{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261012-advance-20261017-r03-relation-0029",
  "pilot_number": "R0015-08379",
  "number": 8379,
  "family_id": "qf-density17-relation-subset-image-fibers",
  "title": {
    "zh-Hant": "第 8379 題：指定來源子集的關係像",
    "en": "Question 8379: Restricted Relation Image and All Target Fibers"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261012-advance-20261017-r03-relation-0029",
    "batch_id": "batch-20261012-advance-20261017-r03-relation-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-relation"
    ],
    "title_zh": "指定來源子集的關係像",
    "stem_zh": "明示Fin來源與目標，有序關係對。來源Fin(2)=[0, 1]，目標Fin(2)=[0, 1]，關係R的列項=[[0, 0], [0, 1], [1, 1]]。來源子集列項=[0]。答 (全正規關係對, 正規子集, R[子集], 限制到子集後按所有目標列的纖維)。",
    "answer": {
      "kind": "expression",
      "value": "([[0, 0], [0, 1], [1, 1]], [0], [0, 1], [[0], [0]])",
      "explanation_zh": "離散結構核對=([[0, 0], [0, 1], [1, 1]], [0], [0, 1], [[0], [0]])；全關係保留在首欄；像與纖維只篩第一分量落於子集的列。；有限精確核對不是一般形式證明；逆關係不自動是反函數；關係不必是總函數；重複列項按集合正規化。"
    },
    "difficulty": {
      "band": "undergraduate_core",
      "level": 1,
      "skills": [
        "指定來源子集的關係像"
      ]
    },
    "variant": {
      "family_id": "qf-density17-relation-subset-image-fibers",
      "seed": 202610173029,
      "parameters": {
        "n": 2,
        "m": 2,
        "pairs": [
          [
            0,
            0
          ],
          [
            0,
            1
          ],
          [
            1,
            1
          ]
        ],
        "subset": [
          0
        ],
        "domain": "Fin_typed_relation",
        "operation": "relation-subset-image-fibers",
        "semantic_template": "qf-density17-relation-subset-image-fibers/{\"m\":2,\"n\":2,\"pairs\":[[0,0],[0,1],[1,1]],\"subset\":[0]}"
      }
    },
    "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": "2026-10-12",
      "contract_version": "ocme-question-factory-v0.1"
    }
  },
  "original_english_edition": {
    "schema_version": "ocme-question-localization-v0.1",
    "question_id": "q-20261012-advance-20261017-r03-relation-0029",
    "batch_id": "batch-20261012-advance-20261017-r03-relation-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "6f6c4b19d616915f8b86ac9f24fbf993cc9f655684ad9856425d04a2e3943fa3",
      "family_id": "qf-density17-relation-subset-image-fibers",
      "seed": 202610173029,
      "semantic_template": "qf-density17-relation-subset-image-fibers/{\"m\":2,\"n\":2,\"pairs\":[[0,0],[0,1],[1,1]],\"subset\":[0]}"
    },
    "fields": {
      "title": "Restricted Relation Image and All Target Fibers",
      "stem": "The explicitly typed relation R⊆Fin(2)×Fin(2) has ordered pairs=[[0, 0], [0, 1], [1, 1]]. Normalize repeated pairs as a set, in lexicographic order. A relation need not be single-valued or total; its converse need not be an inverse function. The source subset is [0]. Give (normalized pairs, sorted source subset, its image, all target fibers restricted to that subset). Normalize repeated subset labels as a set; do not renumber them.",
      "answer": {
        "kind": "expression",
        "value": "([[0, 0], [0, 1], [1, 1]], [0], [0, 1], [[0], [0]])",
        "explanation": "Retain only pairs whose original source lies in the declared subset and enumerate every declared target fiber. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([[0, 0], [0, 1], [1, 1]], [0], [0, 1], [[0], [0]])."
      },
      "skills": [
        "Restricted Relation Image and All Target Fibers"
      ],
      "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": "明示Fin來源與目標，有序關係對。來源Fin(2)=[0, 1]，目標Fin(2)=[0, 1]，關係R的列項=[[0, 0], [0, 1], [1, 1]]。來源子集列項=[0]。答 (全正規關係對, 正規子集, R[子集], 限制到子集後按所有目標列的纖維)。",
        "en": "The explicitly typed relation R⊆Fin(2)×Fin(2) has ordered pairs=[[0, 0], [0, 1], [1, 1]]. Normalize repeated pairs as a set, in lexicographic order. A relation need not be single-valued or total; its converse need not be an inverse function. The source subset is [0]. Give (normalized pairs, sorted source subset, its image, all target fibers restricted to that subset). Normalize repeated subset labels as a set; do not renumber them."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "types=(2,2);raw ordered pairs=[[0, 0], [0, 1], [1, 1]]",
      "after": "normalized relation=[[0, 0], [0, 1], [1, 1]]",
      "relation": "implies",
      "rule_id": "typed_relation_set_normalization",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先核每一有序對的來源與目標型別，再把重複的同一有序對去重。交換座標是逆關係，不是同一對的排序。關係可非單值或非總。",
        "en": "Validate both types of every ordered pair,then deduplicate identical pairs.Swapping coordinates forms the converse,not a sorting of one pair.A relation may be multivalued or partial."
      },
      "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": "raw subset=[0];all pairs=[[0, 0], [0, 1], [1, 1]]",
      "after": "subset=[0];selected pairs=[[0, 0], [0, 1]];image=[0, 1]",
      "relation": "implies",
      "rule_id": "relation_restrict_source_not_renumber",
      "conditions": [],
      "explanation": {
        "zh-Hant": "子集重複標籤按集合去重，只篩第一座標；全原關係仍保留，不重新編號來源。",
        "en": "Deduplicate subset labels as a set and filter only on the first coordinate.Retain the full original relation without renumbering sources."
      },
      "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": "target=0;selected pairs=[[0, 0], [0, 1]]",
      "after": "restricted fiber=[0]",
      "relation": "implies",
      "rule_id": "restricted_relation_target_fiber",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個宣告目標都列限制後全部原像，包含空纖維。",
        "en": "List every restricted preimage at each declared target,including empty fibers."
      },
      "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": "target=1;selected pairs=[[0, 0], [0, 1]]",
      "after": "restricted fiber=[0]",
      "relation": "implies",
      "rule_id": "restricted_relation_target_fiber",
      "conditions": [],
      "explanation": {
        "zh-Hant": "每個宣告目標都列限制後全部原像，包含空纖維。",
        "en": "List every restricted preimage at each declared target,including empty fibers."
      },
      "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": [
      [
        [
          0,
          0
        ],
        [
          0,
          1
        ],
        [
          1,
          1
        ]
      ],
      [
        0
      ],
      [
        0,
        1
      ],
      [
        [
          0
        ],
        [
          0
        ]
      ]
    ]
  },
  "conclusion": {
    "canonical_value": "([[0, 0], [0, 1], [1, 1]], [0], [0, 1], [[0], [0]])",
    "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-20261012-advance-20261017-r03-relation-foundations",
      "batch_path": "public/data/questions/batches/batch-20261012-advance-20261017-r03-relation-foundations/questions.jsonl",
      "batch_sha256": "fa93b079fe3332713b9e2dc835c033d2610769938b4950f23e6348ae0d11f894",
      "original_line": 29,
      "original_line_sha256": "e8bf849752f95c14bb9ef8418972ea045fbbb691e060e02fadf100ba621ac74e",
      "question_compact_sha256": "6f6c4b19d616915f8b86ac9f24fbf993cc9f655684ad9856425d04a2e3943fa3",
      "english_row_compact_sha256": "26dd502196d0ffd462760e75c1e8d129cdef4bacef2c29d9055bcb0e68a1c101",
      "english_manifest_path": "public/data/i18n/editions/batch-20261012-advance-20261017-r03-relation-foundations.en.json",
      "english_rows_path": "public/data/i18n/questions/batch-20261012-advance-20261017-r03-relation-foundations.en.jsonl"
    },
    "mko_bindings": {
      "mko-relation": {
        "sha256": "3788215401bc9a5610691985883d04b506880cc9bd5e2ce0a98054cd5a5f0774",
        "english_sha256": "7d0700fb988776905f5d6ce3bf7399bc7a370ea89a38fe4e6b4c9e8509acdfdb",
        "original_path": "public/data/mko/relation.json",
        "english_path": "public/data/i18n/editions/mko-relation.en.json"
      }
    },
    "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff",
    "selected_part_sha256": "ca1078dc467a0f3769865919aadc46c87712d098f1dfbde90f787471e22bcfec",
    "original_candidate_sha256": "04f022388d4ff79deaf99b8da872a13048b4a153e33610b68ba0d231e8d346ff",
    "original_candidate_line_sha256": "04f022388d4ff79deaf99b8da872a13048b4a153e33610b68ba0d231e8d346ff"
  },
  "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-p16",
    "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": "6ecad9d909199221a07e923f87425e80b3bdec3eed46fe5910d04fca694cae18",
      "question_first_witness_sha256": "e8f00458977a09493b5f9f05ab9e6feec72debaf4c7d1c609d1fc95900a35a1b"
    },
    "status": "second_context_reviewed",
    "mathematical_semantics": true,
    "bilingual_semantics": true,
    "steps_reviewed": 4,
    "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction",
    "exact_question_first_check": {
      "model_sha256": "bc08d6e19f4fe6187dd6c55fdabf851bf51d8264a05d768d9ebed51993f5e4d6",
      "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."
  }
}