{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261008-advance-20261012-r02-counting-principle-0100",
  "pilot_number": "R0009-05400",
  "number": 5400,
  "family_id": "qf-density12-counting-overlap-correction",
  "title": {
    "zh-Hant": "第 5400 題：有重疊選擇的前後計數",
    "en": "Question 5400: Overlap-Corrected Finite Choice Counts"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261008-advance-20261012-r02-counting-principle-0100",
    "batch_id": "batch-20261008-advance-20261012-r02-counting-principle-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-counting-principle"
    ],
    "title_zh": "有重疊選擇的前後計數",
    "stem_zh": "有限選擇協議；整數選擇集合A=[0, 1, 2]、B=[0, 1, 2, 100, 101]。依序求 (|A|+|B|, |A∪B|, |A∩B|)，不要把重疊元素算兩次。",
    "answer": {
      "kind": "expression",
      "value": "(8, 5, 3)",
      "explanation_zh": "重疊前後計數=(8, 5, 3)。按實際元素隸屬枚舉聯交，兩集合大小相加會重列交集；非概率獨立假設。"
    },
    "difficulty": {
      "band": "lower_secondary",
      "level": 3,
      "skills": [
        "有重疊選擇的前後計數"
      ]
    },
    "variant": {
      "family_id": "qf-density12-counting-overlap-correction",
      "seed": 202610122100,
      "parameters": {
        "A": [
          0,
          1,
          2
        ],
        "B": [
          0,
          1,
          2,
          100,
          101
        ],
        "operation": "density12_counting_overlap_correction",
        "domain": "finite_tagged_counting_protocol",
        "semantic_template": "qf-density12-counting-overlap-correction/[[0,2],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": "2026-10-08",
      "contract_version": "ocme-question-factory-v0.1"
    }
  },
  "original_english_edition": {
    "schema_version": "ocme-question-localization-v0.1",
    "question_id": "q-20261008-advance-20261012-r02-counting-principle-0100",
    "batch_id": "batch-20261008-advance-20261012-r02-counting-principle-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "70058c01d89985259fc13f2e516f5fd6f47386b0b7e0a1da5bfc14ec1f80e0b7",
      "family_id": "qf-density12-counting-overlap-correction",
      "seed": 202610122100,
      "semantic_template": "qf-density12-counting-overlap-correction/[[0,2],3]"
    },
    "fields": {
      "title": "Overlap-Corrected Finite Choice Counts",
      "stem": "Finite integer choice sets A=[0, 1, 2], B=[0, 1, 2, 100, 101]. Give (|A|+|B|, |A∪B|, |A∩B|).",
      "answer": {
        "kind": "expression",
        "value": "(8, 5, 3)",
        "explanation": "Enumerate actual union/intersection membership; adding both cardinalities double-counts shared choices. No probability-independence premise is involved. The exact result, retaining the requested order, is (8, 5, 3)."
      },
      "skills": [
        "Overlap-Corrected Finite Choice Counts"
      ],
      "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": "有限選擇協議；整數選擇集合A=[0, 1, 2]、B=[0, 1, 2, 100, 101]。依序求 (|A|+|B|, |A∪B|, |A∩B|)，不要把重疊元素算兩次。",
        "en": "Finite integer choice sets A=[0, 1, 2], B=[0, 1, 2, 100, 101]. Give (|A|+|B|, |A∪B|, |A∩B|)."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "declared finite tagged choices or full-product protocol",
      "after": "combinatorial counts,not probabilities or independence",
      "relation": "implies",
      "rule_id": "finite_counting_protocol_scope",
      "conditions": [],
      "explanation": {
        "zh-Hant": "加法需互斥標籤，乘法需題目明示允許完整跨階段組合；都不是概率獨立假設。",
        "en": "Addition uses disjoint tags and multiplication uses the declared full cross-stage protocol;neither is a probability-independence assumption."
      },
      "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": "choice=0;in A=是;in B=是",
      "after": "shared=是",
      "relation": "implies",
      "rule_id": "counting_overlap_membership",
      "conditions": [],
      "explanation": {
        "zh-Hant": "重疊元素在兩個大小之和出現兩次，但聯集只出現一次。",
        "en": "A shared element appears twice in the sum of sizes but once in the 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
    },
    {
      "id": "S003",
      "uses": [
        "G001",
        "S002"
      ],
      "before": "choice=1;in A=是;in B=是",
      "after": "shared=是",
      "relation": "implies",
      "rule_id": "counting_overlap_membership",
      "conditions": [],
      "explanation": {
        "zh-Hant": "重疊元素在兩個大小之和出現兩次，但聯集只出現一次。",
        "en": "A shared element appears twice in the sum of sizes but once in the 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
    },
    {
      "id": "S004",
      "uses": [
        "G001",
        "S003"
      ],
      "before": "choice=2;in A=是;in B=是",
      "after": "shared=是",
      "relation": "implies",
      "rule_id": "counting_overlap_membership",
      "conditions": [],
      "explanation": {
        "zh-Hant": "重疊元素在兩個大小之和出現兩次，但聯集只出現一次。",
        "en": "A shared element appears twice in the sum of sizes but once in the 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
    },
    {
      "id": "S005",
      "uses": [
        "G001",
        "S004"
      ],
      "before": "choice=100;in A=否;in B=是",
      "after": "shared=否",
      "relation": "implies",
      "rule_id": "counting_overlap_membership",
      "conditions": [],
      "explanation": {
        "zh-Hant": "重疊元素在兩個大小之和出現兩次，但聯集只出現一次。",
        "en": "A shared element appears twice in the sum of sizes but once in the 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
    },
    {
      "id": "S006",
      "uses": [
        "G001",
        "S005"
      ],
      "before": "choice=101;in A=否;in B=是",
      "after": "shared=否",
      "relation": "implies",
      "rule_id": "counting_overlap_membership",
      "conditions": [],
      "explanation": {
        "zh-Hant": "重疊元素在兩個大小之和出現兩次，但聯集只出現一次。",
        "en": "A shared element appears twice in the sum of sizes but once in the 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
    },
    {
      "id": "S007",
      "uses": [
        "G001",
        "S006"
      ],
      "before": "sum of choice-set sizes := sum(3, 5)",
      "after": "sum of choice-set sizes = 8",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先算可能重列的兩集合大小之和。",
        "en": "First add the two set sizes,which may double-count shared choices."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 sum of choice-set sizes，不以小數近似替代。",
          "en": "Recompute sum of choice-set sizes with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sum",
        "inputs": [
          "3",
          "5"
        ],
        "output": "8",
        "label": "sum of choice-set sizes"
      }
    },
    {
      "id": "S008",
      "uses": [
        "G001",
        "S007"
      ],
      "before": "overlap-corrected union size := sub(8, 3)",
      "after": "overlap-corrected union size = 5",
      "relation": "calculation",
      "rule_id": "exact_rational_arithmetic",
      "conditions": [],
      "explanation": {
        "zh-Hant": "扣一次交集，核對實際聯集大小。",
        "en": "Subtract the intersection once and audit the actual union size."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以精確有理數重算 overlap-corrected union size，不以小數近似替代。",
          "en": "Recompute overlap-corrected union size with exact rational arithmetic, not rounded decimals."
        }
      },
      "arithmetic": {
        "operation": "sub",
        "inputs": [
          "8",
          "3"
        ],
        "output": "5",
        "label": "overlap-corrected union size"
      }
    },
    {
      "id": "S009",
      "uses": [
        "G001",
        "S008"
      ],
      "before": "A∪B=[0, 1, 2, 100, 101];A∩B=[0, 1, 2]",
      "after": "union size=5",
      "relation": "implies",
      "rule_id": "finite_choice_inclusion_exclusion",
      "conditions": [],
      "explanation": {
        "zh-Hant": "若選擇未加互斥標籤，必做此重疊修正；不把同一選項視為兩個結果。",
        "en": "Without disjoint tags,apply the overlap correction rather than treating one shared choice as two outcomes."
      },
      "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": [
      "8",
      5,
      3
    ]
  },
  "conclusion": {
    "canonical_value": "(8, 5, 3)",
    "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": {
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    "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff",
    "selected_part_sha256": "3132b1200be7a5de38191badc52fb1727d4dd9d4c2f26818de2e8a76426d2d3f",
    "original_candidate_sha256": "f161d1b3aab855c04d93232f640c0f1d93dad62ffdcb77564e5992cca026c54a",
    "original_candidate_line_sha256": "f161d1b3aab855c04d93232f640c0f1d93dad62ffdcb77564e5992cca026c54a"
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  "historical_source_state": {
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      "source_question_state": "candidate",
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      "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": {
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    "mathematical_semantics": true,
    "bilingual_semantics": true,
    "steps_reviewed": 9,
    "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction",
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    "formalization": "not_formalized",
    "Lean": "NOT_RUN",
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    "authority": "Owner-authorized reviewed reader publication; original question/candidate states remain historical and unchanged."
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}