{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261008-advance-20261010-r02-set-operations-0075",
  "pilot_number": "R0007-04425",
  "number": 4425,
  "family_id": "qf-density10-set-demorgan-sets",
  "title": {
    "zh-Hant": "第 4425 題：明示全集下的兩個補集式",
    "en": "Question 4425: Two Relative-Complement Computations"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261008-advance-20261010-r02-set-operations-0075",
    "batch_id": "batch-20261008-advance-20261010-r02-set-operations-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-set-operations"
    ],
    "title_zh": "明示全集下的兩個補集式",
    "stem_zh": "有限整數集合，全集U=[-24, -1, 0, 1, 24]、A=[0]、B=[-1, 24]，A、B⊆U。依序列 (U\\(A∪B), (U\\A)∩(U\\B))，各列遞增。",
    "answer": {
      "kind": "expression",
      "value": "([-24, 1], [-24, 1])",
      "explanation_zh": "補集兩式=([-24, 1], [-24, 1])。第一式找U內兩側皆不屬者，第二式分別取補後交集；補集只相對明示U。"
    },
    "difficulty": {
      "band": "upper_secondary",
      "level": 3,
      "skills": [
        "明示全集下的兩個補集式"
      ]
    },
    "variant": {
      "family_id": "qf-density10-set-demorgan-sets",
      "seed": 202610102075,
      "parameters": {
        "U": [
          -24,
          -1,
          0,
          1,
          24
        ],
        "A": [
          0
        ],
        "B": [
          -1,
          24
        ],
        "operation": "density10_set_demorgan_sets",
        "domain": "finite_integer_sets",
        "semantic_template": "qf-density10-set-demorgan-sets/[[-24,-1,0,1,24],[-1,0,24]]"
      }
    },
    "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": "2026-10-08",
      "contract_version": "ocme-question-factory-v0.1"
    }
  },
  "original_english_edition": {
    "schema_version": "ocme-question-localization-v0.1",
    "question_id": "q-20261008-advance-20261010-r02-set-operations-0075",
    "batch_id": "batch-20261008-advance-20261010-r02-set-operations-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "f66fc19296282c1877bb610ad2acdb42d64704dc3ca4532c4148b9760e165e86",
      "family_id": "qf-density10-set-demorgan-sets",
      "seed": 202610102075,
      "semantic_template": "qf-density10-set-demorgan-sets/[[-24,-1,0,1,24],[-1,0,24]]"
    },
    "fields": {
      "title": "Two Relative-Complement Computations",
      "stem": "Finite integer sets A=[0] and B=[-1, 24]; repeated elements count once. The explicit universe is U=[-24, -1, 0, 1, 24], with A,B⊆U. Give (U\\(A∪B), (U\\A)∩(U\\B)), each sorted increasingly.",
      "answer": {
        "kind": "expression",
        "value": "([-24, 1], [-24, 1])",
        "explanation": "Compute each route inside the declared universe; complements are not absolute. The exact result, retaining the requested order, is ([-24, 1], [-24, 1])."
      },
      "skills": [
        "Two Relative-Complement Computations"
      ],
      "verification_warning": "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": "有限整數集合，全集U=[-24, -1, 0, 1, 24]、A=[0]、B=[-1, 24]，A、B⊆U。依序列 (U\\(A∪B), (U\\A)∩(U\\B))，各列遞增。",
        "en": "Finite integer sets A=[0] and B=[-1, 24]; repeated elements count once. The explicit universe is U=[-24, -1, 0, 1, 24], with A,B⊆U. Give (U\\(A∪B), (U\\A)∩(U\\B)), each sorted increasingly."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "raw A=[0];raw B=[-1, 24]",
      "after": "A=[0];B=[-1, 24]",
      "relation": "implies",
      "rule_id": "finite_set_extensionality",
      "conditions": [],
      "explanation": {
        "zh-Hant": "整數集合先去重及數值排序；重列或列序不改元素隸屬，原來源清單仍保存。",
        "en": "Deduplicate and numerically sort integer sets;repetition and input order do not change membership.The original lists remain retained."
      },
      "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": "element=-24",
      "after": "in A=否;in B=否",
      "relation": "implies",
      "rule_id": "finite_set_membership_row",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐元素保留兩側隸屬，集合運算由這兩個布林值決定，不重複計數。",
        "en": "Record membership on both sides for each element;these Boolean values determine the set operations without duplicate counting."
      },
      "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": "element=-1",
      "after": "in A=否;in B=是",
      "relation": "implies",
      "rule_id": "finite_set_membership_row",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐元素保留兩側隸屬，集合運算由這兩個布林值決定，不重複計數。",
        "en": "Record membership on both sides for each element;these Boolean values determine the set operations without duplicate counting."
      },
      "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": "element=0",
      "after": "in A=是;in B=否",
      "relation": "implies",
      "rule_id": "finite_set_membership_row",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐元素保留兩側隸屬，集合運算由這兩個布林值決定，不重複計數。",
        "en": "Record membership on both sides for each element;these Boolean values determine the set operations without duplicate counting."
      },
      "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": "element=1",
      "after": "in A=否;in B=否",
      "relation": "implies",
      "rule_id": "finite_set_membership_row",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐元素保留兩側隸屬，集合運算由這兩個布林值決定，不重複計數。",
        "en": "Record membership on both sides for each element;these Boolean values determine the set operations without duplicate counting."
      },
      "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": "element=24",
      "after": "in A=否;in B=是",
      "relation": "implies",
      "rule_id": "finite_set_membership_row",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐元素保留兩側隸屬，集合運算由這兩個布林值決定，不重複計數。",
        "en": "Record membership on both sides for each element;these Boolean values determine the set operations without duplicate counting."
      },
      "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": "S007",
      "uses": [
        "G001",
        "S006"
      ],
      "before": "explicit U=[-24, -1, 0, 1, 24];U\\A=[-24, -1, 1, 24];U\\B=[-24, 0, 1]",
      "after": "U\\(A∪B)=[-24, 1];(U\\A)∩(U\\B)=[-24, 1]",
      "relation": "implies",
      "rule_id": "relative_demorgan_complements",
      "conditions": [],
      "explanation": {
        "zh-Hant": "兩式都逐元素要求不在A且不在B；補集只相對明示U，不能任取域外整數。",
        "en": "Both expressions require membership in neither A nor B;complements are relative only to the declared U,not arbitrary outside integers."
      },
      "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": [
      [
        -24,
        1
      ],
      [
        -24,
        1
      ]
    ],
    "complements": [
      [
        -24,
        -1,
        1,
        24
      ],
      [
        -24,
        0,
        1
      ]
    ],
    "membership_rows": [
      [
        -24,
        false,
        false
      ],
      [
        -1,
        false,
        true
      ],
      [
        0,
        true,
        false
      ],
      [
        1,
        false,
        false
      ],
      [
        24,
        false,
        true
      ]
    ]
  },
  "conclusion": {
    "canonical_value": "([-24, 1], [-24, 1])",
    "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-20261008-advance-20261010-r02-set-operations-foundations",
      "batch_path": "public/data/questions/batches/batch-20261008-advance-20261010-r02-set-operations-foundations/questions.jsonl",
      "batch_sha256": "1eb875be701255297ceef2bd83ebaceff2f7cdec925101d78d161062916bde7d",
      "original_line": 75,
      "original_line_sha256": "c0e84fd8ccd7f84b9bcd4524c2162d5d9b4f29008f88db3fdeed3ae2e66aa0d5",
      "question_compact_sha256": "f66fc19296282c1877bb610ad2acdb42d64704dc3ca4532c4148b9760e165e86",
      "english_row_compact_sha256": "c9e5a5ba0890a2b05ff153897b502c3a6c4a0aa3ec8634ba74cfe2d8d3367a68",
      "english_manifest_path": "public/data/i18n/editions/batch-20261008-advance-20261010-r02-set-operations-foundations.en.json",
      "english_rows_path": "public/data/i18n/questions/batch-20261008-advance-20261010-r02-set-operations-foundations.en.jsonl"
    },
    "mko_bindings": {
      "mko-set-operations": {
        "sha256": "e857b27da045df1a33d58d7cd56b319d6c8ec23162a8cfc3c15f7c70d30f317b",
        "english_sha256": "3cad86e57536464f92b934935e0803aa9aee386b3c235758ae9da575c56f3ac3",
        "original_path": "public/data/mko/set-operations.json",
        "english_path": "public/data/i18n/editions/mko-set-operations.en.json"
      }
    },
    "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff",
    "selected_part_sha256": "de1fb611cf230f243ddd3334fbc53c9205cbe528b78346a10544972dede78b95",
    "original_candidate_sha256": "50861a1c92d37895dc96b4c3fd0953b0c6e43f5375da3f4a7a26c57af2def87f",
    "original_candidate_line_sha256": "50861a1c92d37895dc96b4c3fd0953b0c6e43f5375da3f4a7a26c57af2def87f"
  },
  "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-20261026-004-p8",
    "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": "faf0582c0184613e6fae207374056133eae600883a29c67d8438113bbe3c485a",
      "question_first_witness_sha256": "4b283b6ff424b1d22caa36f3552b0fa21b1b66a0a883d7cbe9cedc8dd77b2e42"
    },
    "status": "second_context_reviewed",
    "mathematical_semantics": true,
    "bilingual_semantics": true,
    "steps_reviewed": 7,
    "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction",
    "exact_question_first_check": {
      "model_sha256": "16540b6604a23cb2ac69abcd687f6569758af19d3544d7497170ee69c26d86cb",
      "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."
  }
}