{
  "schema_version": "ocme-derivation-reader-v1",
  "guide_id": "derivation-mko-inclusion-exclusion-principle-01",
  "mko_id": "mko-inclusion-exclusion-principle",
  "title": {
    "zh-Hant": "三集合全部七個交集項",
    "en": "All seven intersection terms for three sets"
  },
  "contract": {
    "schema_version": "ocme-derivation-guide-v0.1",
    "guide_id": "derivation-mko-inclusion-exclusion-principle-01",
    "source": {
      "mko_id": "mko-inclusion-exclusion-principle",
      "commit": "b1b2f1dcf8c50d67a93fe52d95f8eef7b3abba9f",
      "sha256": "2eb0f4de0d9b0d93d979ced97fcaadb56ea9045d427c0918e35505bde5f2a649",
      "english_sha256": "0aa23ad5de1c87cf782892d43f6bbc7c43d34b422d9ee2ebfaa54cd4c4fd80cf"
    },
    "mode": "theorem_proof",
    "audience_prerequisites": [
      "work"
    ],
    "goal": {
      "zh-Hant": "完整核交替和與每元素的重數。",
      "en": "Audit the full alternating sum and each element's multiplicity."
    },
    "givens": [
      {
        "id": "G1",
        "statement": {
          "zh-Hant": "A={1,2,3}，B={2,3,4}，C={3,4,5}。",
          "en": "A={1,2,3}, B={2,3,4}, C={3,4,5}."
        }
      }
    ],
    "steps": [
      {
        "id": "S01",
        "uses": [
          "G1",
          "work"
        ],
        "before_tex": "A\\cup B\\cup C",
        "after_tex": "\\{1,2,3,4,5\\},\\quad|A\\cup B\\cup C|=5",
        "relation": "implies",
        "rule": "work",
        "conditions": [],
        "explanation": {
          "zh-Hant": "先由完整元素清單直接算union作獨立核對。",
          "en": "First compute the union directly from the complete element lists as an independent check."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "work"
          ],
          "explanation": {
            "zh-Hant": "沒有只靠公式預寫預期結果。",
            "en": "The expected result is not merely assumed from the formula."
          }
        },
        "leaf_basis": "work"
      },
      {
        "id": "S02",
        "uses": [
          "G1",
          "S01",
          "work"
        ],
        "before_tex": "|A|,|B|,|C|",
        "after_tex": "3,3,3",
        "relation": "implies",
        "rule": "work",
        "conditions": [],
        "explanation": {
          "zh-Hant": "三個一階項為正，總和9。",
          "en": "The three positive first-order terms sum to 9."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S01",
            "work"
          ],
          "explanation": {
            "zh-Hant": "indexed entries各算一次。",
            "en": "Each indexed entry is counted once."
          }
        },
        "leaf_basis": "work"
      },
      {
        "id": "S03",
        "uses": [
          "G1",
          "S02",
          "work"
        ],
        "before_tex": "A\\cap B,A\\cap C,B\\cap C",
        "after_tex": "\\{2,3\\},\\{3\\},\\{3,4\\}",
        "relation": "implies",
        "rule": "work",
        "conditions": [],
        "explanation": {
          "zh-Hant": "完整列三個二階交集，大小2、1、2。",
          "en": "List all three second-order intersections, of sizes2,1,2."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S02",
            "work"
          ],
          "explanation": {
            "zh-Hant": "二階項的符號為負。",
            "en": "Second-order terms carry negative signs."
          }
        },
        "leaf_basis": "work"
      },
      {
        "id": "S04",
        "uses": [
          "G1",
          "S03",
          "work"
        ],
        "before_tex": "A\\cap B\\cap C",
        "after_tex": "\\{3\\},\\quad|A\\cap B\\cap C|=1",
        "relation": "implies",
        "rule": "work",
        "conditions": [],
        "explanation": {
          "zh-Hant": "唯一三階交集必須加回。",
          "en": "The sole third-order intersection must be added back."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S03",
            "work"
          ],
          "explanation": {
            "zh-Hant": "不是因pairwise intersections已列就省triple。",
            "en": "Listing pairwise intersections does not permit omitting the triple."
          }
        },
        "leaf_basis": "work"
      },
      {
        "id": "S05",
        "uses": [
          "G1",
          "S04",
          "work"
        ],
        "before_tex": "3+3+3-2-1-2+1",
        "after_tex": "5=|A\\cup B\\cup C|",
        "relation": "implies",
        "rule": "work",
        "conditions": [],
        "explanation": {
          "zh-Hant": "七個nonempty index subsets全部使用。",
          "en": "All seven nonempty index subsets are included."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S04",
            "work"
          ],
          "explanation": {
            "zh-Hant": "3+3+1=7與2³-1一致。",
            "en": "Three singles, three pairs and one triple equal2³-1."
          }
        },
        "leaf_basis": "work"
      },
      {
        "id": "S06",
        "uses": [
          "G1",
          "S05",
          "work"
        ],
        "before_tex": "x\\text{ belongs to }r\\ge1\\text{ indexed sets}",
        "after_tex": "\\sum_{j=1}^{r}(-1)^{j+1}\\binom rj=1-(1-1)^r=1",
        "relation": "implies",
        "rule": "work",
        "conditions": [],
        "explanation": {
          "zh-Hant": "每個在union的元素於交替和中淨算一次，給有限公式的逐元素證明。",
          "en": "Each union element has net multiplicity one, proving the finite formula elementwise."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S05",
            "work"
          ],
          "explanation": {
            "zh-Hant": "不在union的r=0元素貢獻零另處理。",
            "en": "Elements outside the union have r=0 and contribute zero separately."
          }
        },
        "leaf_basis": "work"
      },
      {
        "id": "S07",
        "uses": [
          "G1",
          "S06",
          "work"
        ],
        "before_tex": "\\text{singles minus pairs only}",
        "after_tex": "9-5=4\\ne5",
        "relation": "implies",
        "rule": "work",
        "conditions": [],
        "explanation": {
          "zh-Hant": "此例明確顯示三階不能省，pairwise truncation不是exact式。",
          "en": "This example explicitly shows the missing third-order term; pairwise truncation is not exact."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S06",
            "work"
          ],
          "explanation": {
            "zh-Hant": "若稱Bonferroni bound須另標示，而非等號。",
            "en": "A Bonferroni bound must be separately labeled, not asserted as equality."
          }
        },
        "leaf_basis": "work"
      },
      {
        "id": "S08",
        "uses": [
          "G1",
          "S07",
          "work"
        ],
        "before_tex": "(A,A)\\text{ indexed twice};\\quad n=0",
        "after_tex": "|A|+|A|-|A|=|A|;\\quad|\\bigcup\\varnothing|=0",
        "relation": "implies",
        "rule": "work",
        "conditions": [],
        "explanation": {
          "zh-Hant": "重複家族仍按indices正確消重；空家族sum與union大小皆零。",
          "en": "Indexed repetition still cancels correctly; the empty-family sum and union size are both zero."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S07",
            "work"
          ],
          "explanation": {
            "zh-Hant": "相同set不等於省略一個index責任。",
            "en": "Equal sets do not mean an indexed obligation may be ignored."
          }
        },
        "leaf_basis": "work"
      }
    ],
    "result": {
      "zh-Hant": "全部七項給union size5；每元素淨重數證一般有限家族公式。",
      "en": "All seven terms give union size 5; elementwise multiplicity proves the finite-family formula."
    },
    "negative_controls": [
      {
        "claim": {
          "zh-Hant": "三集合只要singles減pairs就exact。",
          "en": "Singles minus pairs is exact for three sets."
        },
        "expected": "reject",
        "reason": {
          "zh-Hant": "本例4不等union size5。",
          "en": "Here4 differs from union size 5."
        }
      },
      {
        "claim": {
          "zh-Hant": "triple交集應減。",
          "en": "The triple intersection should be subtracted."
        },
        "expected": "reject",
        "reason": {
          "zh-Hant": "三階為奇數，符號正。",
          "en": "Order three is odd and has positive sign."
        }
      },
      {
        "claim": {
          "zh-Hant": "重複兩set就直接把union計兩倍。",
          "en": "Repeated sets make the union twice as large."
        },
        "expected": "reject",
        "reason": {
          "zh-Hant": "二階交集恰扣回一份。",
          "en": "The second-order intersection subtracts the duplicate copy."
        }
      }
    ],
    "status": "candidate",
    "gaps": [],
    "verification": {
      "structure": "checked_local_script",
      "mathematics": "candidate_argument",
      "formalization": "not_formalized"
    },
    "rule_glossary": [
      {
        "id": "mko_definition",
        "kind": "cited",
        "statement": {
          "zh-Hant": "對每個非空 I⊆[n] 使用 (-1)^(|I|+1)|intersection_i in I A_i|；odd-order 正、even-order 負。重複 sets 仍按 indices 處理，空交集對計數和貢獻 0。空家族的聯集為空集，其基數與容斥和皆為 0。只用 singles/pairs 對 n≥3 不是一般的 exact inclusion-exclusion，截斷只能另稱 Bonferroni bound。",
          "en": "For each nonempty index subset I use sign (−1)^(|I|+1): odd-order intersections add and even-order intersections subtract. Equal sets at different indices remain distinct indexed entries, and empty intersections contribute 0. For an empty family, the union is empty and both its cardinality and the inclusion-exclusion sum are 0. Singles-minus-pairs alone is not a general exact formula for three or more sets; it is a separately identified Bonferroni truncation."
        },
        "source": {
          "mko_id": "mko-inclusion-exclusion-principle",
          "sha256": "2eb0f4de0d9b0d93d979ced97fcaadb56ea9045d427c0918e35505bde5f2a649",
          "commit": "b1b2f1dcf8c50d67a93fe52d95f8eef7b3abba9f"
        }
      },
      {
        "id": "work",
        "kind": "prerequisite",
        "statement": {
          "zh-Hant": "集合union與intersection、有限和按元素計重；二項式identity(1-1)^r=sum_j(-1)^j comb(r,j)為明示前提。",
          "en": "Set unions and intersections, finite summation by element multiplicity; the binomial identity(1-1)^r=sum_j(-1)^j comb(r,j) is an explicit prerequisite."
        }
      }
    ],
    "revision_log": [],
    "numeric_checks": [
      {
        "id": "N01",
        "lhs": "3+3+3-2-1-2+1",
        "rhs": "5"
      },
      {
        "id": "N02",
        "lhs": "2**3-1",
        "rhs": "7"
      },
      {
        "id": "N03",
        "lhs": "comb(3,1)-comb(3,2)+comb(3,3)",
        "rhs": "1"
      }
    ]
  },
  "presentation": {
    "md_scope": {
      "zh-Hant": "indexed有限家族，重複sets仍按indices處理。",
      "en": "A finite indexed family, retaining separate indices for repeated sets."
    },
    "md_symbols": [
      [
        "r",
        "元素所屬indexed sets數",
        "number of indexed sets containing an element"
      ]
    ],
    "md_error_controls": [
      {
        "title": "七項 / Seven terms",
        "zh": "按index subsets而非不同集合值枚舉。",
        "en": "Enumerate index subsets, not distinct set values."
      },
      {
        "title": "逐元素 / Elementwise",
        "zh": "r≥1用1-(1-1)^r，r=0另給零。",
        "en": "Use1-(1-1)^r for r≥1 and zero separately for r=0."
      }
    ],
    "md_open_items": []
  },
  "revision": {
    "id": "ocme-new142-reader-adoption-20261021",
    "collection": "coverage-20261021",
    "source_commit": "b1b2f1dcf8c50d67a93fe52d95f8eef7b3abba9f",
    "adopted_source_sha256": "ea75129552baa4d33afae410265a8e2ec2eda5b1a8669c83da8afed4efc43c95",
    "original_source_sha256": "d12d13f29453c49f15f41deea45201369e57e7476d15786a97750ca414dc608a",
    "owner_adopted": true,
    "bilingual_sync": "complete_source_bound",
    "formal_Lean": "NOT_RUN",
    "reader_review": "142_of_142_source_bound",
    "wording_review": "original_Lead_decisions_resolved",
    "glossary_reader_paraphrases": [
      "mko_definition"
    ],
    "independent_AI_concurrence": "SCOPED_LEAD_AND_BILINGUAL_CONTEXT_REVIEW",
    "issue_ids": [
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      "BR142-S134",
      "BR142-P049",
      "BR142-P048",
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    ],
    "parent_manifest_sha256": "d29f4cae8229ebd24f60d496cb1d3cd4ad8c0a4dca9d44daab0c3245c544283a",
    "scope": "Owner-adopted reader revision. Lead and bilingual review concern the recorded text and premises; no new universal or Lean proof, canonical promotion or historical-label rewrite."
  }
}