{
  "schema_version": "ocme-derivation-reader-v1",
  "guide_id": "derivation-mko-event-01",
  "mko_id": "mko-event",
  "title": {
    "zh-Hant": "事件：在完整事件族與較粗事件族中判定事件",
    "en": "Event: deciding what counts as an event under the full event family and under a coarser one"
  },
  "contract": {
    "schema_version": "ocme-derivation-guide-v0.1",
    "guide_id": "derivation-mko-event-01",
    "source": {
      "mko_id": "mko-event",
      "commit": "b28a5edbd46279b1db2a3f3f48bb3f447e18899f",
      "sha256": "5962ae6672cb79c34888bdda4ac66de557a4f9b21243f2445414d7dd326b9c74",
      "english_sha256": "12d8199c90226e08b52a3f9cf27ce44053dc8473e8bab4280145aea146a4c261"
    },
    "mode": "worked_example",
    "audience_prerequisites": [
      "subset_def",
      "set_operations_def",
      "set_extensionality",
      "powerset_def",
      "sigma_algebra_def"
    ],
    "goal": {
      "zh-Hant": "擲一顆骰子一次，S={1,2,3,4,5,6}。(1) 在有限初等模型 F=𝒫(S) 中，說明 E={2,4,6} 是事件，判定觀察結果為 4 與 3 時 E 是否發生；分辨結果 4 與事件 {4}；說明空事件與必然事件。(2) 在一般模型 (S,F₀) 中，F₀={∅,O,E,S}，O={1,3,5}：證明 F₀ 是 S 上的事件族（σ-代數），並證明 {6}∉F₀，所以雖然 S 有限且 {6}⊆S，{6} 在這個模型中不是事件。",
      "en": "Roll a die once, S={1,2,3,4,5,6}. (1) In the finite elementary model F=𝒫(S), show that E={2,4,6} is an event, and decide whether E occurs when the observed outcome is 4 and when it is 3; distinguish the outcome 4 from the event {4}; describe the empty and the sure event. (2) In the general model (S,F₀) with F₀={∅,O,E,S} and O={1,3,5}: prove that F₀ is an event family (σ-algebra) on S, and prove {6}∉F₀, so although S is finite and {6}⊆S, {6} is not an event in this model."
    },
    "givens": [
      {
        "id": "G1",
        "statement": {
          "zh-Hant": "試驗：擲一顆六面骰一次，記錄朝上的點數；樣本空間 S={1,2,3,4,5,6}（假設 a1）。",
          "en": "Experiment: roll a six-sided die once and record the number shown; sample space S={1,2,3,4,5,6} (assumption a1)."
        }
      },
      {
        "id": "G2",
        "statement": {
          "zh-Hant": "模型 M1：有限初等模型，明示事件族 F=𝒫(S)（假設 a2）。",
          "en": "Model M1: the finite elementary model, with the explicit event family F=𝒫(S) (assumption a2)."
        }
      },
      {
        "id": "G3",
        "statement": {
          "zh-Hant": "模型 M2：一般模型 (S,F₀)，明示 F₀={∅,O,E,S}，其中 O={1,3,5}、E={2,4,6}（例如只能觀察點數奇偶的情形）（假設 a2）。",
          "en": "Model M2: the general model (S,F₀), with the explicit F₀={∅,O,E,S}, where O={1,3,5} and E={2,4,6} (for example when only the parity of the roll can be observed) (assumption a2)."
        }
      }
    ],
    "steps": [
      {
        "id": "S01",
        "uses": [
          "G1",
          "G2",
          "G3"
        ],
        "before_tex": "\\text{one roll of a die}",
        "after_tex": "S=\\{1,2,3,4,5,6\\},\\ E=\\{2,4,6\\},\\ O=\\{1,3,5\\}",
        "relation": "definition",
        "rule": "def_intro",
        "conditions": [],
        "explanation": {
          "zh-Hant": "先明示樣本空間 S，再為兩個要用到的子集合取名：E 是偶數結果，O 是奇數結果。依 MKO，列事件之前必須先指定樣本空間與事件族；兩個模型的事件族由 G2、G3 明示。",
          "en": "State the sample space S first, then name the two subsets used below: E, the even outcomes, and O, the odd outcomes. By the MKO, the sample space and the event family must be specified before events are listed; the event families of the two models are stated in G2 and G3."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "G2",
            "G3",
            "def_intro"
          ],
          "explanation": {
            "zh-Hant": "此步只寫出資料並命名，不推出新事實；S 與兩個事件族都已明示，符合假設 a1、a2。",
            "en": "This step only records data and names, deriving no new fact; S and both event families are explicit, as assumptions a1 and a2 require."
          }
        },
        "leaf_basis": "def_intro + mko_event_def"
      },
      {
        "id": "S02",
        "uses": [
          "S01"
        ],
        "before_tex": "E=\\{2,4,6\\},\\ O=\\{1,3,5\\}",
        "after_tex": "E\\subseteq S,\\ O\\subseteq S",
        "relation": "implies",
        "rule": "subset_def",
        "conditions": [],
        "explanation": {
          "zh-Hant": "E 的三個元素 2、4、6 都在 S 中，O 的三個元素 1、3、5 也都在 S 中；這檢查了這兩個有限集合的全部元素，所以依子集合定義 E⊆S、O⊆S。",
          "en": "The three elements 2, 4, 6 of E all lie in S, and so do the three elements 1, 3, 5 of O; this checks every element of these finite sets, so by the subset definition E⊆S and O⊆S."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "subset_def"
          ],
          "explanation": {
            "zh-Hant": "完整窮舉，不是抽樣；反向 S⊆E 不宣稱（1∉E）。",
            "en": "A complete enumeration, not a sample; the converse S⊆E is not claimed (1∉E)."
          }
        },
        "leaf_basis": "subset_def + forall_intro"
      },
      {
        "id": "S03",
        "uses": [
          "G2",
          "S02"
        ],
        "before_tex": "E\\subseteq S",
        "after_tex": "E\\in\\mathcal{P}(S)=F",
        "relation": "equivalent",
        "rule": "powerset_def",
        "conditions": [
          "F=\\mathcal{P}(S)"
        ],
        "explanation": {
          "zh-Hant": "在模型 M1 中 F=𝒫(S)。依冪集定義，X∈𝒫(S) 當且僅當 X⊆S；由 S02，E⊆S，所以 E∈F，E 是 M1 中的事件。依 MKO，只有明示採用 F=𝒫(S) 時，才可以這樣由「是子集合」直接得到「是事件」。",
          "en": "In model M1, F=𝒫(S). By the definition of power set, X∈𝒫(S) exactly when X⊆S; by S02, E⊆S, so E∈F and E is an event in M1. By the MKO, passing directly from 'is a subset' to 'is an event' is allowed only when F=𝒫(S) is explicitly adopted."
        },
        "reverse_audit": {
          "kind": "inverse_equivalence",
          "required_ids": [
            "powerset_def",
            "G2"
          ],
          "explanation": {
            "zh-Hant": "冪集定義是雙向的；但這個等價只在 F=𝒫(S) 的模型中把子集合與事件連起來。",
            "en": "The power-set definition is two-way; but this equivalence links subsets with events only in a model with F=𝒫(S)."
          }
        },
        "leaf_basis": "powerset_def + eq_replace"
      },
      {
        "id": "S04",
        "uses": [
          "S01",
          "S03"
        ],
        "before_tex": "\\omega=4,\\quad \\omega=3",
        "after_tex": "4\\in E\\Rightarrow E\\ \\text{occurs};\\quad 3\\notin E\\Rightarrow E\\ \\text{does not occur}",
        "relation": "implies",
        "rule": "mko_event_def",
        "conditions": [
          "E\\in F"
        ],
        "explanation": {
          "zh-Hant": "依 MKO，試驗結果屬於 E 時稱 E 發生。觀察到 ω=4 時，4 是 E 列舉中的一個，所以 E 發生；觀察到 ω=3 時，3 不在 {2,4,6} 中，所以 E 不發生。事件是否發生，由觀察結果是否屬於它決定。",
          "en": "By the MKO, E occurs when the observed outcome belongs to E. With ω=4, 4 is one of the listed elements of E, so E occurs; with ω=3, 3 is not in {2,4,6}, so E does not occur. Whether an event occurs is decided by whether the observed outcome belongs to it."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "mko_event_def",
            "S03"
          ],
          "explanation": {
            "zh-Hant": "需要 E 已是事件（S03）；「發生」只對事件定義。這裡不宣稱任何機率值。",
            "en": "It needs E to be an event (S03); occurrence is defined only for events. No probability value is claimed."
          }
        },
        "leaf_basis": "mko_event_def + def_unfold"
      },
      {
        "id": "S05",
        "uses": [
          "S01",
          "S03"
        ],
        "before_tex": "4\\in S,\\ \\{4\\}\\subseteq S",
        "after_tex": "4\\ \\text{is an outcome},\\ \\{4\\}\\in F,\\ 4\\neq\\{4\\}",
        "relation": "implies",
        "rule": "powerset_def",
        "conditions": [
          "F=\\mathcal{P}(S)"
        ],
        "explanation": {
          "zh-Hant": "4 是 S 的元素，是一個基本結果；{4} 是 S 的子集合，在 M1 中依冪集定義屬於 F，是一個事件。兩者型別不同：4 是數，{4} 是集合，4≠{4}。觀察到結果 4 時，事件 {4} 發生；E={2,4,6} 含三個結果，所以事件一般不是單一結果。",
          "en": "4 is an element of S, an elementary outcome; {4} is a subset of S and, in M1, belongs to F by the definition of power set, so it is an event. They have different types: 4 is a number, {4} is a set, and 4≠{4}. When the outcome 4 is observed, the event {4} occurs; E={2,4,6} contains three outcomes, so an event is in general not a single outcome."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "powerset_def",
            "subset_def",
            "mko_event_def"
          ],
          "explanation": {
            "zh-Hant": "{4}⊆S 因為它唯一的元素 4 在 S 中；{4}∈F 依賴 M1 的 F=𝒫(S)，在 M2 中不一定成立（S11 處理同類的 {6}）。",
            "en": "{4}⊆S because its only element 4 is in S; {4}∈F relies on F=𝒫(S) in M1 and need not hold in M2 (S11 treats the similar set {6})."
          }
        },
        "leaf_basis": "powerset_def + subset_def + mko_event_def"
      },
      {
        "id": "S06",
        "uses": [
          "S01",
          "G2"
        ],
        "before_tex": "\\varnothing\\subseteq S,\\ S\\subseteq S",
        "after_tex": "\\varnothing,\\ S\\in F;\\ \\varnothing\\ \\text{never occurs},\\ S\\ \\text{always occurs}",
        "relation": "implies",
        "rule": "powerset_def",
        "conditions": [
          "F=\\mathcal{P}(S)"
        ],
        "explanation": {
          "zh-Hant": "∅⊆S（沒有元素需要檢查），S⊆S，所以在 M1 中兩者都是事件。沒有任何 ω 屬於 ∅，所以空事件從不發生；每個觀察結果都屬於 S（樣本空間包含所有可能結果），所以必然事件 S 總是發生。這些名稱並沒有指派機率值。",
          "en": "∅⊆S (there is no element to check) and S⊆S, so both are events in M1. No ω belongs to ∅, so the empty event never occurs; every observed outcome belongs to S (the sample space contains every possible outcome), so the sure event S always occurs. These names do not assign probability values."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "powerset_def",
            "subset_def",
            "mko_event_def",
            "G1"
          ],
          "explanation": {
            "zh-Hant": "S 總是發生需要樣本空間的完整性（G1）；不宣稱 P(∅)=0 或 P(S)=1，那需要另給機率模型。",
            "en": "S always occurring needs the completeness of the sample space (G1); P(∅)=0 or P(S)=1 is not claimed, since that needs a separately given probability model."
          }
        },
        "leaf_basis": "powerset_def + subset_def + mko_event_def"
      },
      {
        "id": "S07",
        "uses": [
          "G3",
          "S02",
          "S06"
        ],
        "before_tex": "F_0=\\{\\varnothing,O,E,S\\}",
        "after_tex": "F_0\\subseteq\\mathcal{P}(S),\\ S\\in F_0",
        "relation": "implies",
        "rule": "sigma_algebra_def",
        "conditions": [],
        "explanation": {
          "zh-Hant": "進入模型 M2，依 σ-代數定義逐條核對。第一條：F₀ 的四個成員 ∅、O、E、S 都是 S 的子集合（S02、S06），所以 F₀ 是 S 的子集合族；而且 S 是 F₀ 的成員之一。",
          "en": "Move to model M2 and check the σ-algebra definition clause by clause. First clause: the four members ∅, O, E, S of F₀ are all subsets of S (S02, S06), so F₀ is a family of subsets of S; and S is one of the members of F₀."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "sigma_algebra_def",
            "S02"
          ],
          "explanation": {
            "zh-Hant": "只核對第一條；其餘兩條在 S08、S10。",
            "en": "Only the first clause is checked here; the other two are in S08 and S10."
          }
        },
        "leaf_basis": "sigma_algebra_def + subset_def"
      },
      {
        "id": "S08",
        "uses": [
          "S07"
        ],
        "before_tex": "S\\setminus\\varnothing,\\ S\\setminus S,\\ S\\setminus E,\\ S\\setminus O",
        "after_tex": "S,\\ \\varnothing,\\ O,\\ E\\ \\in F_0",
        "relation": "implies",
        "rule": "set_operations_def",
        "conditions": [],
        "explanation": {
          "zh-Hant": "逐一計算 F₀ 每個成員在 S 中的補集：S∖∅=S，S∖S=∅；S∖E 是 S 中不屬於 {2,4,6} 的元素，即 {1,3,5}=O；同理 S∖O=E。四個補集都在 F₀ 中，所以 F₀ 對補集封閉，σ-代數的第二條成立。",
          "en": "Compute the complement in S of each member of F₀: S∖∅=S and S∖S=∅; S∖E consists of the elements of S not in {2,4,6}, namely {1,3,5}=O; likewise S∖O=E. All four complements lie in F₀, so F₀ is closed under complement and the second clause of the σ-algebra definition holds."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "set_operations_def",
            "set_extensionality",
            "sigma_algebra_def"
          ],
          "explanation": {
            "zh-Hant": "F₀ 只有四個成員，逐一計算就是完整檢查。",
            "en": "F₀ has only four members, so computing each is a complete check."
          }
        },
        "leaf_basis": "set_operations_def + set_extensionality"
      },
      {
        "id": "S09",
        "uses": [
          "S01"
        ],
        "before_tex": "E=\\{2,4,6\\},\\ O=\\{1,3,5\\}",
        "after_tex": "E\\cup O=S,\\ E\\cap O=\\varnothing",
        "relation": "implies",
        "rule": "set_operations_def",
        "conditions": [],
        "explanation": {
          "zh-Hant": "S 的六個元素中，2、4、6 在 E，1、3、5 在 O，每個元素恰好屬於其中一個。所以 E∪O=S 且 E∩O=∅；個數核對 3+3=6。之後稱 E、O 為 F₀ 的原子。",
          "en": "Of the six elements of S, 2, 4, 6 are in E and 1, 3, 5 are in O, and each element lies in exactly one of them. So E∪O=S and E∩O=∅; count check 3+3=6. From here on E and O are called the atoms of F₀."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "set_operations_def",
            "set_extensionality"
          ],
          "explanation": {
            "zh-Hant": "對 S 的六個元素逐一檢查，是完整窮舉。",
            "en": "Each of the six elements of S is checked, a complete enumeration."
          }
        },
        "leaf_basis": "set_operations_def + set_extensionality + numeric_evaluation"
      },
      {
        "id": "S10",
        "uses": [
          "S07",
          "S08",
          "S09"
        ],
        "before_tex": "(X_i)_{i\\in\\mathbb{N}}\\ \\text{with each}\\ X_i\\in F_0",
        "after_tex": "\\bigcup_{i\\in\\mathbb{N}}X_i\\in F_0;\\ F_0\\ \\text{is an event family on }S",
        "relation": "implies",
        "rule": "L_F0_unions",
        "conditions": [],
        "explanation": {
          "zh-Hant": "依引理 L_F0_unions（在 S10.1 到 S10.3 證明），F₀ 中任意可數多個成員的聯集仍在 F₀ 中；有限聯集是特例（其餘各項取 ∅）。連同 S07、S08，σ-代數定義的三條都成立，所以 F₀ 是 S 上的事件族，M2 是合法的一般模型。",
          "en": "By Lemma L_F0_unions (proved in S10.1 to S10.3), the union of any countably many members of F₀ lies in F₀; finite unions are a special case (take the remaining terms to be ∅). Together with S07 and S08, all three clauses of the σ-algebra definition hold, so F₀ is an event family on S and M2 is a legitimate general model."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "S07",
            "S08",
            "L_F0_unions",
            "sigma_algebra_def"
          ],
          "explanation": {
            "zh-Hant": "三條各由一步提供；不宣稱 F₀ 包含 S 的所有子集合（S11 正好相反）。",
            "en": "Each of the three clauses is supplied by one step; it is not claimed that F₀ contains every subset of S (S11 shows the opposite)."
          }
        },
        "leaf_basis": "sigma_algebra_def + set_operations_def + case_analysis + exists_elim"
      },
      {
        "id": "S10.1",
        "uses": [
          "S09"
        ],
        "before_tex": "X\\in F_0",
        "after_tex": "X=\\bigcup\\{A\\in\\{E,O\\}\\mid A\\subseteq X\\}",
        "relation": "implies",
        "rule": "set_extensionality",
        "conditions": [],
        "explanation": {
          "zh-Hant": "對 F₀ 的四個成員逐一核對：沒有原子包含於 ∅（E、O 都非空），空族的聯集是 ∅；包含於 E 的原子只有 E（1∈O 但 1∉E），聯集是 E；同理，包含於 O 的只有 O；包含於 S 的原子是 E 與 O，聯集 E∪O=S（S09）。所以每個 X∈F₀ 都等於它所包含的原子之聯集。",
          "en": "Check the four members of F₀ one by one: no atom is contained in ∅ (E and O are nonempty), and the union of the empty family is ∅; the only atom contained in E is E (1∈O but 1∉E), with union E; likewise only O is contained in O; the atoms contained in S are E and O, with union E∪O=S (S09). So each X∈F₀ equals the union of the atoms it contains."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "S09",
            "set_extensionality",
            "subset_def"
          ],
          "explanation": {
            "zh-Hant": "F₀ 的四個成員全部檢查，是完整窮舉；這個性質只對 F₀ 的成員宣稱。",
            "en": "All four members of F₀ are checked, a complete enumeration; the property is claimed only for members of F₀."
          }
        },
        "leaf_basis": "set_extensionality + subset_def + set_operations_def"
      },
      {
        "id": "S10.2",
        "uses": [
          "S10.1"
        ],
        "before_tex": "x\\in\\bigcup_{i}X_i",
        "after_tex": "x\\in\\bigcup\\{A\\in\\{E,O\\}\\mid\\exists i,\\ A\\subseteq X_i\\}",
        "relation": "equivalent",
        "rule": "set_operations_def",
        "conditions": [
          "X_i\\in F_0\\ \\text{for all}\\ i"
        ],
        "explanation": {
          "zh-Hant": "x 屬於聯集 ⇔ 有某個 i 使 x∈X_i ⇔（依 S10.1）有某個 i 與某個原子 A⊆X_i 使 x∈A ⇔ x 屬於某個「包含於某個 X_i 的原子」。交換兩個存在量詞的次序不改變真假。所以可數聯集等於這些原子的聯集。",
          "en": "x belongs to the union ⇔ x∈X_i for some i ⇔ (by S10.1) for some i and some atom A⊆X_i, x∈A ⇔ x belongs to some atom that is contained in some X_i. Swapping the order of the two existential quantifiers does not change the truth value. So the countable union equals the union of these atoms."
        },
        "reverse_audit": {
          "kind": "inverse_equivalence",
          "required_ids": [
            "S10.1",
            "set_operations_def",
            "exists_intro",
            "exists_elim"
          ],
          "explanation": {
            "zh-Hant": "每一步都是等價：聯集的定義、S10.1 的集合相等、存在量詞的交換；所以兩邊元素相同。",
            "en": "Each move is an equivalence: the definition of union, the set equality of S10.1, and swapping existential quantifiers; so both sides have the same elements."
          }
        },
        "leaf_basis": "set_operations_def + eq_replace + exists_intro + exists_elim"
      },
      {
        "id": "S10.3",
        "uses": [
          "S10.2",
          "S09"
        ],
        "before_tex": "\\{A\\in\\{E,O\\}\\mid\\exists i,\\ A\\subseteq X_i\\}",
        "after_tex": "\\bigcup_i X_i\\in\\{\\varnothing,E,O,S\\}=F_0",
        "relation": "implies",
        "rule": "case_analysis",
        "conditions": [],
        "explanation": {
          "zh-Hant": "被選出的原子族是 {E,O} 的子族，只有四種可能：空族、{E}、{O}、{E,O}，對應的聯集分別是 ∅、E、O、E∪O=S，全都屬於 F₀。S10.1 到 S10.3 合起來證得引理 L_F0_unions。",
          "en": "The selected family of atoms is a subfamily of {E,O}, so there are only four possibilities: the empty family, {E}, {O} and {E,O}, whose unions are ∅, E, O and E∪O=S, all of which belong to F₀. S10.1 to S10.3 together prove Lemma L_F0_unions."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "S10.2",
            "S09",
            "case_analysis"
          ],
          "explanation": {
            "zh-Hant": "四種情況窮盡 {E,O} 的子族；論證對任意可數族成立，與族中列出多少個集合無關。",
            "en": "The four cases exhaust the subfamilies of {E,O}; the argument holds for any countable family, regardless of how many sets it lists."
          }
        },
        "leaf_basis": "case_analysis + set_operations_def"
      },
      {
        "id": "S11",
        "uses": [
          "S01",
          "G3"
        ],
        "before_tex": "\\{6\\}\\ \\text{compared with}\\ \\varnothing,\\ O,\\ E,\\ S",
        "after_tex": "\\{6\\}\\notin F_0",
        "relation": "implies",
        "rule": "set_extensionality",
        "conditions": [],
        "explanation": {
          "zh-Hant": "{6} 要屬於 F₀，必須等於四個成員之一。{6}≠∅，因為 6∈{6}；{6}≠O，因為 6∉O；{6}≠E，因為 2∈E 但 2∉{6}；{6}≠S，因為 1∈S 但 1∉{6}。四個都不相等，所以 {6}∉F₀。",
          "en": "For {6} to belong to F₀ it must equal one of the four members. {6}≠∅ since 6∈{6}; {6}≠O since 6∉O; {6}≠E since 2∈E but 2∉{6}; {6}≠S since 1∈S but 1∉{6}. None is equal, so {6}∉F₀."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "set_extensionality",
            "G3"
          ],
          "explanation": {
            "zh-Hant": "F₀ 是有限列舉，與每個成員比較就是完整檢查；每個不等都有一個見證元素。",
            "en": "F₀ is a finite list, so comparing with each member is a complete check; each inequality has a witness element."
          }
        },
        "leaf_basis": "set_extensionality + case_analysis"
      },
      {
        "id": "S12",
        "uses": [
          "S10",
          "S11"
        ],
        "before_tex": "\\{6\\}\\subseteq S,\\ |S|=6,\\ \\{6\\}\\notin F_0",
        "after_tex": "\\{6\\}\\ \\text{is not an event of}\\ (S,F_0)",
        "relation": "implies",
        "rule": "mko_event_def",
        "conditions": [
          "F_0\\ \\text{an event family}"
        ],
        "explanation": {
          "zh-Hant": "依 MKO，一般模型 (S,F) 中事件必須屬於 F。{6}⊆S，S 也只有 6 個元素，但 {6}∉F₀（S11），所以在 M2 中 {6} 不是事件；「擲出 6」不能在 M2 中當作事件討論。S 有限並不會讓每個子集合自動成為事件；在 M1 中 {6} 是事件，只因為 M1 明示採用 F=𝒫(S)。",
          "en": "By the MKO, in a general model (S,F) an event must belong to F. {6}⊆S and S has only 6 elements, but {6}∉F₀ (S11), so {6} is not an event in M2; 'the roll is 6' cannot be treated as an event in M2. Finiteness of S does not make every subset an event automatically; in M1, {6} is an event only because M1 explicitly adopts F=𝒫(S)."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "mko_event_def",
            "S10",
            "S11"
          ],
          "explanation": {
            "zh-Hant": "需要 F₀ 是合法事件族（S10），否則 M2 不是模型，結論沒有意義。",
            "en": "It needs F₀ to be a legitimate event family (S10); otherwise M2 would not be a model and the conclusion would be meaningless."
          }
        },
        "leaf_basis": "mko_event_def + powerset_def"
      },
      {
        "id": "S13",
        "uses": [
          "S04",
          "S10"
        ],
        "before_tex": "E\\in F_0,\\ \\omega=4",
        "after_tex": "E\\ \\text{is an event of}\\ (S,F_0)\\ \\text{and occurs}",
        "relation": "implies",
        "rule": "mko_event_def",
        "conditions": [],
        "explanation": {
          "zh-Hant": "E 是 F₀ 的成員，所以在 M2 中 E 也是事件；觀察到 ω=4 時 4∈E，E 發生。同一個集合 E 在兩個模型中都是事件，而 {6} 只在 M1 中是事件：是否為事件取決於事件族，不只取決於 S。",
          "en": "E is a member of F₀, so E is an event in M2 as well; with ω=4, 4∈E and E occurs. The same set E is an event in both models, while {6} is an event only in M1: whether a set is an event depends on the event family, not only on S."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "mko_event_def",
            "S10"
          ],
          "explanation": {
            "zh-Hant": "需要 E∈F₀（G3 的列舉）與 F₀ 合法（S10）；發生與否的判定與 S04 相同。",
            "en": "It needs E∈F₀ (from the list in G3) and F₀ legitimate (S10); occurrence is decided as in S04."
          }
        },
        "leaf_basis": "mko_event_def"
      },
      {
        "id": "S14",
        "uses": [
          "S03",
          "S04",
          "S05",
          "S06",
          "S10",
          "S12",
          "S13"
        ],
        "before_tex": "\\text{models}\\ (S,\\mathcal{P}(S))\\ \\text{and}\\ (S,F_0)",
        "after_tex": "E\\ \\text{event in both};\\ \\{6\\}\\ \\text{event in}\\ (S,\\mathcal{P}(S))\\ \\text{only}",
        "relation": "implies",
        "rule": "and_intro_elim",
        "conditions": [],
        "explanation": {
          "zh-Hant": "彙整：在 M1 中 E 是事件，ω=4 時發生、ω=3 時不發生（S03、S04）；結果 4 不同於事件 {4}（S05）；∅ 從不發生，S 總是發生（S06）。在 M2 中 F₀ 是事件族（S10），E 仍是事件（S13），但 {6} 不是（S12）。",
          "en": "Collect the results: in M1, E is an event that occurs for ω=4 and not for ω=3 (S03, S04); the outcome 4 differs from the event {4} (S05); ∅ never occurs and S always occurs (S06). In M2, F₀ is an event family (S10), E is still an event (S13), but {6} is not (S12)."
        },
        "reverse_audit": {
          "kind": "sufficient_goal_reduction",
          "required_ids": [
            "S03",
            "S04",
            "S05",
            "S06",
            "S10",
            "S12",
            "S13",
            "and_intro_elim"
          ],
          "explanation": {
            "zh-Hant": "目標 G 是題目各項結論的合取；每一項由所列步驟正向證成，此步只做合取，沒有把 G 當作前提。",
            "en": "Goal G is the conjunction of the problem's conclusions; each is proved forwards by the listed steps, and this step only forms the conjunction; G is never used as a premise."
          }
        },
        "leaf_basis": "and_intro_elim"
      }
    ],
    "result": {
      "zh-Hant": "在有限初等模型 F=𝒫(S) 中，E={2,4,6} 是事件；觀察到 4 時 E 發生，觀察到 3 時不發生；結果 4 與事件 {4} 不同；空事件從不發生，必然事件總是發生。在一般模型 (S,F₀) 中，F₀={∅,{1,3,5},{2,4,6},S} 是事件族，E 仍是事件，但 {6}∉F₀，所以雖然 S 有限且 {6}⊆S，{6} 不是事件。",
      "en": "In the finite elementary model F=𝒫(S), E={2,4,6} is an event; it occurs when 4 is observed and not when 3 is observed; the outcome 4 differs from the event {4}; the empty event never occurs and the sure event always occurs. In the general model (S,F₀), F₀={∅,{1,3,5},{2,4,6},S} is an event family and E is still an event, but {6}∉F₀, so although S is finite and {6}⊆S, {6} is not an event."
    },
    "negative_controls": [
      {
        "claim": {
          "zh-Hant": "S 有限，所以在任何模型 (S,F) 中 {6} 都是事件",
          "en": "S is finite, so {6} is an event in every model (S,F)"
        },
        "expected": "reject",
        "reason": {
          "zh-Hant": "在 M2 中 {6}∉F₀（S11、S12）。",
          "en": "In M2, {6}∉F₀ (S11, S12)."
        }
      },
      {
        "claim": {
          "zh-Hant": "事件 E={2,4,6} 就是結果 4",
          "en": "The event E={2,4,6} is the outcome 4"
        },
        "expected": "reject",
        "reason": {
          "zh-Hant": "E 是含三個結果的集合；4 只是其中一個元素（S05）。",
          "en": "E is a set of three outcomes; 4 is only one of its elements (S05)."
        }
      },
      {
        "claim": {
          "zh-Hant": "觀察到 3 時，事件 E={2,4,6} 發生",
          "en": "When 3 is observed, the event E={2,4,6} occurs"
        },
        "expected": "reject",
        "reason": {
          "zh-Hant": "3∉E，所以 E 不發生（S04）。",
          "en": "3∉E, so E does not occur (S04)."
        }
      },
      {
        "claim": {
          "zh-Hant": "E 是事件，所以 P(E)=1/2",
          "en": "E is an event, so P(E)=1/2"
        },
        "expected": "reject",
        "reason": {
          "zh-Hant": "本稿兩個模型都沒有指派機率；是否為事件不決定機率值。",
          "en": "Neither model in this guide assigns probabilities; being an event does not fix a probability value."
        }
      },
      {
        "claim": {
          "zh-Hant": "F₀ 不含 {6}，所以 F₀ 不是事件族",
          "en": "F₀ does not contain {6}, so F₀ is not an event family"
        },
        "expected": "reject",
        "reason": {
          "zh-Hant": "σ-代數的三條在 S07、S08、S10 都已核對；事件族不必包含單點集合。",
          "en": "All three σ-algebra clauses are checked in S07, S08 and S10; an event family need not contain singletons."
        }
      }
    ],
    "status": "candidate",
    "gaps": [],
    "verification": {
      "structure": "checked_local_script",
      "mathematics": "candidate_argument",
      "formalization": "not_formalized"
    },
    "rule_glossary": [
      {
        "id": "subset_def",
        "kind": "prerequisite",
        "statement": {
          "zh-Hant": "子集：A⊆B 表示 A 的每個元素都是 B 的元素，即 ∀x (x∈A⇒x∈B)。",
          "en": "Subset: A⊆B means every element of A is an element of B, i.e. ∀x (x∈A⇒x∈B)."
        }
      },
      {
        "id": "set_operations_def",
        "kind": "prerequisite",
        "statement": {
          "zh-Hant": "集合運算：x∈A∪B ⇔ x∈A 或 x∈B；x∈A∩B ⇔ x∈A 且 x∈B；x∈A∖B ⇔ x∈A 且 x∉B；在全集 U 中 Aᶜ=U∖A。",
          "en": "Set operations: x∈A∪B ⇔ x∈A or x∈B; x∈A∩B ⇔ x∈A and x∈B; x∈A∖B ⇔ x∈A and x∉B; in a universe U, Aᶜ=U∖A."
        }
      },
      {
        "id": "set_extensionality",
        "kind": "prerequisite",
        "statement": {
          "zh-Hant": "集合相等：A=B 當且僅當 A⊆B 且 B⊆A，即兩者有相同元素。",
          "en": "Set equality: A=B if and only if A⊆B and B⊆A, i.e. they have the same elements."
        }
      },
      {
        "id": "def_intro",
        "kind": "prerequisite",
        "statement": {
          "zh-Hant": "引入定義：可為一個命題、式子或物件取名；之後名稱與被定義者可互相替換。",
          "en": "Introducing a definition: a statement, expression or object may be given a name, after which the name and what it stands for are interchangeable."
        }
      },
      {
        "id": "forall_intro",
        "kind": "prerequisite",
        "statement": {
          "zh-Hant": "∀引入（全稱推廣）：若某命題對一個「任意的」元素 x 證成，而證明沒有對 x 加任何額外假設，則它對論域中所有 x 成立。",
          "en": "∀-introduction (generalization): if a statement is proved for an arbitrary element x, with no extra assumption on x, it holds for every x in the domain."
        }
      },
      {
        "id": "eq_replace",
        "kind": "prerequisite",
        "statement": {
          "zh-Hant": "等值代換：命題中的一個式子可換成與它相等的式子，命題的真假不變。",
          "en": "Replacement of equals: an expression inside a statement may be replaced by an equal expression without changing whether the statement is true."
        }
      },
      {
        "id": "def_unfold",
        "kind": "prerequisite",
        "statement": {
          "zh-Hant": "展開定義：把一個已定義的詞換成它的定義內容（或反向收回），命題的真假不變。",
          "en": "Unfolding a definition: replacing a defined term by its definition (or folding it back) does not change whether a statement is true."
        }
      },
      {
        "id": "numeric_evaluation",
        "kind": "prerequisite",
        "statement": {
          "zh-Hant": "明確整數或有理數算式的精確求值；可以獨立重算。",
          "en": "Exact evaluation of an explicit integer or rational expression; independently recomputable."
        }
      },
      {
        "id": "case_analysis",
        "kind": "prerequisite",
        "statement": {
          "zh-Hant": "分情況論證：若 P∨Q 成立，且 P⇒R、Q⇒R 都已證，則 R 成立。所有情況必須涵蓋全部可能。",
          "en": "Proof by cases: if P∨Q holds and both P⇒R and Q⇒R are proved, then R holds. The cases must cover every possibility."
        }
      },
      {
        "id": "exists_elim",
        "kind": "prerequisite",
        "statement": {
          "zh-Hant": "∃消去：由 ∃x∈D, P(x)，可取一個新名字 a 表示某個滿足 P(a) 的元素；只要最後結論不含 a，就可用 a 推理。",
          "en": "∃-elimination: from ∃x∈D, P(x) we may name some element a with P(a) by a fresh name; reasoning with a is allowed as long as the final conclusion does not mention a."
        }
      },
      {
        "id": "exists_intro",
        "kind": "prerequisite",
        "statement": {
          "zh-Hant": "∃引入：若找到 a∈D 使 P(a) 成立，則 ∃x∈D, P(x)。",
          "en": "∃-introduction: if some a∈D satisfies P(a), then ∃x∈D, P(x)."
        }
      },
      {
        "id": "and_intro_elim",
        "kind": "prerequisite",
        "statement": {
          "zh-Hant": "合取：由 P 與 Q 得 P∧Q；由 P∧Q 可分別得 P 與 Q。",
          "en": "Conjunction: from P and Q infer P∧Q; from P∧Q infer P and infer Q."
        }
      },
      {
        "id": "mko_event_def",
        "kind": "cited",
        "source": {
          "mko_id": "mko-event",
          "sha256": "5962ae6672cb79c34888bdda4ac66de557a4f9b21243f2445414d7dd326b9c74",
          "commit": "b28a5edbd46279b1db2a3f3f48bb3f447e18899f"
        },
        "statement": {
          "zh-Hant": "本條目的有限初等模型明示採完整事件族 F=P(S)，因此事件 E 可取 S 的任一子集合；一般指定為 (S,F) 的模型不論 S 有限或無限，都要求事件 E 屬於 F。試驗結果屬於 E 時稱事件 E 發生。",
          "en": "The finite elementary model in this entry explicitly uses the full event family F=P(S), so an event E may be any subset of S. In a general model (S,F), whether S is finite or infinite, an event must belong to F. Event E occurs when the observed outcome belongs to E."
        }
      },
      {
        "id": "powerset_def",
        "kind": "prerequisite",
        "statement": {
          "zh-Hant": "冪集：X∈𝒫(S) 當且僅當 X⊆S。",
          "en": "Power set: X∈𝒫(S) if and only if X⊆S."
        }
      },
      {
        "id": "sigma_algebra_def",
        "kind": "prerequisite",
        "statement": {
          "zh-Hant": "事件族（σ-代數）：S 的子集合族 F 稱為 S 上的事件族，若 S∈F；X∈F 時 S∖X∈F；且 F 中任意可數多個（含有限多個）成員的聯集都屬於 F。",
          "en": "Event family (σ-algebra): a family F of subsets of S is an event family on S if S∈F; S∖X∈F whenever X∈F; and the union of any countably many (including finitely many) members of F belongs to F."
        }
      },
      {
        "id": "L_F0_unions",
        "kind": "lemma",
        "proved_in": [
          "S10.1",
          "S10.2",
          "S10.3"
        ],
        "statement": {
          "zh-Hant": "F₀ 中任意可數多個成員的聯集仍屬於 F₀。",
          "en": "The union of any countably many members of F₀ belongs to F₀."
        }
      }
    ],
    "revision_log": [],
    "numeric_checks": [
      {
        "id": "N1",
        "step": "S04",
        "lhs": "4 % 2",
        "rhs": "0"
      },
      {
        "id": "N2",
        "step": "S04",
        "lhs": "3 % 2",
        "rhs": "1"
      },
      {
        "id": "N3",
        "step": "S09",
        "lhs": "3+3",
        "rhs": "6"
      },
      {
        "id": "N4",
        "step": "S08",
        "lhs": "6-3",
        "rhs": "3"
      },
      {
        "id": "N5",
        "step": "S10.3",
        "lhs": "2**2",
        "rhs": "4"
      }
    ]
  },
  "presentation": {
    "md_scope": {
      "zh-Hant": "在 MKO 的範圍內工作：樣本空間 S 明示（假設 a1），兩個模型的事件族都明示（假設 a2）。一般模型的事件族採用標準的 σ-代數定義（前置知識 sigma_algebra_def）。不指派任何機率值；不處理無限樣本空間或 Lean 的 MeasurableSet；不改動 canonical MKO 或英文 edition。",
      "en": "Works inside the MKO's scope: the sample space S is explicit (assumption a1), and the event family of each model is explicit (assumption a2). The event family of a general model follows the standard σ-algebra definition (prerequisite sigma_algebra_def). No probability values are assigned; infinite sample spaces and Lean's MeasurableSet are not treated; the canonical MKO and its English edition are not changed."
    },
    "md_symbols": [
      [
        "S",
        "擲骰一次的樣本空間",
        "the sample space of one roll"
      ],
      [
        "\\mathcal{P}(S)",
        "S 的所有子集合構成的完整事件族",
        "the full event family of all subsets of S"
      ],
      [
        "F_0",
        "較粗的事件族 {∅,O,E,S}",
        "the coarser event family {∅,O,E,S}"
      ],
      [
        "E,\\ O",
        "偶數結果之集、奇數結果之集",
        "the even outcomes, the odd outcomes"
      ],
      [
        "\\omega",
        "觀察到的結果",
        "the observed outcome"
      ],
      [
        "S\\setminus X",
        "X 在 S 中的補集",
        "the complement of X in S"
      ]
    ],
    "md_error_controls": [
      {
        "title": "事件不是單一結果 / An event is not a single outcome",
        "zh": "MKO 指出常見誤解是把事件當成單一結果。4 是 S 的元素（結果），{4} 是 S 的子集合（事件），4≠{4}；而 E={2,4,6} 含三個結果（S05）。說「E 發生」是說觀察到的結果屬於 E，不是說觀察到的結果等於 E。",
        "en": "The MKO names the mistake of treating an event as a single outcome. 4 is an element of S (an outcome), {4} is a subset of S (an event), and 4≠{4}; E={2,4,6} contains three outcomes (S05). 'E occurs' means the observed outcome belongs to E, not that it equals E."
      },
      {
        "title": "先指定樣本空間與事件族 / Specify the sample space and event family first",
        "zh": "MKO 提醒不要在未指定樣本空間或事件族時列事件。問「{6} 是不是事件」沒有固定答案：在 M1 中是，在 M2 中不是（S12）。所以每次列事件之前，都要說明 S 與 F。",
        "en": "The MKO warns against listing events before the sample space or event family is specified. The question 'is {6} an event?' has no fixed answer: it is in M1 and is not in M2 (S12). So S and F must be stated before any events are listed."
      },
      {
        "title": "S 有限不代表所有子集合都是事件 / A finite S does not make every subset an event",
        "zh": "MKO 指出另一個誤解：樣本空間有限就自動讓所有子集合可測。M2 的 S 只有 6 個元素，F₀ 是合法事件族（S07、S08、S10），但 {6}∉F₀。只有明示採 F=𝒫(S) 時，每個子集合才是事件。",
        "en": "The MKO names another mistake: assuming a finite sample space makes every subset measurable. In M2, S has only 6 elements and F₀ is a legitimate event family (S07, S08, S10), yet {6}∉F₀. Every subset is an event only when F=𝒫(S) is explicitly adopted."
      }
    ],
    "md_open_items": [
      {
        "zh-Hant": "一般模型的事件族採用標準的 σ-代數定義（sigma_algebra_def）作為前置知識；MKO 本身只要求事件屬於明示的 F。",
        "en": "The event family of a general model uses the standard σ-algebra definition (sigma_algebra_def) as a prerequisite; the MKO itself only requires events to belong to the explicit F."
      },
      {
        "zh-Hant": "N1、N2 以餘數核對 4 為偶數、3 為奇數，只是交叉核對；論證依據的是 E 與 O 的列舉。",
        "en": "N1 and N2 check by remainders that 4 is even and 3 is odd; they are cross-checks only, and the argument relies on the lists of E and O."
      }
    ]
  },
  "revision": {
    "id": "production-20261019-adopted-96",
    "source_commit": "b28a5edbd46279b1db2a3f3f48bb3f447e18899f",
    "adopted_source_sha256": "5f67b035b93e0779d5d9d9a91ef38849d1d4aa6bcd00b2c987bd6528a63147cd",
    "original_source_sha256": "5f67b035b93e0779d5d9d9a91ef38849d1d4aa6bcd00b2c987bd6528a63147cd",
    "owner_adopted": true,
    "bilingual_sync": "complete_source_bound",
    "formal_Lean": "NOT_RUN",
    "independent_AI_concurrence": "NOT_MEASURED",
    "issue_ids": [],
    "scope": "Owner-adopted natural-language teaching guide; not a formal proof, canonical MKO rewrite or evidence-status promotion."
  }
}