{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261008-advance-20261010-r02-set-operations-0033",
  "pilot_number": "R0007-04433",
  "number": 4433,
  "family_id": "qf-density10-set-symmetric-difference",
  "title": {
    "zh-Hant": "第 4433 題：兩側差的聯集",
    "en": "Question 4433: Sorted Symmetric Difference"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261008-advance-20261010-r02-set-operations-0033",
    "batch_id": "batch-20261008-advance-20261010-r02-set-operations-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-set-operations"
    ],
    "title_zh": "兩側差的聯集",
    "stem_zh": "有限整數集合 A=[-7, 0, 7]、B=[0, 8]。列 (A\\B)∪(B\\A)，按整數遞增且不重列。",
    "answer": {
      "kind": "expression",
      "value": "[-7, 7, 8]",
      "explanation_zh": "兩側差聯集=[-7, 7, 8]。保留僅屬一側的元素；兩側都屬的元素排除，並非一般聯集。"
    },
    "difficulty": {
      "band": "upper_secondary",
      "level": 1,
      "skills": [
        "兩側差的聯集"
      ]
    },
    "variant": {
      "family_id": "qf-density10-set-symmetric-difference",
      "seed": 202610102033,
      "parameters": {
        "A": [
          -7,
          0,
          7
        ],
        "B": [
          0,
          8
        ],
        "operation": "density10_set_symmetric_difference",
        "domain": "finite_integer_sets",
        "semantic_template": "qf-density10-set-symmetric-difference/[[-7,0,7],[0,8]]"
      }
    },
    "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-0033",
    "batch_id": "batch-20261008-advance-20261010-r02-set-operations-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "f711175bf10d466caff0f9dd93e1a51ae27e7763b30aad974c4ddba2ddea862a",
      "family_id": "qf-density10-set-symmetric-difference",
      "seed": 202610102033,
      "semantic_template": "qf-density10-set-symmetric-difference/[[-7,0,7],[0,8]]"
    },
    "fields": {
      "title": "Sorted Symmetric Difference",
      "stem": "Finite integer sets A=[-7, 0, 7] and B=[0, 8]; repeated elements count once. List (A\\B)∪(B\\A) in increasing order without repetition.",
      "answer": {
        "kind": "expression",
        "value": "[-7, 7, 8]",
        "explanation": "Retain elements belonging to exactly one side; this is not the ordinary union. The exact result, retaining the requested order, is [-7, 7, 8]."
      },
      "skills": [
        "Sorted Symmetric Difference"
      ],
      "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 A=[-7, 0, 7]、B=[0, 8]。列 (A\\B)∪(B\\A)，按整數遞增且不重列。",
        "en": "Finite integer sets A=[-7, 0, 7] and B=[0, 8]; repeated elements count once. List (A\\B)∪(B\\A) in increasing order without repetition."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "raw A=[-7, 0, 7];raw B=[0, 8]",
      "after": "A=[-7, 0, 7];B=[0, 8]",
      "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=-7",
      "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": "S003",
      "uses": [
        "G001",
        "S002"
      ],
      "before": "element=0",
      "after": "in A=是;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=7",
      "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=8",
      "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": "S006",
      "uses": [
        "G001",
        "S005"
      ],
      "before": "A\\B=[-7, 7];B\\A=[8]",
      "after": "[-7, 7, 8]",
      "relation": "implies",
      "rule_id": "symmetric_difference_exclusive_membership",
      "conditions": [],
      "explanation": {
        "zh-Hant": "只屬其中一側才保留；兩側都屬的共有元素必須排除，不是一般聯集。",
        "en": "Retain elements belonging to exactly one side;shared elements must be excluded,unlike ordinary union."
      },
      "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": [
      -7,
      7,
      8
    ],
    "membership_rows": [
      [
        -7,
        true,
        false
      ],
      [
        0,
        true,
        true
      ],
      [
        7,
        true,
        false
      ],
      [
        8,
        false,
        true
      ]
    ]
  },
  "conclusion": {
    "canonical_value": "[-7, 7, 8]",
    "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": 33,
      "original_line_sha256": "2fd1a295ccaeaec67d345325ab444ef2bff8ba6f8c98d86c106f890412914f44",
      "question_compact_sha256": "f711175bf10d466caff0f9dd93e1a51ae27e7763b30aad974c4ddba2ddea862a",
      "english_row_compact_sha256": "ed2a25002051124882d7e73502c8c396e671d0531eb579b6a2a357393fb2a41f",
      "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": "8cc400066ae7399a9bbf14a857c1df6562181b4a62bf3fc34051393869e3ee37",
    "original_candidate_line_sha256": "8cc400066ae7399a9bbf14a857c1df6562181b4a62bf3fc34051393869e3ee37"
  },
  "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": 6,
    "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction",
    "exact_question_first_check": {
      "model_sha256": "cd02232c2fe5131be5ba858b0687449ca0e437d9fbc35abead6e5eb3c46ce244",
      "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."
  }
}