{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261008-advance-20261011-r04-set-0043",
  "pilot_number": "R0009-05043",
  "number": 5043,
  "family_id": "qf-density11-set-equality-defect",
  "title": {
    "zh-Hant": "第 5043 題：集合相等與雙向差",
    "en": "Question 5043: Set Equality with Both Difference Witnesses"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261008-advance-20261011-r04-set-0043",
    "batch_id": "batch-20261008-advance-20261011-r04-set-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-set"
    ],
    "title_zh": "集合相等與雙向差",
    "stem_zh": "有限整數集合；A=[-17, 0, 17]、B=[0, 17, 18]。依序答 (A與B是否相等, A\\B, B\\A)，差集遞增列舉。",
    "answer": {
      "kind": "expression",
      "value": "(否, [-17], [18])",
      "explanation_zh": "相等資格與雙向差=(否, [-17], [18])。先逐元素取兩側差；兩側差都空才相等，只看大小相同不足。"
    },
    "difficulty": {
      "band": "lower_secondary",
      "level": 3,
      "skills": [
        "集合相等與雙向差"
      ]
    },
    "variant": {
      "family_id": "qf-density11-set-equality-defect",
      "seed": 202610114043,
      "parameters": {
        "A": [
          -17,
          0,
          17
        ],
        "B": [
          0,
          17,
          18
        ],
        "operation": "density11_set_equality_defect",
        "domain": "finite_integer_set_slice",
        "semantic_template": "qf-density11-set-equality-defect/[[-17],[18]]"
      }
    },
    "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-20261011-r04-set-0043",
    "batch_id": "batch-20261008-advance-20261011-r04-set-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "c788c0db9f6bfe1a88c117a50f95f0f1c667856da84307673b7ef0ecb65322d8",
      "family_id": "qf-density11-set-equality-defect",
      "seed": 202610114043,
      "semantic_template": "qf-density11-set-equality-defect/[[-17],[18]]"
    },
    "fields": {
      "title": "Set Equality with Both Difference Witnesses",
      "stem": "Finite integer sets A=[-17, 0, 17] and B=[0, 17, 18]; repeated elements count once. Give (are A and B equal?, A\\B, B\\A), with increasing difference lists.",
      "answer": {
        "kind": "expression",
        "value": "(No, [-17], [18])",
        "explanation": "Equality requires both differences to be empty; equal cardinalities alone are insufficient. The exact result, retaining the requested order, is (No, [-17], [18])."
      },
      "skills": [
        "Set Equality with Both Difference Witnesses"
      ],
      "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=[-17, 0, 17]、B=[0, 17, 18]。依序答 (A與B是否相等, A\\B, B\\A)，差集遞增列舉。",
        "en": "Finite integer sets A=[-17, 0, 17] and B=[0, 17, 18]; repeated elements count once. Give (are A and B equal?, A\\B, B\\A), with increasing difference lists."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "raw integer list=[-17, 0, 17]",
      "after": "extensional set=[-17, 0, 17]",
      "relation": "implies",
      "rule_id": "set_extensionality_normalization",
      "conditions": [],
      "explanation": {
        "zh-Hant": "集合只看有哪些元素；去重及數值排序不改隸屬，並保留原始清單作來源。",
        "en": "A set depends on its elements;deduplication and numeric ordering do not change membership.The original list is retained as source."
      },
      "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=-17;in A=是;in B=否",
      "after": "in A\\B=是;in B\\A=否",
      "relation": "implies",
      "rule_id": "two_sided_set_difference_row",
      "conditions": [],
      "explanation": {
        "zh-Hant": "分別查兩側差，兩邊元素數相同不保證元素相同。",
        "en": "Check both directional differences;equal cardinalities alone do not imply equal elements."
      },
      "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;in A=是;in B=是",
      "after": "in A\\B=否;in B\\A=否",
      "relation": "implies",
      "rule_id": "two_sided_set_difference_row",
      "conditions": [],
      "explanation": {
        "zh-Hant": "分別查兩側差，兩邊元素數相同不保證元素相同。",
        "en": "Check both directional differences;equal cardinalities alone do not imply equal elements."
      },
      "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=17;in A=是;in B=是",
      "after": "in A\\B=否;in B\\A=否",
      "relation": "implies",
      "rule_id": "two_sided_set_difference_row",
      "conditions": [],
      "explanation": {
        "zh-Hant": "分別查兩側差，兩邊元素數相同不保證元素相同。",
        "en": "Check both directional differences;equal cardinalities alone do not imply equal elements."
      },
      "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=18;in A=否;in B=是",
      "after": "in A\\B=否;in B\\A=是",
      "relation": "implies",
      "rule_id": "two_sided_set_difference_row",
      "conditions": [],
      "explanation": {
        "zh-Hant": "分別查兩側差，兩邊元素數相同不保證元素相同。",
        "en": "Check both directional differences;equal cardinalities alone do not imply equal elements."
      },
      "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=[-17];B\\A=[18]",
      "after": "sets equal=否",
      "relation": "implies",
      "rule_id": "set_equality_two_inclusions",
      "conditions": [],
      "explanation": {
        "zh-Hant": "兩側差都空，才同時A⊆B及B⊆A，因外延性得到相等；只一側空不夠。",
        "en": "Both differences must be empty to establish A⊆B and B⊆A,and thus extensional equality.One empty difference alone is insufficient."
      },
      "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": [
      false,
      [
        -17
      ],
      [
        18
      ]
    ]
  },
  "conclusion": {
    "canonical_value": "(否, [-17], [18])",
    "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-20261011-r04-set-foundations",
      "batch_path": "public/data/questions/batches/batch-20261008-advance-20261011-r04-set-foundations/questions.jsonl",
      "batch_sha256": "f470bac7d0a7bba63071eb9254603f66dd92d4c55f5ceb13a0d1f7850edeb98e",
      "original_line": 43,
      "original_line_sha256": "67c40c0b96ca30b2ccbc44f7f8ad547912319a0f592d30cdeca9567b0053ac9a",
      "question_compact_sha256": "c788c0db9f6bfe1a88c117a50f95f0f1c667856da84307673b7ef0ecb65322d8",
      "english_row_compact_sha256": "58074a995421d92ef10c2a4a16f3adf6bb3066d00c8898929c2a861971275809",
      "english_manifest_path": "public/data/i18n/editions/batch-20261008-advance-20261011-r04-set-foundations.en.json",
      "english_rows_path": "public/data/i18n/questions/batch-20261008-advance-20261011-r04-set-foundations.en.jsonl"
    },
    "mko_bindings": {
      "mko-set": {
        "sha256": "4f736510823c29158d78264f57fa9f96c5c2107ddc99999622656fdacc7a5728",
        "english_sha256": "f1024a1b7bcb63048a9ba5c6ca166a8236e5d37f63c4c64d3985f769c7ae742c",
        "original_path": "public/data/mko/set.json",
        "english_path": "public/data/i18n/editions/mko-set.en.json"
      }
    },
    "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff",
    "selected_part_sha256": "3132b1200be7a5de38191badc52fb1727d4dd9d4c2f26818de2e8a76426d2d3f",
    "original_candidate_sha256": "55ed219ec6bfaa84ee390ed3b9aa456340f1d96419e9e54bcd28636749bd3ad8",
    "original_candidate_line_sha256": "55ed219ec6bfaa84ee390ed3b9aa456340f1d96419e9e54bcd28636749bd3ad8"
  },
  "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-p10",
    "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": "71e7db519df31f3a164533e472e293cb64f001b475a8d002b4a5c2740a237b56",
      "question_first_witness_sha256": "5a43f4f81b6256281f8d139a4010bbde127be803c528e5878857269c8f854462"
    },
    "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": "c6eb60b02758b51ed55e5dda6515dd04f68d1f6a0c32d2623bab9148a897f2f3",
      "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."
  }
}