐一核對完整有序對及方向，不能自行交換兩分量。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["二元關係的有序對成員"]},"variant":{"family_id":"qf-relation-pair-membership","seed":202610043011,"parameters":{"A":[-2,-1,0],"B":[2,3,4],"R":[[-2,2],[-1,3],[0,4]],"operation":"pair_membership","x":2,"y":-2,"semantic_template":"qf-relation-pair-membership/{\"A\":[-2,-1,0],\"B\":[2,3,4],\"R\":[[-2,2],[-1,3],[0,4]],\"operation\":\"pair_membership\",\"x\":2,\"y\":-2}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r03-relation-0012","batch_id":"batch-20261003-advance-20261004-r03-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-relation"],"title_zh":"二元關係的有序對成員","stem_zh":"二元關係的來源 A={-1, 0, 1}、目標 B={3, 4, 5}，R={(0, 3), (1, 4)}⊆A×B。有序對 (1, 3) 是否屬於 R？","answer":{"kind":"short_text","value":"否","explanation_zh":"(1, 3)∈R=否。逐一核對完整有序對及方向，不能自行交換兩分量。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["二元關係的有序對成員"]},"variant":{"family_id":"qf-relation-pair-membership","seed":202610043012,"parameters":{"A":[-1,0,1],"B":[3,4,5],"R":[[0,3],[1,4]],"operation":"pair_membership","x":1,"y":3,"semantic_template":"qf-relation-pair-membership/{\"A\":[-1,0,1],\"B\":[3,4,5],\"R\":[[0,3],[1,4]],\"operation\":\"pair_membership\",\"x\":1,\"y\":3}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r03-relation-0013","batch_id":"batch-20261003-advance-20261004-r03-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-relation"],"title_zh":"二元關係的有序對成員","stem_zh":"二元關係的來源 A={0, 1, 2}、目標 B={4, 5, 6}，R=∅⊆A×B。有序對 (0, 4) 是否屬於 R？","answer":{"kind":"short_text","value":"否","explanation_zh":"(0, 4)∈R=否。逐一核對完整有序對及方向，不能自行交換兩分量。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["二元關係的有序對成員"]},"variant":{"family_id":"qf-relation-pair-membership","seed":202610043013,"parameters":{"A":[0,1,2],"B":[4,5,6],"R":[],"operation":"pair_membership","x":0,"y":4,"semantic_template":"qf-relation-pair-membership/{\"A\":[0,1,2],\"B\":[4,5,6],\"R\":[],\"operation\":\"pair_membership\",\"x\":0,\"y\":4}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r03-relation-0014","batch_id":"batch-20261003-advance-20261004-r03-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-relation"],"title_zh":"二元關係的有序對成員","stem_zh":"二元關係的來源 A={1, 2, 3}、目標 B={5, 6, 7}，R={(1, 5)}⊆A×B。有序對 (5, 1) 是否屬於 R？","answer":{"kind":"short_text","value":"否","explanation_zh":"(5, 1)∈R=否。逐一核對完整有序對及方向，不能自行交換兩分量。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["二元關係的有序對成員"]},"variant":{"family_id":"qf-relation-pair-membership","seed":202610043014,"parameters":{"A":[1,2,3],"B":[5,6,7],"R":[[1,5]],"operation":"pair_membership","x":5,"y":1,"semantic_template":"qf-relation-pair-membership/{\"A\":[1,2,3],\"B\":[5,6,7],\"R\":[[1,5]],\"operation\":\"pair_membership\",\"x\":5,\"y\":1}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r03-relation-0015","batch_id":"batch-20261003-advance-20261004-r03-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-relation"],"title_zh":"二元關係的有序對成員","stem_zh":"二元關係的來源 A={2, 3, 4}、目標 B={6, 7, 8}，R={(2, 6), (2, 7)}⊆A×B。有序對 (4, 6) 是否屬於 R？","answer":{"kind":"short_text","value":"否","explanation_zh":"(4, 6)∈R=否。逐一核對完整有序對及方向，不能自行交換兩分量。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["二元關係的有序對成員"]},"variant":{"family_id":"qf-relation-pair-membership","seed":202610043015,"parameters":{"A":[2,3,4],"B":[6,7,8],"R":[[2,6],[2,7]],"operation":"pair_membership","x":4,"y":6,"semantic_template":"qf-relation-pair-membership/{\"A\":[2,3,4],\"B\":[6,7,8],\"R\":[[2,6],[2,7]],\"operation\":\"pair_membership\",\"x\":4,\"y\":6}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r03-relation-0016","batch_id":"batch-20261003-advance-20261004-r03-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-relation"],"title_zh":"二元關係的有序對成員","stem_zh":"二元關係的來源 A={3, 4, 5}、目標 B={7, 8, 9}，R={(3, 8), (4, 8)}⊆A×B。有序對 (3, 8) 是否屬於 R？","answer":{"kind":"short_text","value":"是","explanation_zh":"(3, 8)∈R=是。逐一核對完整有序對及方向，不能自行交換兩分量。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["二元關係的有序對成員"]},"variant":{"family_id":"qf-relation-pair-membership","seed":202610043016,"parameters":{"A":[3,4,5],"B":[7,8,9],"R":[[3,8],[4,8]],"operation":"pair_membership","x":3,"y":8,"semantic_template":"qf-relation-pair-membership/{\"A\":[3,4,5],\"B\":[7,8,9],\"R\":[[3,8],[4,8]],\"operation\":\"pair_membership\",\"x\":3,\"y\":8}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r03-relation-0017","batch_id":"batch-20261003-advance-20261004-r03-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-relation"],"title_zh":"二元關係的有序對成員","stem_zh":"二元關係的來源 A={4, 5, 6}、目標 B={8, 9, 10}，R={(4, 8), (5, 9), (6, 10)}⊆A×B。有序對 (8, 4) 是否屬於 R？","answer":{"kind":"short_text","value":"否","explanation_zh":"(8, 4)∈R=否。逐一核對完整有序對及方向，不能自行交換兩分量。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["二元關係的有序對成員"]},"variant":{"family_id":"qf-relation-pair-membership","seed":202610043017,"parameters":{"A":[4,5,6],"B":[8,9,10],"R":[[4,8],[5,9],[6,10]],"operation":"pair_membership","x":8,"y":4,"semantic_template":"qf-relation-pair-membership/{\"A\":[4,5,6],\"B\":[8,9,10],\"R\":[[4,8],[5,9],[6,10]],\"operation\":\"pair_membership\",\"x\":8,\"y\":4}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r03-relation-0018","batch_id":"batch-20261003-advance-20261004-r03-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-relation"],"title_zh":"二元關係的有序對成員","stem_zh":"二元關係的來源 A={5, 6, 7}、目標 B={9, 10, 11}，R={(6, 9), (7, 10)}⊆A×B。有序對 (7, 9) 是否屬於 R？","answer":{"kind":"short_text","value":"否","explanation_zh":"(7, 9)∈R=否。逐一核對完整有序對及方向，不能自行交換兩分量。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["二元關係的有序對成員"]},"variant":{"family_id":"qf-relation-pair-membership","seed":202610043018,"parameters":{"A":[5,6,7],"B":[9,10,11],"R":[[6,9],[7,10]],"operation":"pair_membership","x":7,"y":9,"semantic_template":"qf-relation-pair-membership/{\"A\":[5,6,7],\"B\":[9,10,11],\"R\":[[6,9],[7,10]],\"operation\":\"pair_membership\",\"x\":7,\"y\":9}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r03-relation-0019","batch_id":"batch-20261003-advance-20261004-r03-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-relation"],"title_zh":"二元關係的有序對成員","stem_zh":"二元關係的來源 A={6, 7, 8}、目標 B={10, 11, 12}，R=∅⊆A×B。有序對 (6, 10) 是否屬於 R？","answer":{"kind":"short_text","value":"否","explanation_zh":"(6, 10)∈R=否。逐一核對完整有序對及方向，不能自行交換兩分量。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["二元關係的有序對成員"]},"variant":{"family_id":"qf-relation-pair-membership","seed":202610043019,"parameters":{"A":[6,7,8],"B":[10,11,12],"R":[],"operation":"pair_membership","x":6,"y":10,"semantic_template":"qf-relation-pair-membership/{\"A\":[6,7,8],\"B\":[10,11,12],\"R\":[],\"operation\":\"pair_membership\",\"x\":6,\"y\":10}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r03-relation-0020","batch_id":"batch-20261003-advance-20261004-r03-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-relation"],"title_zh":"二元關係的有序對成員","stem_zh":"二元關係的來源 A={7, 8, 9}、目標 B={11, 12, 13}，R={(7, 11)}⊆A×B。有序對 (11, 7) 是否屬於 R？","answer":{"kind":"short_text","value":"否","explanation_zh":"(11, 7)∈R=否。逐一核對完整有序對及方向，不能自行交換兩分量。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["二元關係的有序對成員"]},"variant":{"family_id":"qf-relation-pair-membership","seed":202610043020,"parameters":{"A":[7,8,9],"B":[11,12,13],"R":[[7,11]],"operation":"pair_membership","x":11,"y":7,"semantic_template":"qf-relation-pair-membership/{\"A\":[7,8,9],\"B\":[11,12,13],\"R\":[[7,11]],\"operation\":\"pair_membership\",\"x\":11,\"y\":7}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r03-relation-0021","batch_id":"batch-20261003-advance-20261004-r03-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-relation"],"title_zh":"二元關係的有序對成員","stem_zh":"二元關係的來源 A={8, 9, 10}、目標 B={12, 13, 14}，R={(8, 12), (8, 13)}⊆A×B。有序對 (10, 12) 是否屬於 R？","answer":{"kind":"short_text","value":"否","explanation_zh":"(10, 12)∈R=否。逐一核對完整有序對及方向，不能自行交換兩分量。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["二元關係的有序對成員"]},"variant":{"family_id":"qf-relation-pair-membership","seed":202610043021,"parameters":{"A":[8,9,10],"B":[12,13,14],"R":[[8,12],[8,13]],"operation":"pair_membership","x":10,"y":12,"semantic_template":"qf-relation-pair-membership/{\"A\":[8,9,10],\"B\":[12,13,14],\"R\":[[8,12],[8,13]],\"operation\":\"pair_membership\",\"x\":10,\"y\":12}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r03-relation-0022","batch_id":"batch-20261003-advance-20261004-r03-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-relation"],"title_zh":"二元關係的有序對成員","stem_zh":"二元關係的來源 A={9, 10, 11}、目標 B={13, 14, 15}，R={(9, 14), (10, 14)}⊆A×B。有序對 (9, 14) 是否屬於 R？","answer":{"kind":"short_text","value":"是","explanation_zh":"(9, 14)∈R=是。逐一核對完整有序對及方向，不能自行交換兩分量。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["二元關係的有序對成員"]},"variant":{"family_id":"qf-relation-pair-membership","seed":202610043022,"parameters":{"A":[9,10,11],"B":[13,14,15],"R":[[9,14],[10,14]],"operation":"pair_membership","x":9,"y":14,"semantic_template":"qf-relation-pair-membership/{\"A\":[9,10,11],\"B\":[13,14,15],\"R\":[[9,14],[10,14]],\"operation\":\"pair_membership\",\"x\":9,\"y\":14}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r03-relation-0023","batch_id":"batch-20261003-advance-20261004-r03-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-relation"],"title_zh":"二元關係的有序對成員","stem_zh":"二元關係的來源 A={10, 11, 12}、目標 B={14, 15, 16}，R={(10, 14), (11, 15), (12, 16)}⊆A×B。有序對 (14, 10) 是否屬於 R？","answer":{"kind":"short_text","value":"否","explanation_zh":"(14, 10)∈R=否。逐一核對完整有序對及方向，不能自行交換兩分量。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["二元關係的有序對成員"]},"variant":{"family_id":"qf-relation-pair-membership","seed":202610043023,"parameters":{"A":[10,11,12],"B":[14,15,16],"R":[[10,14],[11,15],[12,16]],"operation":"pair_membership","x":14,"y":10,"semantic_template":"qf-relation-pair-membership/{\"A\":[10,11,12],\"B\":[14,15,16],\"R\":[[10,14],[11,15],[12,16]],\"operation\":\"pair_membership\",\"x\":14,\"y\":10}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r03-relation-0024","batch_id":"batch-20261003-advance-20261004-r03-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-relation"],"title_zh":"二元關係的有序對成員","stem_zh":"二元關係的來源 A={11, 12, 13}、目標 B={15, 16, 17}，R={(12, 15), (13, 16)}⊆A×B。有序對 (13, 15) 是否屬於 R？","answer":{"kind":"short_text","value":"否","explanation_zh":"(13, 15)∈R=否。逐一核對完整有序對及方向，不能自行交換兩分量。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["二元關係的有序對成員"]},"variant":{"family_id":"qf-relation-pair-membership","seed":202610043024,"parameters":{"A":[11,12,13],"B":[15,16,17],"R":[[12,15],[13,16]],"operation":"pair_membership","x":13,"y":15,"semantic_template":"qf-relation-pair-membership/{\"A\":[11,12,13],\"B\":[15,16,17],\"R\":[[12,15],[13,16]],\"operation\":\"pair_membership\",\"x\":13,\"y\":15}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r03-relation-0025","batch_id":"batch-20261003-advance-20261004-r03-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-relation"],"title_zh":"二元關係的有序對成員","stem_zh":"二元關係的來源 A={12, 13, 14}、目標 B={16, 17, 18}，R=∅⊆A×B。有序對 (12, 16) 是否屬於 R？","answer":{"kind":"short_text","value":"否","explanation_zh":"(12, 16)∈R=否。逐一核對完整有序對及方向，不能自行交換兩分量。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["二元關係的有序對成員"]},"variant":{"family_id":"qf-relation-pair-membership","seed":202610043025,"parameters":{"A":[12,13,14],"B":[16,17,18],"R":[],"operation":"pair_membership","x":12,"y":16,"semantic_template":"qf-relation-pair-membership/{\"A\":[12,13,14],\"B\":[16,17,18],\"R\":[],\"operation\":\"pair_membership\",\"x\":12,\"y\":16}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r03-relation-0026","batch_id":"batch-20261003-advance-20261004-r03-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-relation"],"title_zh":"二元關係的第一投影","stem_zh":"二元關係的來源 A=∅、目標 B=∅，R=∅⊆A×B。求實際關係定義域 Dom(R)，輸出遞增且元素只列一次。","answer":{"kind":"expression","value":"∅","explanation_zh":"Dom(R)=∅。只收集 R 中實際出現的第一分量，不把宣告來源 A 自動當成第一投影。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["二元關係的第一投影"]},"variant":{"family_id":"qf-relation-domain-projection","seed":202610043026,"parameters":{"A":[],"B":[],"R":[],"operation":"domain_projection","semantic_template":"qf-relation-domain-projection/{\"A\":[],\"B\":[],\"R\":[],\"operation\":\"domain_projection\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r03-relation-0027","batch_id":"batch-20261003-advance-20261004-r03-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-relation"],"title_zh":"二元關係的第一投影","stem_zh":"二元關係的來源 A={0}、目標 B={1}，R=∅⊆A×B。求實際關係定義域 Dom(R)，輸出遞增且元素只列一次。","answer":{"kind":"expression","value":"∅","explanation_zh":"Dom(R)=∅。只收集 R 中實際出現的第一分量，不把宣告來源 A 自動當成第一投影。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["二元關係的第一投影"]},"variant":{"family_id":"qf-relation-domain-projection","seed":202610043027,"parameters":{"A":[0],"B":[1],"R":[],"operation":"domain_projection","semantic_template":"qf-relation-domain-projection/{\"A\":[0],\"B\":[1],\"R\":[],\"operation\":\"domain_projection\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r03-relation-0028","batch_id":"batch-20261003-advance-20261004-r03-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-relation"],"title_zh":"二元關係的第一投影","stem_zh":"二元關係的來源 A={-10, -9, -8}、目標 B={-6, -5, -4}，R={(-10, -6), (-10, -5)}⊆A×B。求實際關係定義域 Dom(R)，輸出遞增且元素只列一次。","answer":{"kind":"expression","value":"{-10}","explanation_zh":"Dom(R)={-10}。只收集 R 中實際出現的第一分量，不把宣告來源 A 自動當成第一投影。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["二元關係的第一投影"]},"variant":{"family_id":"qf-relation-domain-projection","seed":202610043028,"parameters":{"A":[-10,-9,-8],"B":[-6,-5,-4],"R":[[-10,-6],[-10,-5]],"operation":"domain_projection","semantic_template":"qf-relation-domain-projection/{\"A\":[-10,-9,-8],\"B\":[-6,-5,-4],\"R\":[[-10,-6],[-10,-5]],\"operation\":\"domain_projection\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r03-relation-0029","batch_id":"batch-20261003-advance-20261004-r03-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-relation"],"title_zh":"二元關係的第一投影","stem_zh":"二元關係的來源 A={-9, -8, -7}、目標 B={-5, -4, -3}，R={(-9, -4), (-8, -4)}⊆A×B。求實際關係定義域 Dom(R)，輸出遞增且元素只列一次。","answer":{"kind":"expression","value":"{-9, -8}","explanation_zh":"Dom(R)={-9, -8}。只收集 R 中實際出現的第一分量，不把宣告來源 A 自動當成第一投影。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["二元關係的第一投影"]},"variant":{"family_id":"qf-relation-domain-projection","seed":202610043029,"parameters":{"A":[-9,-8,-7],"B":[-5,-4,-3],"R":[[-9,-4],[-8,-4]],"operation":"domain_projection","semantic_template":"qf-relation-domain-projection/{\"A\":[-9,-8,-7],\"B\":[-5,-4,-3],\"R\":[[-9,-4],[-8,-4]],\"operation\":\"domain_projection\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r03-relation-0030","batch_id":"batch-20261003-advance-20261004-r03-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-relation"],"title_zh":"二元關係的第一投影","stem_zh":"二元關係的來源 A={-8, -7, -6}、目標 B={-4, -3, -2}，R={(-8, -4), (-7, -3), (-6, -2)}⊆A×B。求實際關係定義域 Dom(R)，輸出遞增且元素只列一次。","answer":{"kind":"expression","value":"{-8, -7, -6}","explanation_zh":"Dom(R)={-8, -7, -6}。只收集 R 中實際出現的第一分量，不把宣告來源 A 自動當成第一投影。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["二元關係的第一投影"]},"variant":{"family_id":"qf-relation-domain-projection","seed":202610043030,"parameters":{"A":[-8,-7,-6],"B":[-4,-3,-2],"R":[[-8,-4],[-7,-3],[-6,-2]],"operation":"domain_projection","semantic_template":"qf-relation-domain-projection/{\"A\":[-8,-7,-6],\"B\":[-4,-3,-2],\"R\":[[-8,-4],[-7,-3],[-6,-2]],\"operation\":\"domain_projection\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r03-relation-0031","batch_id":"batch-20261003-advance-20261004-r03-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-relation"],"title_zh":"二元關係的第一投影","stem_zh":"二元關係的來源 A={-7, -6, -5}、目標 B={-3, -2, -1}，R={(-6, -3), (-5, -2)}⊆A×B。求實際關係定義域 Dom(R)，輸出遞增且元素只列一次。","answer":{"kind":"expression","value":"{-6, -5}","explanation_zh":"Dom(R)={-6, -5}。只收集 R 中實際出現的第一分量，不把宣告來源 A 自動當成第一投影。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["二元關係的第一投影"]},"variant":{"family_id":"qf-relation-domain-projection","seed":202610043031,"parameters":{"A":[-7,-6,-5],"B":[-3,-2,-1],"R":[[-6,-3],[-5,-2]],"operation":"domain_projection","semantic_template":"qf-relation-domain-projection/{\"A\":[-7,-6,-5],\"B\":[-3,-2,-1],\"R\":[[-6,-3],[-5,-2]],\"operation\":\"domain_projection\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r03-relation-0032","batch_id":"batch-20261003-advance-20261004-r03-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-relation"],"title_zh":"二元關係的第一投影","stem_zh":"二元關係的來源 A={-6, -5, -4}、目標 B={-2, -1, 0}，R=∅⊆A×B。求實際關係定義域 Dom(R)，輸出遞增且元素只列一次。","answer":{"kind":"expression","value":"∅","explanation_zh":"Dom(R)=∅。只收集 R 中實際出現的第一分量，不把宣告來源 A 自動當成第一投影。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["二元關係的第一投影"]},"variant":{"family_id":"qf-relation-domain-projection","seed":202610043032,"parameters":{"A":[-6,-5,-4],"B":[-2,-1,0],"R":[],"operation":"domain_projection","semantic_template":"qf-relation-domain-projection/{\"A\":[-6,-5,-4],\"B\":[-2,-1,0],\"R\":[],\"operation\":\"domain_projection\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r03-relation-0033","batch_id":"batch-20261003-advance-20261004-r03-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-relation"],"title_zh":"二元關係的第一投影","stem_zh":"二元關係的來源 A={-5, -4, -3}、目標 B={-1, 0, 1}，R={(-5, -1)}⊆A×B。求實際關係定義域 Dom(R)，輸出遞增且元素只列一次。","answer":{"kind":"expression","value":"{-5}","explanation_zh":"Dom(R)={-5}。只收集 R 中實際出現的第一分量，不把宣告來源 A 自動當成第一投影。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["二元關係的第一投影"]},"variant":{"family_id":"qf-relation-domain-projection","seed":202610043033,"parameters":{"A":[-5,-4,-3],"B":[-1,0,1],"R":[[-5,-1]],"operation":"domain_projection","semantic_template":"qf-relation-domain-projection/{\"A\":[-5,-4,-3],\"B\":[-1,0,1],\"R\":[[-5,-1]],\"operation\":\"domain_projection\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r03-relation-0034","batch_id":"batch-20261003-advance-20261004-r03-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-relation"],"title_zh":"二元關係的第一投影","stem_zh":"二元關係的來源 A={-4, -3, -2}、目標 B={0, 1, 2}，R={(-4, 0), (-4, 1)}⊆A×B。求實際關係定義域 Dom(R)，輸出遞增且元素只列一次。","answer":{"kind":"expression","value":"{-4}","explanation_zh":"Dom(R)={-4}。只收集 R 中實際出現的第一分量，不把宣告來源 A 自動當成第一投影。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["二元關係的第一投影"]},"variant":{"family_id":"qf-relation-domain-projection","seed":202610043034,"parameters":{"A":[-4,-3,-2],"B":[0,1,2],"R":[[-4,0],[-4,1]],"operation":"domain_projection","semantic_template":"qf-relation-domain-projection/{\"A\":[-4,-3,-2],\"B\":[0,1,2],\"R\":[[-4,0],[-4,1]],\"operation\":\"domain_projection\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r03-relation-0035","batch_id":"batch-20261003-advance-20261004-r03-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-relation"],"title_zh":"二元關係的第一投影","stem_zh":"二元關係的來源 A={-3, -2, -1}、目標 B={1, 2, 3}，R={(-3, 2), (-2, 2)}⊆A×B。求實際關係定義域 Dom(R)，輸出遞增且元素只列一次。","answer":{"kind":"expression","value":"{-3, -2}","explanation_zh":"Dom(R)={-3, -2}。只收集 R 中實際出現的第一分量，不把宣告來源 A 自動當成第一投影。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["二元關係的第一投影"]},"variant":{"family_id":"qf-relation-domain-projection","seed":202610043035,"parameters":{"A":[-3,-2,-1],"B":[1,2,3],"R":[[-3,2],[-2,2]],"operation":"domain_projection","semantic_template":"qf-relation-domain-projection/{\"A\":[-3,-2,-1],\"B\":[1,2,3],\"R\":[[-3,2],[-2,2]],\"operation\":\"domain_projection\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r03-relation-0036","batch_id":"batch-20261003-advance-20261004-r03-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-relation"],"title_zh":"二元關係的第一投影","stem_zh":"二元關係的來源 A={-2, -1, 0}、目標 B={2, 3, 4}，R={(-2, 2), (-1, 3), (0, 4)}⊆A×B。求實際關係定義域 Dom(R)，輸出遞增且元素只列一次。","answer":{"kind":"expression","value":"{-2, -1, 0}","explanation_zh":"Dom(R)={-2, -1, 0}。只收集 R 中實際出現的第一分量，不把宣告來源 A 自動當成第一投影。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["二元關係的第一投影"]},"variant":{"family_id":"qf-relation-domain-projection","seed":202610043036,"parameters":{"A":[-2,-1,0],"B":[2,3,4],"R":[[-2,2],[-1,3],[0,4]],"operation":"domain_projection","semantic_template":"qf-relation-domain-projection/{\"A\":[-2,-1,0],\"B\":[2,3,4],\"R\":[[-2,2],[-1,3],[0,4]],\"operation\":\"domain_projection\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r03-relation-0037","batch_id":"batch-20261003-advance-20261004-r03-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-relation"],"title_zh":"二元關係的第一投影","stem_zh":"二元關係的來源 A={-1, 0, 1}、目標 B={3, 4, 5}，R={(0, 3), (1, 4)}⊆A×B。求實際關係定義域 Dom(R)，輸出遞增且元素只列一次。","answer":{"kind":"expression","value":"{0, 1}","explanation_zh":"Dom(R)={0, 1}。只收集 R 中實際出現的第一分量，不把宣告來源 A 自動當成第一投影。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["二元關係的第一投影"]},"variant":{"family_id":"qf-relation-domain-projection","seed":202610043037,"parameters":{"A":[-1,0,1],"B":[3,4,5],"R":[[0,3],[1,4]],"operation":"domain_projection","semantic_template":"qf-relation-domain-projection/{\"A\":[-1,0,1],\"B\":[3,4,5],\"R\":[[0,3],[1,4]],\"operation\":\"domain_projection\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r03-relation-0038","batch_id":"batch-20261003-advance-20261004-r03-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-relation"],"title_zh":"二元關係的第一投影","stem_zh":"二元關係的來源 A={0, 1, 2}、目標 B={4, 5, 6}，R=∅⊆A×B。求實際關係定義域 Dom(R)，輸出遞增且元素只列一次。","answer":{"kind":"expression","value":"∅","explanation_zh":"Dom(R)=∅。只收集 R 中實際出現的第一分量，不把宣告來源 A 自動當成第一投影。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["二元關係的第一投影"]},"variant":{"family_id":"qf-relation-domain-projection","seed":202610043038,"parameters":{"A":[0,1,2],"B":[4,5,6],"R":[],"operation":"domain_projection","semantic_template":"qf-relation-domain-projection/{\"A\":[0,1,2],\"B\":[4,5,6],\"R\":[],\"operation\":\"domain_projection\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r03-relation-0039","batch_id":"batch-20261003-advance-20261004-r03-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-relation"],"title_zh":"二元關係的第一投影","stem_zh":"二元關係的來源 A={1, 2, 3}、目標 B={5, 6, 7}，R={(1, 5)}⊆A×B。求實際關係定義域 Dom(R)，輸出遞增且元素只列一次。","answer":{"kind":"expression","value":"{1}","explanation_zh":"Dom(R)={1}。只收集 R 中實際出現的第一分量，不把宣告來源 A 自動當成第一投影。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["二元關係的第一投影"]},"variant":{"family_id":"qf-relation-domain-projection","seed":202610043039,"parameters":{"A":[1,2,3],"B":[5,6,7],"R":[[1,5]],"operation":"domain_projection","semantic_template":"qf-relation-domain-projection/{\"A\":[1,2,3],\"B\":[5,6,7],\"R\":[[1,5]],\"operation\":\"domain_projection\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r03-relation-0040","batch_id":"batch-20261003-advance-20261004-r03-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-relation"],"title_zh":"二元關係的第一投影","stem_zh":"二元關係的來源 A={2, 3, 4}、目標 B={6, 7, 8}，R={(2, 6), (2, 7)}⊆A×B。求實際關係定義域 Dom(R)，輸出遞增且元素只列一次。","answer":{"kind":"expression","value":"{2}","explanation_zh":"Dom(R)={2}。只收集 R 中實際出現的第一分量，不把宣告來源 A 自動當成第一投影。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["二元關係的第一投影"]},"variant":{"family_id":"qf-relation-domain-projection","seed":202610043040,"parameters":{"A":[2,3,4],"B":[6,7,8],"R":[[2,6],[2,7]],"operation":"domain_projection","semantic_template":"qf-relation-domain-projection/{\"A\":[2,3,4],\"B\":[6,7,8],\"R\":[[2,6],[2,7]],\"operation\":\"domain_projection\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r03-relation-0041","batch_id":"batch-20261003-advance-20261004-r03-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-relation"],"title_zh":"二元關係的第一投影","stem_zh":"二元關係的來源 A={3, 4, 5}、目標 B={7, 8, 9}，R={(3, 8), (4, 8)}⊆A×B。求實際關係定義域 Dom(R)，輸出遞增且元素只列一次。","answer":{"kind":"expression","value":"{3, 4}","explanation_zh":"Dom(R)={3, 4}。只收集 R 中實際出現的第一分量，不把宣告來源 A 自動當成第一投影。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["二元關係的第一投影"]},"variant":{"family_id":"qf-relation-domain-projection","seed":202610043041,"parameters":{"A":[3,4,5],"B":[7,8,9],"R":[[3,8],[4,8]],"operation":"domain_projection","semantic_template":"qf-relation-domain-projection/{\"A\":[3,4,5],\"B\":[7,8,9],\"R\":[[3,8],[4,8]],\"operation\":\"domain_projection\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r03-relation-0042","batch_id":"batch-20261003-advance-20261004-r03-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-relation"],"title_zh":"二元關係的第一投影","stem_zh":"二元關係的來源 A={4, 5, 6}、目標 B={8, 9, 10}，R={(4, 8), (5, 9), (6, 10)}⊆A×B。求實際關係定義域 Dom(R)，輸出遞增且元素只列一次。","answer":{"kind":"expression","value":"{4, 5, 6}","explanation_zh":"Dom(R)={4, 5, 6}。只收集 R 中實際出現的第一分量，不把宣告來源 A 自動當成第一投影。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["二元關係的第一投影"]},"variant":{"family_id":"qf-relation-domain-projection","seed":202610043042,"parameters":{"A":[4,5,6],"B":[8,9,10],"R":[[4,8],[5,9],[6,10]],"operation":"domain_projection","semantic_template":"qf-relation-domain-projection/{\"A\":[4,5,6],\"B\":[8,9,10],\"R\":[[4,8],[5,9],[6,10]],\"operation\":\"domain_projection\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r03-relation-0043","batch_id":"batch-20261003-advance-20261004-r03-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-relation"],"title_zh":"二元關係的第一投影","stem_zh":"二元關係的來源 A={5, 6, 7}、目標 B={9, 10, 11}，R={(6, 9), (7, 10)}⊆A×B。求實際關係定義域 Dom(R)，輸出遞增且元素只列一次。","answer":{"kind":"expression","value":"{6, 7}","explanation_zh":"Dom(R)={6, 7}。只收集 R 中實際出現的第一分量，不把宣告來源 A 自動當成第一投影。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["二元關係的第一投影"]},"variant":{"family_id":"qf-relation-domain-projection","seed":202610043043,"parameters":{"A":[5,6,7],"B":[9,10,11],"R":[[6,9],[7,10]],"operation":"domain_projection","semantic_template":"qf-relation-domain-projection/{\"A\":[5,6,7],\"B\":[9,10,11],\"R\":[[6,9],[7,10]],\"operation\":\"domain_projection\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r03-relation-0044","batch_id":"batch-20261003-advance-20261004-r03-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-relation"],"title_zh":"二元關係的第一投影","stem_zh":"二元關係的來源 A={6, 7, 8}、目標 B={10, 11, 12}，R=∅⊆A×B。求實際關係定義域 Dom(R)，輸出遞增且元素只列一次。","answer":{"kind":"expression","value":"∅","explanation_zh":"Dom(R)=∅。只收集 R 中實際出現的第一分量，不把宣告來源 A 自動當成第一投影。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["二元關係的第一投影"]},"variant":{"family_id":"qf-relation-domain-projection","seed":202610043044,"parameters":{"A":[6,7,8],"B":[10,11,12],"R":[],"operation":"domain_projection","semantic_template":"qf-relation-domain-projection/{\"A\":[6,7,8],\"B\":[10,11,12],\"R\":[],\"operation\":\"domain_projection\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r03-relation-0045","batch_id":"batch-20261003-advance-20261004-r03-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-relation"],"title_zh":"二元關係的第一投影","stem_zh":"二元關係的來源 A={7, 8, 9}、目標 B={11, 12, 13}，R={(7, 11)}⊆A×B。求實際關係定義域 Dom(R)，輸出遞增且元素只列一次。","answer":{"kind":"expression","value":"{7}","explanation_zh":"Dom(R)={7}。只收集 R 中實際出現的第一分量，不把宣告來源 A 自動當成第一投影。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["二元關係的第一投影"]},"variant":{"family_id":"qf-relation-domain-projection","seed":202610043045,"parameters":{"A":[7,8,9],"B":[11,12,13],"R":[[7,11]],"operation":"domain_projection","semantic_template":"qf-relation-domain-projection/{\"A\":[7,8,9],\"B\":[11,12,13],\"R\":[[7,11]],\"operation\":\"domain_projection\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r03-relation-0046","batch_id":"batch-20261003-advance-20261004-r03-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-relation"],"title_zh":"二元關係的第一投影","stem_zh":"二元關係的來源 A={8, 9, 10}、目標 B={12, 13, 14}，R={(8, 12), (8, 13)}⊆A×B。求實際關係定義域 Dom(R)，輸出遞增且元素只列一次。","answer":{"kind":"expression","value":"{8}","explanation_zh":"Dom(R)={8}。只收集 R 中實際出現的第一分量，不把宣告來源 A 自動當成第一投影。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["二元關係的第一投影"]},"variant":{"family_id":"qf-relation-domain-projection","seed":202610043046,"parameters":{"A":[8,9,10],"B":[12,13,14],"R":[[8,12],[8,13]],"operation":"domain_projection","semantic_template":"qf-relation-domain-projection/{\"A\":[8,9,10],\"B\":[12,13,14],\"R\":[[8,12],[8,13]],\"operation\":\"domain_projection\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r03-relation-0047","batch_id":"batch-20261003-advance-20261004-r03-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-relation"],"title_zh":"二元關係的第一投影","stem_zh":"二元關係的來源 A={9, 10, 11}、目標 B={13, 14, 15}，R={(9, 14), (10, 14)}⊆A×B。求實際關係定義域 Dom(R)，輸出遞增且元素只列一次。","answer":{"kind":"expression","value":"{9, 10}","explanation_zh":"Dom(R)={9, 10}。只收集 R 中實際出現的第一分量，不把宣告來源 A 自動當成第一投影。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["二元關係的第一投影"]},"variant":{"family_id":"qf-relation-domain-projection","seed":202610043047,"parameters":{"A":[9,10,11],"B":[13,14,15],"R":[[9,14],[10,14]],"operation":"domain_projection","semantic_template":"qf-relation-domain-projection/{\"A\":[9,10,11],\"B\":[13,14,15],\"R\":[[9,14],[10,14]],\"operation\":\"domain_projection\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r03-relation-0048","batch_id":"batch-20261003-advance-20261004-r03-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-relation"],"title_zh":"二元關係的第一投影","stem_zh":"二元關係的來源 A={10, 11, 12}、目標 B={14, 15, 16}，R={(10, 14), (11, 15), (12, 16)}⊆A×B。求實際關係定義域 Dom(R)，輸出遞增且元素只列一次。","answer":{"kind":"expression","value":"{10, 11, 12}","explanation_zh":"Dom(R)={10, 11, 12}。只收集 R 中實際出現的第一分量，不把宣告來源 A 自動當成第一投影。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["二元關係的第一投影"]},"variant":{"family_id":"qf-relation-domain-projection","seed":202610043048,"parameters":{"A":[10,11,12],"B":[14,15,16],"R":[[10,14],[11,15],[12,16]],"operation":"domain_projection","semantic_template":"qf-relation-domain-projection/{\"A\":[10,11,12],\"B\":[14,15,16],\"R\":[[10,14],[11,15],[12,16]],\"operation\":\"domain_projection\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r03-relation-0049","batch_id":"batch-20261003-advance-20261004-r03-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-relation"],"title_zh":"二元關係的第一投影","stem_zh":"二元關係的來源 A={11, 12, 13}、目標 B={15, 16, 17}，R={(12, 15), (13, 16)}⊆A×B。求實際關係定義域 Dom(R)，輸出遞增且元素只列一次。","answer":{"kind":"expression","value":"{12, 13}","explanation_zh":"Dom(R)={12, 13}。只收集 R 中實際出現的第一分量，不把宣告來源 A 自動當成第一投影。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["二元關係的第一投影"]},"variant":{"family_id":"qf-relation-domain-projection","seed":202610043049,"parameters":{"A":[11,12,13],"B":[15,16,17],"R":[[12,15],[13,16]],"operation":"domain_projection","semantic_template":"qf-relation-domain-projection/{\"A\":[11,12,13],\"B\":[15,16,17],\"R\":[[12,15],[13,16]],\"operation\":\"domain_projection\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r03-relation-0050","batch_id":"batch-20261003-advance-20261004-r03-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-relation"],"title_zh":"二元關係的第一投影","stem_zh":"二元關係的來源 A={12, 13, 14}、目標 B={16, 17, 18}，R=∅⊆A×B。求實際關係定義域 Dom(R)，輸出遞增且元素只列一次。","answer":{"kind":"expression","value":"∅","explanation_zh":"Dom(R)=∅。只收集 R 中實際出現的第一分量，不把宣告來源 A 自動當成第一投影。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["二元關係的第一投影"]},"variant":{"family_id":"qf-relation-domain-projection","seed":202610043050,"parameters":{"A":[12,13,14],"B":[16,17,18],"R":[],"operation":"domain_projection","semantic_template":"qf-relation-domain-projection/{\"A\":[12,13,14],\"B\":[16,17,18],\"R\":[],\"operation\":\"domain_projection\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r03-relation-0051","batch_id":"batch-20261003-advance-20261004-r03-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-relation"],"title_zh":"二元關係的子集合像","stem_zh":"二元關係的來源 A=∅、目標 B=∅，R=∅⊆A×B。明示 C=∅⊆A，求 R[C] 的遞增、無重複集合。","answer":{"kind":"expression","value":"∅","explanation_zh":"R[C]=∅。選第一分量在 C 的所有關係對，再收集第二分量；不要求每個輸入只有一個輸出。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["二元關係的子集合像"]},"variant":{"family_id":"qf-relation-subset-image","seed":202610043051,"parameters":{"A":[],"B":[],"R":[],"operation":"subset_image","C":[],"semantic_template":"qf-relation-subset-image/{\"A\":[],\"B\":[],\"C\":[],\"R\":[],\"operation\":\"subset_image\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r03-relation-0052","batch_id":"batch-20261003-advance-20261004-r03-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-relation"],"title_zh":"二元關係的子集合像","stem_zh":"二元關係的來源 A={0}、目標 B={1}，R=∅⊆A×B。明示 C={0}⊆A，求 R[C] 的遞增、無重複集合。","answer":{"kind":"expression","value":"∅","explanation_zh":"R[C]=∅。選第一分量在 C 的所有關係對，再收集第二分量；不要求每個輸入只有一個輸出。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["二元關係的子集合像"]},"variant":{"family_id":"qf-relation-subset-image","seed":202610043052,"parameters":{"A":[0],"B":[1],"R":[],"operation":"subset_image","C":[0],"semantic_template":"qf-relation-subset-image/{\"A\":[0],\"B\":[1],\"C\":[0],\"R\":[],\"operation\":\"subset_image\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r03-relation-0053","batch_id":"batch-20261003-advance-20261004-r03-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-relation"],"title_zh":"二元關係的子集合像","stem_zh":"二元關係的來源 A={-10, -9, -8}、目標 B={-6, -5, -4}，R={(-10, -6), (-10, -5)}⊆A×B。明示 C={-10}⊆A，求 R[C] 的遞增、無重複集合。","answer":{"kind":"expression","value":"{-6, -5}","explanation_zh":"R[C]={-6, -5}。選第一分量在 C 的所有關係對，再收集第二分量；不要求每個輸入只有一個輸出。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["二元關係的子集合像"]},"variant":{"family_id":"qf-relation-subset-image","seed":202610043053,"parameters":{"A":[-10,-9,-8],"B":[-6,-5,-4],"R":[[-10,-6],[-10,-5]],"operation":"subset_image","C":[-10],"semantic_template":"qf-relation-subset-image/{\"A\":[-10,-9,-8],\"B\":[-6,-5,-4],\"C\":[-10],\"R\":[[-10,-6],[-10,-5]],\"operation\":\"subset_image\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r03-relation-0054","batch_id":"batch-20261003-advance-20261004-r03-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-relation"],"title_zh":"二元關係的子集合像","stem_zh":"二元關係的來源 A={-9, -8, -7}、目標 B={-5, -4, -3}，R={(-9, -4), (-8, -4)}⊆A×B。明示 C={-8, -7}⊆A，求 R[C] 的遞增、無重複集合。","answer":{"kind":"expression","value":"{-4}","explanation_zh":"R[C]={-4}。選第一分量在 C 的所有關係對，再收集第二分量；不要求每個輸入只有一個輸出。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["二元關係的子集合像"]},"variant":{"family_id":"qf-relation-subset-image","seed":202610043054,"parameters":{"A":[-9,-8,-7],"B":[-5,-4,-3],"R":[[-9,-4],[-8,-4]],"operation":"subset_image","C":[-8,-7],"semantic_template":"qf-relation-subset-image/{\"A\":[-9,-8,-7],\"B\":[-5,-4,-3],\"C\":[-8,-7],\"R\":[[-9,-4],[-8,-4]],\"operation\":\"subset_image\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r03-relation-0055","batch_id":"batch-20261003-advance-20261004-r03-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-relation"],"title_zh":"二元關係的子集合像","stem_zh":"二元關係的來源 A={-8, -7, -6}、目標 B={-4, -3, -2}，R={(-8, -4), (-7, -3), (-6, -2)}⊆A×B。明示 C=∅⊆A，求 R[C] 的遞增、無重複集合。","answer":{"kind":"expression","value":"∅","explanation_zh":"R[C]=∅。選第一分量在 C 的所有關係對，再收集第二分量；不要求每個輸入只有一個輸出。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["二元關係的子集合像"]},"variant":{"family_id":"qf-relation-subset-image","seed":202610043055,"parameters":{"A":[-8,-7,-6],"B":[-4,-3,-2],"R":[[-8,-4],[-7,-3],[-6,-2]],"operation":"subset_image","C":[],"semantic_template":"qf-relation-subset-image/{\"A\":[-8,-7,-6],\"B\":[-4,-3,-2],\"C\":[],\"R\":[[-8,-4],[-7,-3],[-6,-2]],\"operation\":\"subset_image\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r03-relation-0056","batch_id":"batch-20261003-advance-20261004-r03-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-relation"],"title_zh":"二元關係的子集合像","stem_zh":"二元關係的來源 A={-7, -6, -5}、目標 B={-3, -2, -1}，R={(-6, -3), (-5, -2)}⊆A×B。明示 C={-7, -6, -5}⊆A，求 R[C] 的遞增、無重複集合。","answer":{"kind":"expression","value":"{-3, -2}","explanation_zh":"R[C]={-3, -2}。選第一分量在 C 的所有關係對，再收集第二分量；不要求每個輸入只有一個輸出。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["二元關係的子集合像"]},"variant":{"family_id":"qf-relation-subset-image","seed":202610043056,"parameters":{"A":[-7,-6,-5],"B":[-3,-2,-1],"R":[[-6,-3],[-5,-2]],"operation":"subset_image","C":[-7,-6,-5],"semantic_template":"qf-relation-subset-image/{\"A\":[-7,-6,-5],\"B\":[-3,-2,-1],\"C\":[-7,-6,-5],\"R\":[[-6,-3],[-5,-2]],\"operation\":\"subset_image\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r03-relation-0057","batch_id":"batch-20261003-advance-20261004-r03-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-relation"],"title_zh":"二元關係的子集合像","stem_zh":"二元關係的來源 A={-6, -5, -4}、目標 B={-2, -1, 0}，R=∅⊆A×B。明示 C={-6}⊆A，求 R[C] 的遞增、無重複集合。","answer":{"kind":"expression","value":"∅","explanation_zh":"R[C]=∅。選第一分量在 C 的所有關係對，再收集第二分量；不要求每個輸入只有一個輸出。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["二元關係的子集合像"]},"variant":{"family_id":"qf-relation-subset-image","seed":202610043057,"parameters":{"A":[-6,-5,-4],"B":[-2,-1,0],"R":[],"operation":"subset_image","C":[-6],"semantic_template":"qf-relation-subset-image/{\"A\":[-6,-5,-4],\"B\":[-2,-1,0],\"C\":[-6],\"R\":[],\"operation\":\"subset_image\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r03-relation-0058","batch_id":"batch-20261003-advance-20261004-r03-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-relation"],"title_zh":"二元關係的子集合像","stem_zh":"二元關係的來源 A={-5, -4, -3}、目標 B={-1, 0, 1}，R={(-5, -1)}⊆A×B。明示 C={-4, -3}⊆A，求 R[C] 的遞增、無重複集合。","answer":{"kind":"expression","value":"∅","explanation_zh":"R[C]=∅。選第一分量在 C 的所有關係對，再收集第二分量；不要求每個輸入只有一個輸出。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["二元關係的子集合像"]},"variant":{"family_id":"qf-relation-subset-image","seed":202610043058,"parameters":{"A":[-5,-4,-3],"B":[-1,0,1],"R":[[-5,-1]],"operation":"subset_image","C":[-4,-3],"semantic_template":"qf-relation-subset-image/{\"A\":[-5,-4,-3],\"B\":[-1,0,1],\"C\":[-4,-3],\"R\":[[-5,-1]],\"operation\":\"subset_image\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r03-relation-0059","batch_id":"batch-20261003-advance-20261004-r03-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-relation"],"title_zh":"二元關係的子集合像","stem_zh":"二元關係的來源 A={-4, -3, -2}、目標 B={0, 1, 2}，R={(-4, 0), (-4, 1)}⊆A×B。明示 C=∅⊆A，求 R[C] 的遞增、無重複集合。","answer":{"kind":"expression","value":"∅","explanation_zh":"R[C]=∅。選第一分量在 C 的所有關係對，再收集第二分量；不要求每個輸入只有一個輸出。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["二元關係的子集合像"]},"variant":{"family_id":"qf-relation-subset-image","seed":202610043059,"parameters":{"A":[-4,-3,-2],"B":[0,1,2],"R":[[-4,0],[-4,1]],"operation":"subset_image","C":[],"semantic_template":"qf-relation-subset-image/{\"A\":[-4,-3,-2],\"B\":[0,1,2],\"C\":[],\"R\":[[-4,0],[-4,1]],\"operation\":\"subset_image\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r03-relation-0060","batch_id":"batch-20261003-advance-20261004-r03-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-relation"],"title_zh":"二元關係的子集合像","stem_zh":"二元關係的來源 A={-3, -2, -1}、目標 B={1, 2, 3}，R={(-3, 2), (-2, 2)}⊆A×B。明示 C={-3, -2, -1}⊆A，求 R[C] 的遞增、無重複集合。","answer":{"kind":"expression","value":"{2}","explanation_zh":"R[C]={2}。選第一分量在 C 的所有關係對，再收集第二分量；不要求每個輸入只有一個輸出。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["二元關係的子集合像"]},"variant":{"family_id":"qf-relation-subset-image","seed":202610043060,"parameters":{"A":[-3,-2,-1],"B":[1,2,3],"R":[[-3,2],[-2,2]],"operation":"subset_image","C":[-3,-2,-1],"semantic_template":"qf-relation-subset-image/{\"A\":[-3,-2,-1],\"B\":[1,2,3],\"C\":[-3,-2,-1],\"R\":[[-3,2],[-2,2]],\"operation\":\"subset_image\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r03-relation-0061","batch_id":"batch-20261003-advance-20261004-r03-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-relation"],"title_zh":"二元關係的子集合像","stem_zh":"二元關係的來源 A={-2, -1, 0}、目標 B={2, 3, 4}，R={(-2, 2), (-1, 3), (0, 4)}⊆A×B。明示 C={-2}⊆A，求 R[C] 的遞增、無重複集合。","answer":{"kind":"expression","value":"{2}","explanation_zh":"R[C]={2}。選第一分量在 C 的所有關係對，再收集第二分量；不要求每個輸入只有一個輸出。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["二元關係的子集合像"]},"variant":{"family_id":"qf-relation-subset-image","seed":202610043061,"parameters":{"A":[-2,-1,0],"B":[2,3,4],"R":[[-2,2],[-1,3],[0,4]],"operation":"subset_image","C":[-2],"semantic_template":"qf-relation-subset-image/{\"A\":[-2,-1,0],\"B\":[2,3,4],\"C\":[-2],\"R\":[[-2,2],[-1,3],[0,4]],\"operation\":\"subset_image\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r03-relation-0062","batch_id":"batch-20261003-advance-20261004-r03-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-relation"],"title_zh":"二元關係的子集合像","stem_zh":"二元關係的來源 A={-1, 0, 1}、目標 B={3, 4, 5}，R={(0, 3), (1, 4)}⊆A×B。明示 C={0, 1}⊆A，求 R[C] 的遞增、無重複集合。","answer":{"kind":"expression","value":"{3, 4}","explanation_zh":"R[C]={3, 4}。選第一分量在 C 的所有關係對，再收集第二分量；不要求每個輸入只有一個輸出。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["二元關係的子集合像"]},"variant":{"family_id":"qf-relation-subset-image","seed":202610043062,"parameters":{"A":[-1,0,1],"B":[3,4,5],"R":[[0,3],[1,4]],"operation":"subset_image","C":[0,1],"semantic_template":"qf-relation-subset-image/{\"A\":[-1,0,1],\"B\":[3,4,5],\"C\":[0,1],\"R\":[[0,3],[1,4]],\"operation\":\"subset_image\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r03-relation-0063","batch_id":"batch-20261003-advance-20261004-r03-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-relation"],"title_zh":"二元關係的子集合像","stem_zh":"二元關係的來源 A={0, 1, 2}、目標 B={4, 5, 6}，R=∅⊆A×B。明示 C=∅⊆A，求 R[C] 的遞增、無重複集合。","answer":{"kind":"expression","value":"∅","explanation_zh":"R[C]=∅。選第一分量在 C 的所有關係對，再收集第二分量；不要求每個輸入只有一個輸出。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["二元關係的子集合像"]},"variant":{"family_id":"qf-relation-subset-image","seed":202610043063,"parameters":{"A":[0,1,2],"B":[4,5,6],"R":[],"operation":"subset_image","C":[],"semantic_template":"qf-relation-subset-image/{\"A\":[0,1,2],\"B\":[4,5,6],\"C\":[],\"R\":[],\"operation\":\"subset_image\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r03-relation-0064","batch_id":"batch-20261003-advance-20261004-r03-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-relation"],"title_zh":"二元關係的子集合像","stem_zh":"二元關係的來源 A={1, 2, 3}、目標 B={5, 6, 7}，R={(1, 5)}⊆A×B。明示 C={1, 2, 3}⊆A，求 R[C] 的遞增、無重複集合。","answer":{"kind":"expression","value":"{5}","explanation_zh":"R[C]={5}。選第一分量在 C 的所有關係對，再收集第二分量；不要求每個輸入只有一個輸出。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["二元關係的子集合像"]},"variant":{"family_id":"qf-relation-subset-image","seed":202610043064,"parameters":{"A":[1,2,3],"B":[5,6,7],"R":[[1,5]],"operation":"subset_image","C":[1,2,3],"semantic_template":"qf-relation-subset-image/{\"A\":[1,2,3],\"B\":[5,6,7],\"C\":[1,2,3],\"R\":[[1,5]],\"operation\":\"subset_image\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r03-relation-0065","batch_id":"batch-20261003-advance-20261004-r03-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-relation"],"title_zh":"二元關係的子集合像","stem_zh":"二元關係的來源 A={2, 3, 4}、目標 B={6, 7, 8}，R={(2, 6), (2, 7)}⊆A×B。明示 C={2}⊆A，求 R[C] 的遞增、無重複集合。","answer":{"kind":"expression","value":"{6, 7}","explanation_zh":"R[C]={6, 7}。選第一分量在 C 的所有關係對，再收集第二分量；不要求每個輸入只有一個輸出。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["二元關係的子集合像"]},"variant":{"family_id":"qf-relation-subset-image","seed":202610043065,"parameters":{"A":[2,3,4],"B":[6,7,8],"R":[[2,6],[2,7]],"operation":"subset_image","C":[2],"semantic_template":"qf-relation-subset-image/{\"A\":[2,3,4],\"B\":[6,7,8],\"C\":[2],\"R\":[[2,6],[2,7]],\"operation\":\"subset_image\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r03-relation-0066","batch_id":"batch-20261003-advance-20261004-r03-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-relation"],"title_zh":"二元關係的子集合像","stem_zh":"二元關係的來源 A={3, 4, 5}、目標 B={7, 8, 9}，R={(3, 8), (4, 8)}⊆A×B。明示 C={4, 5}⊆A，求 R[C] 的遞增、無重複集合。","answer":{"kind":"expression","value":"{8}","explanation_zh":"R[C]={8}。選第一分量在 C 的所有關係對，再收集第二分量；不要求每個輸入只有一個輸出。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["二元關係的子集合像"]},"variant":{"family_id":"qf-relation-subset-image","seed":202610043066,"parameters":{"A":[3,4,5],"B":[7,8,9],"R":[[3,8],[4,8]],"operation":"subset_image","C":[4,5],"semantic_template":"qf-relation-subset-image/{\"A\":[3,4,5],\"B\":[7,8,9],\"C\":[4,5],\"R\":[[3,8],[4,8]],\"operation\":\"subset_image\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r03-relation-0067","batch_id":"batch-20261003-advance-20261004-r03-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-relation"],"title_zh":"二元關係的子集合像","stem_zh":"二元關係的來源 A={4, 5, 6}、目標 B={8, 9, 10}，R={(4, 8), (5, 9), (6, 10)}⊆A×B。明示 C=∅⊆A，求 R[C] 的遞增、無重複集合。","answer":{"kind":"expression","value":"∅","explanation_zh":"R[C]=∅。選第一分量在 C 的所有關係對，再收集第二分量；不要求每個輸入只有一個輸出。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["二元關係的子集合像"]},"variant":{"family_id":"qf-relation-subset-image","seed":202610043067,"parameters":{"A":[4,5,6],"B":[8,9,10],"R":[[4,8],[5,9],[6,10]],"operation":"subset_image","C":[],"semantic_template":"qf-relation-subset-image/{\"A\":[4,5,6],\"B\":[8,9,10],\"C\":[],\"R\":[[4,8],[5,9],[6,10]],\"operation\":\"subset_image\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r03-relation-0068","batch_id":"batch-20261003-advance-20261004-r03-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-relation"],"title_zh":"二元關係的子集合像","stem_zh":"二元關係的來源 A={5, 6, 7}、目標 B={9, 10, 11}，R={(6, 9), (7, 10)}⊆A×B。明示 C={5, 6, 7}⊆A，求 R[C] 的遞增、無重複集合。","answer":{"kind":"expression","value":"{9, 10}","explanation_zh":"R[C]={9, 10}。選第一分量在 C 的所有關係對，再收集第二分量；不要求每個輸入只有一個輸出。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["二元關係的子集合像"]},"variant":{"family_id":"qf-relation-subset-image","seed":202610043068,"parameters":{"A":[5,6,7],"B":[9,10,11],"R":[[6,9],[7,10]],"operation":"subset_image","C":[5,6,7],"semantic_template":"qf-relation-subset-image/{\"A\":[5,6,7],\"B\":[9,10,11],\"C\":[5,6,7],\"R\":[[6,9],[7,10]],\"operation\":\"subset_image\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r03-relation-0069","batch_id":"batch-20261003-advance-20261004-r03-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-relation"],"title_zh":"二元關係的子集合像","stem_zh":"二元關係的來源 A={6, 7, 8}、目標 B={10, 11, 12}，R=∅⊆A×B。明示 C={6}⊆A，求 R[C] 的遞增、無重複集合。","answer":{"kind":"expression","value":"∅","explanation_zh":"R[C]=∅。選第一分量在 C 的所有關係對，再收集第二分量；不要求每個輸入只有一個輸出。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["二元關係的子集合像"]},"variant":{"family_id":"qf-relation-subset-image","seed":202610043069,"parameters":{"A":[6,7,8],"B":[10,11,12],"R":[],"operation":"subset_image","C":[6],"semantic_template":"qf-relation-subset-image/{\"A\":[6,7,8],\"B\":[10,11,12],\"C\":[6],\"R\":[],\"operation\":\"subset_image\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r03-relation-0070","batch_id":"batch-20261003-advance-20261004-r03-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-relation"],"title_zh":"二元關係的子集合像","stem_zh":"二元關係的來源 A={7, 8, 9}、目標 B={11, 12, 13}，R={(7, 11)}⊆A×B。明示 C={8, 9}⊆A，求 R[C] 的遞增、無重複集合。","answer":{"kind":"expression","value":"∅","explanation_zh":"R[C]=∅。選第一分量在 C 的所有關係對，再收集第二分量；不要求每個輸入只有一個輸出。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["二元關係的子集合像"]},"variant":{"family_id":"qf-relation-subset-image","seed":202610043070,"parameters":{"A":[7,8,9],"B":[11,12,13],"R":[[7,11]],"operation":"subset_image","C":[8,9],"semantic_template":"qf-relation-subset-image/{\"A\":[7,8,9],\"B\":[11,12,13],\"C\":[8,9],\"R\":[[7,11]],\"operation\":\"subset_image\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r03-relation-0071","batch_id":"batch-20261003-advance-20261004-r03-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-relation"],"title_zh":"二元關係的子集合像","stem_zh":"二元關係的來源 A={8, 9, 10}、目標 B={12, 13, 14}，R={(8, 12), (8, 13)}⊆A×B。明示 C=∅⊆A，求 R[C] 的遞增、無重複集合。","answer":{"kind":"expression","value":"∅","explanation_zh":"R[C]=∅。選第一分量在 C 的所有關係對，再收集第二分量；不要求每個輸入只有一個輸出。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["二元關係的子集合像"]},"variant":{"family_id":"qf-relation-subset-image","seed":202610043071,"parameters":{"A":[8,9,10],"B":[12,13,14],"R":[[8,12],[8,13]],"operation":"subset_image","C":[],"semantic_template":"qf-relation-subset-image/{\"A\":[8,9,10],\"B\":[12,13,14],\"C\":[],\"R\":[[8,12],[8,13]],\"operation\":\"subset_image\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r03-relation-0072","batch_id":"batch-20261003-advance-20261004-r03-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-relation"],"title_zh":"二元關係的子集合像","stem_zh":"二元關係的來源 A={9, 10, 11}、目標 B={13, 14, 15}，R={(9, 14), (10, 14)}⊆A×B。明示 C={9, 10, 11}⊆A，求 R[C] 的遞增、無重複集合。","answer":{"kind":"expression","value":"{14}","explanation_zh":"R[C]={14}。選第一分量在 C 的所有關係對，再收集第二分量；不要求每個輸入只有一個輸出。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["二元關係的子集合像"]},"variant":{"family_id":"qf-relation-subset-image","seed":202610043072,"parameters":{"A":[9,10,11],"B":[13,14,15],"R":[[9,14],[10,14]],"operation":"subset_image","C":[9,10,11],"semantic_template":"qf-relation-subset-image/{\"A\":[9,10,11],\"B\":[13,14,15],\"C\":[9,10,11],\"R\":[[9,14],[10,14]],\"operation\":\"subset_image\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r03-relation-0073","batch_id":"batch-20261003-advance-20261004-r03-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-relation"],"title_zh":"二元關係的子集合像","stem_zh":"二元關係的來源 A={10, 11, 12}、目標 B={14, 15, 16}，R={(10, 14), (11, 15), (12, 16)}⊆A×B。明示 C={10}⊆A，求 R[C] 的遞增、無重複集合。","answer":{"kind":"expression","value":"{14}","explanation_zh":"R[C]={14}。選第一分量在 C 的所有關係對，再收集第二分量；不要求每個輸入只有一個輸出。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["二元關係的子集合像"]},"variant":{"family_id":"qf-relation-subset-image","seed":202610043073,"parameters":{"A":[10,11,12],"B":[14,15,16],"R":[[10,14],[11,15],[12,16]],"operation":"subset_image","C":[10],"semantic_template":"qf-relation-subset-image/{\"A\":[10,11,12],\"B\":[14,15,16],\"C\":[10],\"R\":[[10,14],[11,15],[12,16]],\"operation\":\"subset_image\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r03-relation-0074","batch_id":"batch-20261003-advance-20261004-r03-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-relation"],"title_zh":"二元關係的子集合像","stem_zh":"二元關係的來源 A={11, 12, 13}、目標 B={15, 16, 17}，R={(12, 15), (13, 16)}⊆A×B。明示 C={12, 13}⊆A，求 R[C] 的遞增、無重複集合。","answer":{"kind":"expression","value":"{15, 16}","explanation_zh":"R[C]={15, 16}。選第一分量在 C 的所有關係對，再收集第二分量；不要求每個輸入只有一個輸出。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["二元關係的子集合像"]},"variant":{"family_id":"qf-relation-subset-image","seed":202610043074,"parameters":{"A":[11,12,13],"B":[15,16,17],"R":[[12,15],[13,16]],"operation":"subset_image","C":[12,13],"semantic_template":"qf-relation-subset-image/{\"A\":[11,12,13],\"B\":[15,16,17],\"C\":[12,13],\"R\":[[12,15],[13,16]],\"operation\":\"subset_image\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r03-relation-0075","batch_id":"batch-20261003-advance-20261004-r03-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-relation"],"title_zh":"二元關係的子集合像","stem_zh":"二元關係的來源 A={12, 13, 14}、目標 B={16, 17, 18}，R=∅⊆A×B。明示 C=∅⊆A，求 R[C] 的遞增、無重複集合。","answer":{"kind":"expression","value":"∅","explanation_zh":"R[C]=∅。選第一分量在 C 的所有關係對，再收集第二分量；不要求每個輸入只有一個輸出。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["二元關係的子集合像"]},"variant":{"family_id":"qf-relation-subset-image","seed":202610043075,"parameters":{"A":[12,13,14],"B":[16,17,18],"R":[],"operation":"subset_image","C":[],"semantic_template":"qf-relation-subset-image/{\"A\":[12,13,14],\"B\":[16,17,18],\"C\":[],\"R\":[],\"operation\":\"subset_image\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r03-relation-0076","batch_id":"batch-20261003-advance-20261004-r03-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-relation"],"title_zh":"二元關係的逆關係","stem_zh":"二元關係的來源 A=∅、目標 B=∅，R=∅⊆A×B。交換來源與目標後，逆關係 R⁻¹ 的規範有序對集合為何？","answer":{"kind":"expression","value":"∅","explanation_zh":"R⁻¹=∅。每個 (x,y) 換成 (y,x)，按第一、第二分量排序；逆關係不等於反函數。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["二元關係的逆關係"]},"variant":{"family_id":"qf-relation-converse-set","seed":202610043076,"parameters":{"A":[],"B":[],"R":[],"operation":"converse","semantic_template":"qf-relation-converse-set/{\"A\":[],\"B\":[],\"R\":[],\"operation\":\"converse\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r03-relation-0077","batch_id":"batch-20261003-advance-20261004-r03-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-relation"],"title_zh":"二元關係的逆關係","stem_zh":"二元關係的來源 A={0}、目標 B={1}，R=∅⊆A×B。交換來源與目標後，逆關係 R⁻¹ 的規範有序對集合為何？","answer":{"kind":"expression","value":"∅","explanation_zh":"R⁻¹=∅。每個 (x,y) 換成 (y,x)，按第一、第二分量排序；逆關係不等於反函數。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["二元關係的逆關係"]},"variant":{"family_id":"qf-relation-converse-set","seed":202610043077,"parameters":{"A":[0],"B":[1],"R":[],"operation":"converse","semantic_template":"qf-relation-converse-set/{\"A\":[0],\"B\":[1],\"R\":[],\"operation\":\"converse\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r03-relation-0078","batch_id":"batch-20261003-advance-20261004-r03-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-relation"],"title_zh":"二元關係的逆關係","stem_zh":"二元關係的來源 A={-10, -9, -8}、目標 B={-6, -5, -4}，R={(-10, -6), (-10, -5)}⊆A×B。交換來源與目標後，逆關係 R⁻¹ 的規範有序對集合為何？","answer":{"kind":"expression","value":"{(-6, -10), (-5, -10)}","explanation_zh":"R⁻¹={(-6, -10), (-5, -10)}。每個 (x,y) 換成 (y,x)，按第一、第二分量排序；逆關係不等於反函數。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["二元關係的逆關係"]},"variant":{"family_id":"qf-relation-converse-set","seed":202610043078,"parameters":{"A":[-10,-9,-8],"B":[-6,-5,-4],"R":[[-10,-6],[-10,-5]],"operation":"converse","semantic_template":"qf-relation-converse-set/{\"A\":[-10,-9,-8],\"B\":[-6,-5,-4],\"R\":[[-10,-6],[-10,-5]],\"operation\":\"converse\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r03-relation-0079","batch_id":"batch-20261003-advance-20261004-r03-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-relation"],"title_zh":"二元關係的逆關係","stem_zh":"二元關係的來源 A={-9, -8, -7}、目標 B={-5, -4, -3}，R={(-9, -4), (-8, -4)}⊆A×B。交換來源與目標後，逆關係 R⁻¹ 的規範有序對集合為何？","answer":{"kind":"expression","value":"{(-4, -9), (-4, -8)}","explanation_zh":"R⁻¹={(-4, -9), (-4, -8)}。每個 (x,y) 換成 (y,x)，按第一、第二分量排序；逆關係不等於反函數。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["二元關係的逆關係"]},"variant":{"family_id":"qf-relation-converse-set","seed":202610043079,"parameters":{"A":[-9,-8,-7],"B":[-5,-4,-3],"R":[[-9,-4],[-8,-4]],"operation":"converse","semantic_template":"qf-relation-converse-set/{\"A\":[-9,-8,-7],\"B\":[-5,-4,-3],\"R\":[[-9,-4],[-8,-4]],\"operation\":\"converse\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r03-relation-0080","batch_id":"batch-20261003-advance-20261004-r03-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-relation"],"title_zh":"二元關係的逆關係","stem_zh":"二元關係的來源 A={-8, -7, -6}、目標 B={-4, -3, -2}，R={(-8, -4), (-7, -3), (-6, -2)}⊆A×B。交換來源與目標後，逆關係 R⁻¹ 的規範有序對集合為何？","answer":{"kind":"expression","value":"{(-4, -8), (-3, -7), (-2, -6)}","explanation_zh":"R⁻¹={(-4, -8), (-3, -7), (-2, -6)}。每個 (x,y) 換成 (y,x)，按第一、第二分量排序；逆關係不等於反函數。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["二元關係的逆關係"]},"variant":{"family_id":"qf-relation-converse-set","seed":202610043080,"parameters":{"A":[-8,-7,-6],"B":[-4,-3,-2],"R":[[-8,-4],[-7,-3],[-6,-2]],"operation":"converse","semantic_template":"qf-relation-converse-set/{\"A\":[-8,-7,-6],\"B\":[-4,-3,-2],\"R\":[[-8,-4],[-7,-3],[-6,-2]],\"operation\":\"converse\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r03-relation-0081","batch_id":"batch-20261003-advance-20261004-r03-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-relation"],"title_zh":"二元關係的逆關係","stem_zh":"二元關係的來源 A={-7, -6, -5}、目標 B={-3, -2, -1}，R={(-6, -3), (-5, -2)}⊆A×B。交換來源與目標後，逆關係 R⁻¹ 的規範有序對集合為何？","answer":{"kind":"expression","value":"{(-3, -6), (-2, -5)}","explanation_zh":"R⁻¹={(-3, -6), (-2, -5)}。每個 (x,y) 換成 (y,x)，按第一、第二分量排序；逆關係不等於反函數。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["二元關係的逆關係"]},"variant":{"family_id":"qf-relation-converse-set","seed":202610043081,"parameters":{"A":[-7,-6,-5],"B":[-3,-2,-1],"R":[[-6,-3],[-5,-2]],"operation":"converse","semantic_template":"qf-relation-converse-set/{\"A\":[-7,-6,-5],\"B\":[-3,-2,-1],\"R\":[[-6,-3],[-5,-2]],\"operation\":\"converse\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r03-relation-0082","batch_id":"batch-20261003-advance-20261004-r03-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-relation"],"title_zh":"二元關係的逆關係","stem_zh":"二元關係的來源 A={-6, -5, -4}、目標 B={-2, -1, 0}，R=∅⊆A×B。交換來源與目標後，逆關係 R⁻¹ 的規範有序對集合為何？","answer":{"kind":"expression","value":"∅","explanation_zh":"R⁻¹=∅。每個 (x,y) 換成 (y,x)，按第一、第二分量排序；逆關係不等於反函數。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["二元關係的逆關係"]},"variant":{"family_id":"qf-relation-converse-set","seed":202610043082,"parameters":{"A":[-6,-5,-4],"B":[-2,-1,0],"R":[],"operation":"converse","semantic_template":"qf-relation-converse-set/{\"A\":[-6,-5,-4],\"B\":[-2,-1,0],\"R\":[],\"operation\":\"converse\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r03-relation-0083","batch_id":"batch-20261003-advance-20261004-r03-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-relation"],"title_zh":"二元關係的逆關係","stem_zh":"二元關係的來源 A={-5, -4, -3}、目標 B={-1, 0, 1}，R={(-5, -1)}⊆A×B。交換來源與目標後，逆關係 R⁻¹ 的規範有序對集合為何？","answer":{"kind":"expression","value":"{(-1, -5)}","explanation_zh":"R⁻¹={(-1, -5)}。每個 (x,y) 換成 (y,x)，按第一、第二分量排序；逆關係不等於反函數。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["二元關係的逆關係"]},"variant":{"family_id":"qf-relation-converse-set","seed":202610043083,"parameters":{"A":[-5,-4,-3],"B":[-1,0,1],"R":[[-5,-1]],"operation":"converse","semantic_template":"qf-relation-converse-set/{\"A\":[-5,-4,-3],\"B\":[-1,0,1],\"R\":[[-5,-1]],\"operation\":\"converse\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r03-relation-0084","batch_id":"batch-20261003-advance-20261004-r03-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-relation"],"title_zh":"二元關係的逆關係","stem_zh":"二元關係的來源 A={-4, -3, -2}、目標 B={0, 1, 2}，R={(-4, 0), (-4, 1)}⊆A×B。交換來源與目標後，逆關係 R⁻¹ 的規範有序對集合為何？","answer":{"kind":"expression","value":"{(0, -4), (1, -4)}","explanation_zh":"R⁻¹={(0, -4), (1, -4)}。每個 (x,y) 換成 (y,x)，按第一、第二分量排序；逆關係不等於反函數。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["二元關係的逆關係"]},"variant":{"family_id":"qf-relation-converse-set","seed":202610043084,"parameters":{"A":[-4,-3,-2],"B":[0,1,2],"R":[[-4,0],[-4,1]],"operation":"converse","semantic_template":"qf-relation-converse-set/{\"A\":[-4,-3,-2],\"B\":[0,1,2],\"R\":[[-4,0],[-4,1]],\"operation\":\"converse\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r03-relation-0085","batch_id":"batch-20261003-advance-20261004-r03-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-relation"],"title_zh":"二元關係的逆關係","stem_zh":"二元關係的來源 A={-3, -2, -1}、目標 B={1, 2, 3}，R={(-3, 2), (-2, 2)}⊆A×B。交換來源與目標後，逆關係 R⁻¹ 的規範有序對集合為何？","answer":{"kind":"expression","value":"{(2, -3), (2, -2)}","explanation_zh":"R⁻¹={(2, -3), (2, -2)}。每個 (x,y) 換成 (y,x)，按第一、第二分量排序；逆關係不等於反函數。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["二元關係的逆關係"]},"variant":{"family_id":"qf-relation-converse-set","seed":202610043085,"parameters":{"A":[-3,-2,-1],"B":[1,2,3],"R":[[-3,2],[-2,2]],"operation":"converse","semantic_template":"qf-relation-converse-set/{\"A\":[-3,-2,-1],\"B\":[1,2,3],\"R\":[[-3,2],[-2,2]],\"operation\":\"converse\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r03-relation-0086","batch_id":"batch-20261003-advance-20261004-r03-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-relation"],"title_zh":"二元關係的逆關係","stem_zh":"二元關係的來源 A={-2, -1, 0}、目標 B={2, 3, 4}，R={(-2, 2), (-1, 3), (0, 4)}⊆A×B。交換來源與目標後，逆關係 R⁻¹ 的規範有序對集合為何？","answer":{"kind":"expression","value":"{(2, -2), (3, -1), (4, 0)}","explanation_zh":"R⁻¹={(2, -2), (3, -1), (4, 0)}。每個 (x,y) 換成 (y,x)，按第一、第二分量排序；逆關係不等於反函數。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["二元關係的逆關係"]},"variant":{"family_id":"qf-relation-converse-set","seed":202610043086,"parameters":{"A":[-2,-1,0],"B":[2,3,4],"R":[[-2,2],[-1,3],[0,4]],"operation":"converse","semantic_template":"qf-relation-converse-set/{\"A\":[-2,-1,0],\"B\":[2,3,4],\"R\":[[-2,2],[-1,3],[0,4]],\"operation\":\"converse\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r03-relation-0087","batch_id":"batch-20261003-advance-20261004-r03-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-relation"],"title_zh":"二元關係的逆關係","stem_zh":"二元關係的來源 A={-1, 0, 1}、目標 B={3, 4, 5}，R={(0, 3), (1, 4)}⊆A×B。交換來源與目標後，逆關係 R⁻¹ 的規範有序對集合為何？","answer":{"kind":"expression","value":"{(3, 0), (4, 1)}","explanation_zh":"R⁻¹={(3, 0), (4, 1)}。每個 (x,y) 換成 (y,x)，按第一、第二分量排序；逆關係不等於反函數。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["二元關係的逆關係"]},"variant":{"family_id":"qf-relation-converse-set","seed":202610043087,"parameters":{"A":[-1,0,1],"B":[3,4,5],"R":[[0,3],[1,4]],"operation":"converse","semantic_template":"qf-relation-converse-set/{\"A\":[-1,0,1],\"B\":[3,4,5],\"R\":[[0,3],[1,4]],\"operation\":\"converse\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r03-relation-0088","batch_id":"batch-20261003-advance-20261004-r03-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-relation"],"title_zh":"二元關係的逆關係","stem_zh":"二元關係的來源 A={0, 1, 2}、目標 B={4, 5, 6}，R=∅⊆A×B。交換來源與目標後，逆關係 R⁻¹ 的規範有序對集合為何？","answer":{"kind":"expression","value":"∅","explanation_zh":"R⁻¹=∅。每個 (x,y) 換成 (y,x)，按第一、第二分量排序；逆關係不等於反函數。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["二元關係的逆關係"]},"variant":{"family_id":"qf-relation-converse-set","seed":202610043088,"parameters":{"A":[0,1,2],"B":[4,5,6],"R":[],"operation":"converse","semantic_template":"qf-relation-converse-set/{\"A\":[0,1,2],\"B\":[4,5,6],\"R\":[],\"operation\":\"converse\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r03-relation-0089","batch_id":"batch-20261003-advance-20261004-r03-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-relation"],"title_zh":"二元關係的逆關係","stem_zh":"二元關係的來源 A={1, 2, 3}、目標 B={5, 6, 7}，R={(1, 5)}⊆A×B。交換來源與目標後，逆關係 R⁻¹ 的規範有序對集合為何？","answer":{"kind":"expression","value":"{(5, 1)}","explanation_zh":"R⁻¹={(5, 1)}。每個 (x,y) 換成 (y,x)，按第一、第二分量排序；逆關係不等於反函數。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["二元關係的逆關係"]},"variant":{"family_id":"qf-relation-converse-set","seed":202610043089,"parameters":{"A":[1,2,3],"B":[5,6,7],"R":[[1,5]],"operation":"converse","semantic_template":"qf-relation-converse-set/{\"A\":[1,2,3],\"B\":[5,6,7],\"R\":[[1,5]],\"operation\":\"converse\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r03-relation-0090","batch_id":"batch-20261003-advance-20261004-r03-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-relation"],"title_zh":"二元關係的逆關係","stem_zh":"二元關係的來源 A={2, 3, 4}、目標 B={6, 7, 8}，R={(2, 6), (2, 7)}⊆A×B。交換來源與目標後，逆關係 R⁻¹ 的規範有序對集合為何？","answer":{"kind":"expression","value":"{(6, 2), (7, 2)}","explanation_zh":"R⁻¹={(6, 2), (7, 2)}。每個 (x,y) 換成 (y,x)，按第一、第二分量排序；逆關係不等於反函數。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["二元關係的逆關係"]},"variant":{"family_id":"qf-relation-converse-set","seed":202610043090,"parameters":{"A":[2,3,4],"B":[6,7,8],"R":[[2,6],[2,7]],"operation":"converse","semantic_template":"qf-relation-converse-set/{\"A\":[2,3,4],\"B\":[6,7,8],\"R\":[[2,6],[2,7]],\"operation\":\"converse\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r03-relation-0091","batch_id":"batch-20261003-advance-20261004-r03-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-relation"],"title_zh":"二元關係的逆關係","stem_zh":"二元關係的來源 A={3, 4, 5}、目標 B={7, 8, 9}，R={(3, 8), (4, 8)}⊆A×B。交換來源與目標後，逆關係 R⁻¹ 的規範有序對集合為何？","answer":{"kind":"expression","value":"{(8, 3), (8, 4)}","explanation_zh":"R⁻¹={(8, 3), (8, 4)}。每個 (x,y) 換成 (y,x)，按第一、第二分量排序；逆關係不等於反函數。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["二元關係的逆關係"]},"variant":{"family_id":"qf-relation-converse-set","seed":202610043091,"parameters":{"A":[3,4,5],"B":[7,8,9],"R":[[3,8],[4,8]],"operation":"converse","semantic_template":"qf-relation-converse-set/{\"A\":[3,4,5],\"B\":[7,8,9],\"R\":[[3,8],[4,8]],\"operation\":\"converse\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r03-relation-0092","batch_id":"batch-20261003-advance-20261004-r03-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-relation"],"title_zh":"二元關係的逆關係","stem_zh":"二元關係的來源 A={4, 5, 6}、目標 B={8, 9, 10}，R={(4, 8), (5, 9), (6, 10)}⊆A×B。交換來源與目標後，逆關係 R⁻¹ 的規範有序對集合為何？","answer":{"kind":"expression","value":"{(8, 4), (9, 5), (10, 6)}","explanation_zh":"R⁻¹={(8, 4), (9, 5), (10, 6)}。每個 (x,y) 換成 (y,x)，按第一、第二分量排序；逆關係不等於反函數。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["二元關係的逆關係"]},"variant":{"family_id":"qf-relation-converse-set","seed":202610043092,"parameters":{"A":[4,5,6],"B":[8,9,10],"R":[[4,8],[5,9],[6,10]],"operation":"converse","semantic_template":"qf-relation-converse-set/{\"A\":[4,5,6],\"B\":[8,9,10],\"R\":[[4,8],[5,9],[6,10]],\"operation\":\"converse\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r03-relation-0093","batch_id":"batch-20261003-advance-20261004-r03-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-relation"],"title_zh":"二元關係的逆關係","stem_zh":"二元關係的來源 A={5, 6, 7}、目標 B={9, 10, 11}，R={(6, 9), (7, 10)}⊆A×B。交換來源與目標後，逆關係 R⁻¹ 的規範有序對集合為何？","answer":{"kind":"expression","value":"{(9, 6), (10, 7)}","explanation_zh":"R⁻¹={(9, 6), (10, 7)}。每個 (x,y) 換成 (y,x)，按第一、第二分量排序；逆關係不等於反函數。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["二元關係的逆關係"]},"variant":{"family_id":"qf-relation-converse-set","seed":202610043093,"parameters":{"A":[5,6,7],"B":[9,10,11],"R":[[6,9],[7,10]],"operation":"converse","semantic_template":"qf-relation-converse-set/{\"A\":[5,6,7],\"B\":[9,10,11],\"R\":[[6,9],[7,10]],\"operation\":\"converse\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r03-relation-0094","batch_id":"batch-20261003-advance-20261004-r03-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-relation"],"title_zh":"二元關係的逆關係","stem_zh":"二元關係的來源 A={6, 7, 8}、目標 B={10, 11, 12}，R=∅⊆A×B。交換來源與目標後，逆關係 R⁻¹ 的規範有序對集合為何？","answer":{"kind":"expression","value":"∅","explanation_zh":"R⁻¹=∅。每個 (x,y) 換成 (y,x)，按第一、第二分量排序；逆關係不等於反函數。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["二元關係的逆關係"]},"variant":{"family_id":"qf-relation-converse-set","seed":202610043094,"parameters":{"A":[6,7,8],"B":[10,11,12],"R":[],"operation":"converse","semantic_template":"qf-relation-converse-set/{\"A\":[6,7,8],\"B\":[10,11,12],\"R\":[],\"operation\":\"converse\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r03-relation-0095","batch_id":"batch-20261003-advance-20261004-r03-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-relation"],"title_zh":"二元關係的逆關係","stem_zh":"二元關係的來源 A={7, 8, 9}、目標 B={11, 12, 13}，R={(7, 11)}⊆A×B。交換來源與目標後，逆關係 R⁻¹ 的規範有序對集合為何？","answer":{"kind":"expression","value":"{(11, 7)}","explanation_zh":"R⁻¹={(11, 7)}。每個 (x,y) 換成 (y,x)，按第一、第二分量排序；逆關係不等於反函數。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["二元關係的逆關係"]},"variant":{"family_id":"qf-relation-converse-set","seed":202610043095,"parameters":{"A":[7,8,9],"B":[11,12,13],"R":[[7,11]],"operation":"converse","semantic_template":"qf-relation-converse-set/{\"A\":[7,8,9],\"B\":[11,12,13],\"R\":[[7,11]],\"operation\":\"converse\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r03-relation-0096","batch_id":"batch-20261003-advance-20261004-r03-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-relation"],"title_zh":"二元關係的逆關係","stem_zh":"二元關係的來源 A={8, 9, 10}、目標 B={12, 13, 14}，R={(8, 12), (8, 13)}⊆A×B。交換來源與目標後，逆關係 R⁻¹ 的規範有序對集合為何？","answer":{"kind":"expression","value":"{(12, 8), (13, 8)}","explanation_zh":"R⁻¹={(12, 8), (13, 8)}。每個 (x,y) 換成 (y,x)，按第一、第二分量排序；逆關係不等於反函數。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["二元關係的逆關係"]},"variant":{"family_id":"qf-relation-converse-set","seed":202610043096,"parameters":{"A":[8,9,10],"B":[12,13,14],"R":[[8,12],[8,13]],"operation":"converse","semantic_template":"qf-relation-converse-set/{\"A\":[8,9,10],\"B\":[12,13,14],\"R\":[[8,12],[8,13]],\"operation\":\"converse\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r03-relation-0097","batch_id":"batch-20261003-advance-20261004-r03-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-relation"],"title_zh":"二元關係的逆關係","stem_zh":"二元關係的來源 A={9, 10, 11}、目標 B={13, 14, 15}，R={(9, 14), (10, 14)}⊆A×B。交換來源與目標後，逆關係 R⁻¹ 的規範有序對集合為何？","answer":{"kind":"expression","value":"{(14, 9), (14, 10)}","explanation_zh":"R⁻¹={(14, 9), (14, 10)}。每個 (x,y) 換成 (y,x)，按第一、第二分量排序；逆關係不等於反函數。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["二元關係的逆關係"]},"variant":{"family_id":"qf-relation-converse-set","seed":202610043097,"parameters":{"A":[9,10,11],"B":[13,14,15],"R":[[9,14],[10,14]],"operation":"converse","semantic_template":"qf-relation-converse-set/{\"A\":[9,10,11],\"B\":[13,14,15],\"R\":[[9,14],[10,14]],\"operation\":\"converse\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r03-relation-0098","batch_id":"batch-20261003-advance-20261004-r03-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-relation"],"title_zh":"二元關係的逆關係","stem_zh":"二元關係的來源 A={10, 11, 12}、目標 B={14, 15, 16}，R={(10, 14), (11, 15), (12, 16)}⊆A×B。交換來源與目標後，逆關係 R⁻¹ 的規範有序對集合為何？","answer":{"kind":"expression","value":"{(14, 10), (15, 11), (16, 12)}","explanation_zh":"R⁻¹={(14, 10), (15, 11), (16, 12)}。每個 (x,y) 換成 (y,x)，按第一、第二分量排序；逆關係不等於反函數。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["二元關係的逆關係"]},"variant":{"family_id":"qf-relation-converse-set","seed":202610043098,"parameters":{"A":[10,11,12],"B":[14,15,16],"R":[[10,14],[11,15],[12,16]],"operation":"converse","semantic_template":"qf-relation-converse-set/{\"A\":[10,11,12],\"B\":[14,15,16],\"R\":[[10,14],[11,15],[12,16]],\"operation\":\"converse\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r03-relation-0099","batch_id":"batch-20261003-advance-20261004-r03-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-relation"],"title_zh":"二元關係的逆關係","stem_zh":"二元關係的來源 A={11, 12, 13}、目標 B={15, 16, 17}，R={(12, 15), (13, 16)}⊆A×B。交換來源與目標後，逆關係 R⁻¹ 的規範有序對集合為何？","answer":{"kind":"expression","value":"{(15, 12), (16, 13)}","explanation_zh":"R⁻¹={(15, 12), (16, 13)}。每個 (x,y) 換成 (y,x)，按第一、第二分量排序；逆關係不等於反函數。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["二元關係的逆關係"]},"variant":{"family_id":"qf-relation-converse-set","seed":202610043099,"parameters":{"A":[11,12,13],"B":[15,16,17],"R":[[12,15],[13,16]],"operation":"converse","semantic_template":"qf-relation-converse-set/{\"A\":[11,12,13],\"B\":[15,16,17],\"R\":[[12,15],[13,16]],\"operation\":\"converse\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261003-advance-20261004-r03-relation-0100","batch_id":"batch-20261003-advance-20261004-r03-relation-foundations","language":"zh-Hant","source_mko_ids":["mko-relation"],"title_zh":"二元關係的逆關係","stem_zh":"二元關係的來源 A={12, 13, 14}、目標 B={16, 17, 18}，R=∅⊆A×B。交換來源與目標後，逆關係 R⁻¹ 的規範有序對集合為何？","answer":{"kind":"expression","value":"∅","explanation_zh":"R⁻¹=∅。每個 (x,y) 換成 (y,x)，按第一、第二分量排序；逆關係不等於反函數。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["二元關係的逆關係"]},"variant":{"family_id":"qf-relation-converse-set","seed":202610043100,"parameters":{"A":[12,13,14],"B":[16,17,18],"R":[],"operation":"converse","semantic_template":"qf-relation-converse-set/{\"A\":[12,13,14],\"B\":[16,17,18],\"R\":[],\"operation\":\"converse\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication"],"warning_zh":"僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"Codex independent peer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-03","contract_version":"ocme-question-factory-v0.1"}}