{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20261003-advance-20261004-r03-relation-0064",
  "pilot_number": "R0069-35039",
  "number": 35039,
  "family_id": "qf-relation-subset-image",
  "title": {
    "zh-Hant": "第 35039 題：二元關係的子集合像",
    "en": "Question 35039: Image of a Subset under a Binary Relation"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20261003-advance-20261004-r03-relation-0064",
    "batch_id": "batch-20261003-advance-20261004-r03-relation-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-relation"
    ],
    "title_zh": "二元關係的子集合像",
    "stem_zh": "二元關係的來源 A={1, 2, 3}、目標 B={5, 6, 7}，R={(1, 5)}⊆A×B。明示 C={1, 2, 3}⊆A，求 R[C] 的遞增、無重複集合。",
    "answer": {
      "kind": "expression",
      "value": "{5}",
      "explanation_zh": "R[C]={5}。選第一分量在 C 的所有關係對，再收集第二分量；不要求每個輸入只有一個輸出。"
    },
    "difficulty": {
      "band": "undergraduate_core",
      "level": 2,
      "skills": [
        "二元關係的子集合像"
      ]
    },
    "variant": {
      "family_id": "qf-relation-subset-image",
      "seed": 202610043064,
      "parameters": {
        "A": [
          1,
          2,
          3
        ],
        "B": [
          5,
          6,
          7
        ],
        "R": [
          [
            1,
            5
          ]
        ],
        "operation": "subset_image",
        "C": [
          1,
          2,
          3
        ],
        "semantic_template": "qf-relation-subset-image/{\"A\":[1,2,3],\"B\":[5,6,7],\"C\":[1,2,3],\"R\":[[1,5]],\"operation\":\"subset_image\"}"
      }
    },
    "verification": {
      "status": "candidate",
      "checks": [
        "producer_exact_integer_rational_arithmetic",
        "producer_finite_enumeration",
        "producer_source_snapshot_check",
        "producer_substantive_template_deduplication"
      ],
      "warning_zh": "僅為產生端候選及自檢；主架構者須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"
    },
    "provenance": {
      "generator_role": "question_producer",
      "generator_model": "Codex independent peer (inherited model; exact runtime model not exposed)",
      "generated_at": "2026-10-03",
      "contract_version": "ocme-question-factory-v0.1"
    }
  },
  "original_english_edition": {
    "schema_version": "ocme-question-localization-v0.1",
    "question_id": "q-20261003-advance-20261004-r03-relation-0064",
    "batch_id": "batch-20261003-advance-20261004-r03-relation-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "dae84b64aff2b5cbe6b92d0987e04d92bf4c01c06736f8cdc45b19feef8a9d22",
      "family_id": "qf-relation-subset-image",
      "seed": 202610043064,
      "semantic_template": "qf-relation-subset-image/{\"A\":[1,2,3],\"B\":[5,6,7],\"C\":[1,2,3],\"R\":[[1,5]],\"operation\":\"subset_image\"}"
    },
    "fields": {
      "title": "Image of a Subset under a Binary Relation",
      "stem": "The binary relation has source A={1, 2, 3}, target B={5, 6, 7}, and R={(1, 5)}⊆A×B. For the explicit C={1, 2, 3}⊆A, list R[C] in increasing order with no duplicates.",
      "answer": {
        "kind": "expression",
        "value": "{5}",
        "explanation": "Select relation pairs whose first component lies in C and collect their second components. R[C]={5}; a relation need not have a unique output for each input."
      },
      "skills": [
        "Image of a Subset under a Binary Relation"
      ],
      "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": "二元關係的來源 A={1, 2, 3}、目標 B={5, 6, 7}，R={(1, 5)}⊆A×B。明示 C={1, 2, 3}⊆A，求 R[C] 的遞增、無重複集合。",
        "en": "The binary relation has source A={1, 2, 3}, target B={5, 6, 7}, and R={(1, 5)}⊆A×B. For the explicit C={1, 2, 3}⊆A, list R[C] in increasing order with no duplicates."
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "source roles = [qf-relation-subset-image, [A, B, C, R, operation]]",
      "after": "derived = [subset_image, no-invented-domain]",
      "relation": "implies",
      "rule_id": "R69_SOURCE_specific_keys_operation_domain_AND_binary_contract",
      "conditions": [],
      "explanation": {
        "zh-Hant": "各bank的keys/operation/domain獨立：LDL/PULSE/function legacy為表內underscore，TREE是完整tree_*與明定domain，BASE signed較寬不偷加op/domain。所有算術保持binary primitive契約。",
        "en": "Keep EACH bank’s keys/operation/domain distinct:LDL/PULSE/function legacy table underscores,TREE full tree_* with its declared domain,and permissive BASE signed without invented op/domain.All arithmetic respects the binary-primitive contract."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "反查本題原資料、型別、角色、完整前提與欄序；單向推論不倒當等價，有限表核對不冒充無限論域/形式證明。原題/英文/MKO/AST/狀態不變。",
          "en": "Reverse-audit THIS question’s ORIGINAL data,types,roles,complete premises and field order.Do not invert implication or turn finite-table checks into infinite-domain/formal proof.Original Q/English/MKO/AST/status remain unchanged."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S002",
      "uses": [
        "G001",
        "S001"
      ],
      "before": "source roles = [relation A ORIGINAL, 100, [1, 2, 3]]",
      "after": "derived = [[1, 2, 3]]",
      "relation": "implies",
      "rule_id": "Finite_RAW_ALL_SAFE_distinct_labels_then_derived_sort",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先驗全部raw SAFE元素、不重複及此bank個數界，空集合法；只把衍生表示排序，原JSON不修改。TREE與LDL的上限不可互搬。",
        "en": "Type EVERY RAW SAFE label,distinctness and THIS bank’s cardinality bound;empty is allowed.Sort ONLY the derived representation,not original JSON.Do not exchange TREE and LDL limits."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "反查本題原資料、型別、角色、完整前提與欄序；單向推論不倒當等價，有限表核對不冒充無限論域/形式證明。原題/英文/MKO/AST/狀態不變。",
          "en": "Reverse-audit THIS question’s ORIGINAL data,types,roles,complete premises and field order.Do not invert implication or turn finite-table checks into infinite-domain/formal proof.Original Q/English/MKO/AST/status remain unchanged."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S003",
      "uses": [
        "G001",
        "S002"
      ],
      "before": "source roles = [relation B ORIGINAL, 100, [5, 6, 7]]",
      "after": "derived = [[5, 6, 7]]",
      "relation": "implies",
      "rule_id": "Finite_RAW_ALL_SAFE_distinct_labels_then_derived_sort",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先驗全部raw SAFE元素、不重複及此bank個數界，空集合法；只把衍生表示排序，原JSON不修改。TREE與LDL的上限不可互搬。",
        "en": "Type EVERY RAW SAFE label,distinctness and THIS bank’s cardinality bound;empty is allowed.Sort ONLY the derived representation,not original JSON.Do not exchange TREE and LDL limits."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "反查本題原資料、型別、角色、完整前提與欄序；單向推論不倒當等價，有限表核對不冒充無限論域/形式證明。原題/英文/MKO/AST/狀態不變。",
          "en": "Reverse-audit THIS question’s ORIGINAL data,types,roles,complete premises and field order.Do not invert implication or turn finite-table checks into infinite-domain/formal proof.Original Q/English/MKO/AST/status remain unchanged."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S004",
      "uses": [
        "G001",
        "S003"
      ],
      "before": "source roles = [relation R ORIGINAL, 0, [1, 5]]",
      "after": "derived = [[1, 5]]",
      "relation": "implies",
      "rule_id": "Pairs_EVERY_RAW_two_SAFE_cells_BEFORE_any_dedup",
      "conditions": [],
      "explanation": {
        "zh-Hant": "去重之前逐raw pair驗恰兩個SAFE分量；相同row可作表示重複，但不能先丟掉未驗欄位。",
        "en": "BEFORE deduplication,type EACH RAW pair as EXACTLY TWO SAFE components.Identical rows may be representation duplicates,but no unvalidated cell may be silently discarded."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "反查本題原資料、型別、角色、完整前提與欄序；單向推論不倒當等價，有限表核對不冒充無限論域/形式證明。原題/英文/MKO/AST/狀態不變。",
          "en": "Reverse-audit THIS question’s ORIGINAL data,types,roles,complete premises and field order.Do not invert implication or turn finite-table checks into infinite-domain/formal proof.Original Q/English/MKO/AST/status remain unchanged."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S005",
      "uses": [
        "G001",
        "S004"
      ],
      "before": "source roles = [[1, 2, 3], [5, 6, 7], [[1, 5]]]",
      "after": "derived = [是]",
      "relation": "implies",
      "rule_id": "Relation_WHOLE_R_subset_declared_AcrossB_NO_function_premise",
      "conditions": [],
      "explanation": {
        "zh-Hant": "完整R先合格為A×B子集，關係可一對多/多對一，不要求total或function。raw重複pair在型別檢查後可正規化。",
        "en": "FIRST qualify the WHOLE R as a subset of A×B.A relation may be one-to-many/many-to-one and need not be total or a function.Typed raw duplicate pairs may normalize."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "反查本題原資料、型別、角色、完整前提與欄序；單向推論不倒當等價，有限表核對不冒充無限論域/形式證明。原題/英文/MKO/AST/狀態不變。",
          "en": "Reverse-audit THIS question’s ORIGINAL data,types,roles,complete premises and field order.Do not invert implication or turn finite-table checks into infinite-domain/formal proof.Original Q/English/MKO/AST/status remain unchanged."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S006",
      "uses": [
        "G001",
        "S005"
      ],
      "before": "source roles = [relation image C ORIGINAL, 100, [1, 2, 3]]",
      "after": "derived = [[1, 2, 3]]",
      "relation": "implies",
      "rule_id": "Finite_RAW_ALL_SAFE_distinct_labels_then_derived_sort",
      "conditions": [],
      "explanation": {
        "zh-Hant": "先驗全部raw SAFE元素、不重複及此bank個數界，空集合法；只把衍生表示排序，原JSON不修改。TREE與LDL的上限不可互搬。",
        "en": "Type EVERY RAW SAFE label,distinctness and THIS bank’s cardinality bound;empty is allowed.Sort ONLY the derived representation,not original JSON.Do not exchange TREE and LDL limits."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "反查本題原資料、型別、角色、完整前提與欄序；單向推論不倒當等價，有限表核對不冒充無限論域/形式證明。原題/英文/MKO/AST/狀態不變。",
          "en": "Reverse-audit THIS question’s ORIGINAL data,types,roles,complete premises and field order.Do not invert implication or turn finite-table checks into infinite-domain/formal proof.Original Q/English/MKO/AST/status remain unchanged."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S007",
      "uses": [
        "G001",
        "S006"
      ],
      "before": "source roles = [1, 5, [1, 2, 3]]",
      "after": "derived = [是]",
      "relation": "implies",
      "rule_id": "Relation_IMAGE_EACH_pair_SOURCE_membership_then_ALL_outputs",
      "conditions": [],
      "explanation": {
        "zh-Hant": "逐pair先核第一分量是否屬C，符合者收全部第二分量；不能只取一個output或把C本身當image。",
        "en": "For EACH pair,test whether its FIRST coordinate belongs to C and collect ALL qualifying SECOND coordinates.Do not select only one output or substitute C itself."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "反查本題原資料、型別、角色、完整前提與欄序；單向推論不倒當等價，有限表核對不冒充無限論域/形式證明。原題/英文/MKO/AST/狀態不變。",
          "en": "Reverse-audit THIS question’s ORIGINAL data,types,roles,complete premises and field order.Do not invert implication or turn finite-table checks into infinite-domain/formal proof.Original Q/English/MKO/AST/status remain unchanged."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S008",
      "uses": [
        "G001",
        "S007"
      ],
      "before": "source roles = [[1, 2, 3], [5, 6, 7], [1, 2, 3], [[1, 5]]]",
      "after": "derived = [[5], {5}]",
      "relation": "implies",
      "rule_id": "Relation_IMAGE_complete_selected_pairs_dedup_numeric_output",
      "conditions": [],
      "explanation": {
        "zh-Hant": "C先為A子集，R[C]在B內且去重排序；C空或無關聯時image空，並不因此否定其他C的像。",
        "en": "FIRST require C⊆A.The complete image R[C]lies in B and is deduplicated/numerically sorted.Empty C or no selected edges gives an empty image,not a claim about other subsets."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "反查本題原資料、型別、角色、完整前提與欄序；單向推論不倒當等價，有限表核對不冒充無限論域/形式證明。原題/英文/MKO/AST/狀態不變。",
          "en": "Reverse-audit THIS question’s ORIGINAL data,types,roles,complete premises and field order.Do not invert implication or turn finite-table checks into infinite-domain/formal proof.Original Q/English/MKO/AST/status remain unchanged."
        }
      },
      "arithmetic": null
    }
  ],
  "math_model": {
    "result": "{5}"
  },
  "conclusion": {
    "canonical_value": "{5}",
    "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-20261003-advance-20261004-r03-relation-foundations",
      "batch_path": "public/data/questions/batches/batch-20261003-advance-20261004-r03-relation-foundations/questions.jsonl",
      "batch_sha256": "dafc3bbe4cc84e28621dd86797ac13842574a4a0040141c8131ecd0112d1b189",
      "original_line": 64,
      "original_line_sha256": "06cd50b9c0e02edf40dd35ed3288ffdb3826dcd158aabd9cd4139da59d4ac945",
      "question_compact_sha256": "dae84b64aff2b5cbe6b92d0987e04d92bf4c01c06736f8cdc45b19feef8a9d22",
      "english_row_compact_sha256": "95f2580cc2868fa1dd77bb6ae13f0ed49d37d71cb6dc0a382d935da919702df4",
      "english_manifest_path": "public/data/i18n/editions/batch-20261003-advance-20261004-r03-relation-foundations.en.json",
      "english_rows_path": "public/data/i18n/questions/batch-20261003-advance-20261004-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": "63155b87d4994521b4a60f8f179750f57cbbd094e66cc42652608d1bbab47a9f",
    "original_candidate_sha256": "888811e3881230deb3f51a5a439577b617f60786973050c834425b5e8bb21634",
    "original_candidate_line_sha256": "888811e3881230deb3f51a5a439577b617f60786973050c834425b5e8bb21634"
  },
  "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-20261026-012-p70",
    "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": "ccf93c72638d6abf6a412514da5838d55ecb09b71d0e4fdec2b559dc9fdee14b",
      "question_first_witness_sha256": "11af9a338b5a610342b2cfb2ee295683131b3139ac402dd19b1ba1725976d518"
    },
    "status": "second_context_reviewed",
    "mathematical_semantics": true,
    "bilingual_semantics": true,
    "steps_reviewed": 8,
    "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction",
    "exact_question_first_check": {
      "model_sha256": "eceb1be927f46da00d6cecf7b6bdabc407bc766feb1ba1f565af9b3827ec26d3",
      "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."
  }
}