{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261009-advance-20261013-r06-event-0038",
  "pilot_number": "R0011-06013",
  "number": 6013,
  "family_id": "qf-density13-event-boolean-algebra-pair",
  "title": {
    "zh-Hant": "第 6013 題：聯交事件與聯集外補",
    "en": "Question 6013: Event Union, Intersection, and Outside Complement"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261009-advance-20261013-r06-event-0038",
    "batch_id": "batch-20261009-advance-20261013-r06-event-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-event"
    ],
    "title_zh": "聯交事件與聯集外補",
    "stem_zh": "S=Fin(n)，n=5；F=P(S)；A=[0, 1, 3]、B=[]均在S內。依序列 (A∪B, A∩B, S\\(A∪B))，各去重遞增。",
    "answer": {
      "kind": "expression",
      "value": "([0, 1, 3], [], [2, 4])",
      "explanation_zh": "聯交與外補事件=([0, 1, 3], [], [2, 4])。枚舉同一S的聯交與外補，不混同裸結果與事件；有限S不自動等於F=P(S)，集合資料不給概率或獨立性。"
    },
    "difficulty": {
      "band": "upper_secondary",
      "level": 2,
      "skills": [
        "聯交事件與聯集外補"
      ]
    },
    "variant": {
      "family_id": "qf-density13-event-boolean-algebra-pair",
      "seed": 202610136038,
      "parameters": {
        "n": 5,
        "A": [
          0,
          1,
          3
        ],
        "B": [],
        "event_model": "full_powerset",
        "operation": "density13_event_boolean_algebra_pair",
        "domain": "finite_declared_event_family",
        "semantic_template": "qf-density13-event-boolean-algebra-pair/[5,[0,1,3],[]]"
      }
    },
    "verification": {
      "status": "candidate",
      "checks": [
        "producer_exact_integer_rational_arithmetic",
        "producer_finite_enumeration",
        "producer_source_snapshot_check",
        "producer_substantive_template_deduplication",
        "producer_json_schema"
      ],
      "warning_zh": "僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"
    },
    "provenance": {
      "generator_role": "question_producer",
      "generator_model": "model_unknown",
      "generated_at": "2026-10-09",
      "contract_version": "ocme-question-factory-v0.1"
    }
  },
  "original_english_edition": {
    "schema_version": "ocme-question-localization-v0.1",
    "question_id": "q-20261009-advance-20261013-r06-event-0038",
    "batch_id": "batch-20261009-advance-20261013-r06-event-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "dcd92c15979f05311f1f21c9d853f71a98f7c0bd2fc425cdf0bf14912ae637fd",
      "family_id": "qf-density13-event-boolean-algebra-pair",
      "seed": 202610136038,
      "semantic_template": "qf-density13-event-boolean-algebra-pair/[5,[0,1,3],[]]"
    },
    "fields": {
      "title": "Event Union, Intersection, and Outside Complement",
      "stem": "S=Fin(5)={0,…,4}; the event family is explicitly F=P(S). A=[0, 1, 3], B=[] are subsets of S. Give (A∪B, A∩B, S\\(A∪B)), each deduplicated and increasing.",
      "answer": {
        "kind": "expression",
        "value": "([0, 1, 3], [], [2, 4])",
        "explanation": "Enumerate all three sets in the same sample space without confusing an outcome with an event. A finite S alone does not declare F=P(S); these set data assign neither probabilities nor independence. The exact result, retaining the requested order, is ([0, 1, 3], [], [2, 4])."
      },
      "skills": [
        "Event Union, Intersection, and Outside Complement"
      ],
      "verification_warning": "This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."
    }
  },
  "givens": [
    {
      "id": "G001",
      "text": {
        "zh-Hant": "S=Fin(n)，n=5；F=P(S)；A=[0, 1, 3]、B=[]均在S內。依序列 (A∪B, A∩B, S\\(A∪B))，各去重遞增。",
        "en": "S=Fin(5)={0,…,4}; the event family is explicitly F=P(S). A=[0, 1, 3], B=[] are subsets of S. Give (A∪B, A∩B, S\\(A∪B)), each deduplicated and increasing."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "S=Fin(5)=[0, 1, 2, 3, 4];declared event model=full_powerset",
      "after": "finite S alone does not specify its event family or a probability law",
      "relation": "implies",
      "rule_id": "declared_finite_event_family",
      "conditions": [],
      "explanation": {
        "zh-Hant": "有限樣本空間不自動指定F=P(S)；此處依題目宣告的事件族運算，集合資料不給概率、等可能性或獨立性。",
        "en": "A finite sample space does not automatically choose F=P(S).Use the declared event family;set data alone give no probabilities,equiprobability or independence."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S002",
      "uses": [
        "G001",
        "S001"
      ],
      "before": "outcome=0;in A=是;in B=否",
      "after": "in union=是;in intersection=否;outside union=否",
      "relation": "implies",
      "rule_id": "event_boolean_membership_row",
      "conditions": [],
      "explanation": {
        "zh-Hant": "在同一S逐元素作含括聯集、交集及聯集外補，不把單一基本結果與事件集合混同。",
        "en": "Within the same S,evaluate inclusive union,intersection and the union's complement for each element;an elementary outcome is not an event set."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S003",
      "uses": [
        "G001",
        "S002"
      ],
      "before": "outcome=1;in A=是;in B=否",
      "after": "in union=是;in intersection=否;outside union=否",
      "relation": "implies",
      "rule_id": "event_boolean_membership_row",
      "conditions": [],
      "explanation": {
        "zh-Hant": "在同一S逐元素作含括聯集、交集及聯集外補，不把單一基本結果與事件集合混同。",
        "en": "Within the same S,evaluate inclusive union,intersection and the union's complement for each element;an elementary outcome is not an event set."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S004",
      "uses": [
        "G001",
        "S003"
      ],
      "before": "outcome=2;in A=否;in B=否",
      "after": "in union=否;in intersection=否;outside union=是",
      "relation": "implies",
      "rule_id": "event_boolean_membership_row",
      "conditions": [],
      "explanation": {
        "zh-Hant": "在同一S逐元素作含括聯集、交集及聯集外補，不把單一基本結果與事件集合混同。",
        "en": "Within the same S,evaluate inclusive union,intersection and the union's complement for each element;an elementary outcome is not an event set."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S005",
      "uses": [
        "G001",
        "S004"
      ],
      "before": "outcome=3;in A=是;in B=否",
      "after": "in union=是;in intersection=否;outside union=否",
      "relation": "implies",
      "rule_id": "event_boolean_membership_row",
      "conditions": [],
      "explanation": {
        "zh-Hant": "在同一S逐元素作含括聯集、交集及聯集外補，不把單一基本結果與事件集合混同。",
        "en": "Within the same S,evaluate inclusive union,intersection and the union's complement for each element;an elementary outcome is not an event set."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S006",
      "uses": [
        "G001",
        "S005"
      ],
      "before": "outcome=4;in A=否;in B=否",
      "after": "in union=否;in intersection=否;outside union=是",
      "relation": "implies",
      "rule_id": "event_boolean_membership_row",
      "conditions": [],
      "explanation": {
        "zh-Hant": "在同一S逐元素作含括聯集、交集及聯集外補，不把單一基本結果與事件集合混同。",
        "en": "Within the same S,evaluate inclusive union,intersection and the union's complement for each element;an elementary outcome is not an event set."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。",
          "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence."
        }
      },
      "arithmetic": null
    }
  ],
  "math_model": {
    "result": [
      [
        0,
        1,
        3
      ],
      [],
      [
        2,
        4
      ]
    ]
  },
  "conclusion": {
    "canonical_value": "([0, 1, 3], [], [2, 4])",
    "explanation": {
      "zh-Hant": "依上列完整代值及條件核對得到原題答案；原題編號、格式與狀態保持不變。",
      "en": "The full substitutions and premise checks give the original answer; source identity,format and status are unchanged."
    }
  },
  "definition_note": {
    "zh-Hant": "沿用原題明示的論域、假設與來源 MKO 定義。本題已完成第二對話數理與中英語義覆核，並由原始參數逐題獨立重算模型、答案與每個算術步驟；原題仍為 candidate。",
    "en": "Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate."
  },
  "source": {
    "input_commit": "0f7be61e2c22317a4075561dc822c87d881a50db",
    "question_binding": {
      "batch_id": "batch-20261009-advance-20261013-r06-event-foundations",
      "batch_path": "public/data/questions/batches/batch-20261009-advance-20261013-r06-event-foundations/questions.jsonl",
      "batch_sha256": "2ed3cc2f0adb9a12f8e2c4de29b320306a65fe7af8296a1f1c5d0b14bc7776d9",
      "original_line": 38,
      "original_line_sha256": "73b0b8d09b367a31130c065ae94b9042e0168ea4560fb69a8f4c00f2a69254f5",
      "question_compact_sha256": "dcd92c15979f05311f1f21c9d853f71a98f7c0bd2fc425cdf0bf14912ae637fd",
      "english_row_compact_sha256": "2fbbd2abea32de0b21af974bba9267d8d133cbe0cd9b4992dc5b0888e655b15b",
      "english_manifest_path": "public/data/i18n/editions/batch-20261009-advance-20261013-r06-event-foundations.en.json",
      "english_rows_path": "public/data/i18n/questions/batch-20261009-advance-20261013-r06-event-foundations.en.jsonl"
    },
    "mko_bindings": {
      "mko-event": {
        "sha256": "5962ae6672cb79c34888bdda4ac66de557a4f9b21243f2445414d7dd326b9c74",
        "english_sha256": "12d8199c90226e08b52a3f9cf27ce44053dc8473e8bab4280145aea146a4c261",
        "original_path": "public/data/mko/event.json",
        "english_path": "public/data/i18n/editions/mko-event.en.json"
      }
    },
    "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff",
    "selected_part_sha256": "5b2ce631281eb24b279908f47c6de66fa60d326520852879d324c4f3ac653970",
    "original_candidate_sha256": "47596b4990a78a7472f9e5c2d04bdafedce9b95702fa7d3dca9701928ac39a8f",
    "original_candidate_line_sha256": "47596b4990a78a7472f9e5c2d04bdafedce9b95702fa7d3dca9701928ac39a8f"
  },
  "historical_source_state": {
    "review_state": {
      "derivation": "candidate_authored_and_locally_checked",
      "source_question_state": "candidate",
      "independent_AI_or_human_review": "NOT_PERFORMED",
      "formalization": "not_formalized",
      "Lean": "NOT_RUN"
    },
    "scope_note": {
      "zh-Hant": "來源綁定教學推導候選；精確核對不等於獨立AI／人工審查或一般形式證明。原題、答案、MKO、公式、AST與歷史不變。",
      "en": "Source-bound pedagogical derivation candidate.Exact checking is not independent AI/human review or general formal proof.Original questions,answers,MKOs,formulas,ASTs and history remain unchanged."
    },
    "label": "Frozen producer metadata; current per-question review is separate below."
  },
  "revision": {
    "id": "review-20261026-005-p12",
    "review_date": "2026-10-26",
    "owner_authorized_reader_publication": true,
    "scope": "One hundred sixty family prototypes and their mathematical/bilingual reasoning variants, stems, boundary statements and every distinct explanation shape were reviewed in a context independent of the producer; parameter substitutions, source bindings, full math models, both answer strings and every arithmetic step were independently reconstructed per question from the original parameters only, with negative controls. This grouped method does not claim a separate literal AI reading of every repeated sentence or an extra AI voter.",
    "review_receipts": {
      "semantic_review_sha256": "b9bbc1c947f477a9c4bb5a579368ac5265f2da4572ce612c30f3633ecaf4ae80",
      "question_first_witness_sha256": "93f720df3ab3af13a792a3e6b443728e3c2a56276df839911f36569333236c55"
    },
    "status": "second_context_reviewed",
    "mathematical_semantics": true,
    "bilingual_semantics": true,
    "steps_reviewed": 6,
    "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction",
    "exact_question_first_check": {
      "model_sha256": "8329d5697fcec72fdb24557fd3b1beb29cf435d2b8d5bfd97a9e8aa70f937b99",
      "arithmetic_steps": 0
    },
    "uses_answer_as_premise": false,
    "human_review": "NOT_PERFORMED",
    "formalization": "not_formalized",
    "Lean": "NOT_RUN",
    "Canonical_truth_promotion": false,
    "authority": "Owner-authorized reviewed reader publication; original question/candidate states remain historical and unchanged."
  }
}