{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r03-relation-0001","batch_id":"batch-20261012-advance-20261017-r03-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7900c95f74543fbc4b1372e4c3feb3b1b7e2c9b0462cf3a75870fda27a202400","family_id":"qf-density17-relation-full-typed-inventory","seed":202610173001,"semantic_template":"qf-density17-relation-full-typed-inventory/{\"m\":0,\"n\":0,\"pairs\":[]}"},"fields":{"title":"Typed Relation Projections and Missing Sources","stem":"The explicitly typed relation R⊆Fin(0)×Fin(0) has ordered pairs=[]. 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. Give ([n,m], normalized pairs, source projection, target projection, pair count, missing source labels).","answer":{"kind":"expression","value":"([0, 0], [], [], [], 0, [])","explanation":"Project each coordinate of the complete pair set; missing sources are legal for a relation. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 0], [], [], [], 0, [])."},"skills":["Typed Relation Projections and Missing Sources"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r03-relation-0002","batch_id":"batch-20261012-advance-20261017-r03-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ae54e15964771f90446f4f09e3ae30c9fd44c9d2c57350c8fbf5fc275fcfeede","family_id":"qf-density17-relation-full-typed-inventory","seed":202610173002,"semantic_template":"qf-density17-relation-full-typed-inventory/{\"m\":0,\"n\":2,\"pairs\":[]}"},"fields":{"title":"Typed Relation Projections and Missing Sources","stem":"The explicitly typed relation R⊆Fin(2)×Fin(0) has ordered pairs=[]. 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. Give ([n,m], normalized pairs, source projection, target projection, pair count, missing source labels).","answer":{"kind":"expression","value":"([2, 0], [], [], [], 0, [0, 1])","explanation":"Project each coordinate of the complete pair set; missing sources are legal for a relation. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([2, 0], [], [], [], 0, [0, 1])."},"skills":["Typed Relation Projections and Missing Sources"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r03-relation-0003","batch_id":"batch-20261012-advance-20261017-r03-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6c153c1c4615c5baaa3674a58e7f6aed38af52992c0d7b8dc27b358a725e2ebb","family_id":"qf-density17-relation-full-typed-inventory","seed":202610173003,"semantic_template":"qf-density17-relation-full-typed-inventory/{\"m\":2,\"n\":0,\"pairs\":[]}"},"fields":{"title":"Typed Relation Projections and Missing Sources","stem":"The explicitly typed relation R⊆Fin(0)×Fin(2) has ordered pairs=[]. 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. Give ([n,m], normalized pairs, source projection, target projection, pair count, missing source labels).","answer":{"kind":"expression","value":"([0, 2], [], [], [], 0, [])","explanation":"Project each coordinate of the complete pair set; missing sources are legal for a relation. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 2], [], [], [], 0, [])."},"skills":["Typed Relation Projections and Missing Sources"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r03-relation-0004","batch_id":"batch-20261012-advance-20261017-r03-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"32ebc95b72e9e783b87c81015c0ce35cef17963b15cd2d7304c0a3c896ce0e7a","family_id":"qf-density17-relation-full-typed-inventory","seed":202610173004,"semantic_template":"qf-density17-relation-full-typed-inventory/{\"m\":2,\"n\":2,\"pairs\":[[0,0],[1,1]]}"},"fields":{"title":"Typed Relation Projections and Missing Sources","stem":"The explicitly typed relation R⊆Fin(2)×Fin(2) has ordered pairs=[[0, 0], [1, 1], [0, 0]]. 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. Give ([n,m], normalized pairs, source projection, target projection, pair count, missing source labels).","answer":{"kind":"expression","value":"([2, 2], [[0, 0], [1, 1]], [0, 1], [0, 1], 2, [])","explanation":"Project each coordinate of the complete pair set; missing sources are legal for a relation. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([2, 2], [[0, 0], [1, 1]], [0, 1], [0, 1], 2, [])."},"skills":["Typed Relation Projections and Missing Sources"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r03-relation-0005","batch_id":"batch-20261012-advance-20261017-r03-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ca8d1b431dbe323dedbcbec12119af641fcf417f5b000ec3c064c7ad40a7be86","family_id":"qf-density17-relation-full-typed-inventory","seed":202610173005,"semantic_template":"qf-density17-relation-full-typed-inventory/{\"m\":2,\"n\":2,\"pairs\":[[0,0],[0,1]]}"},"fields":{"title":"Typed Relation Projections and Missing Sources","stem":"The explicitly typed relation R⊆Fin(2)×Fin(2) has ordered pairs=[[0, 0], [0, 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. Give ([n,m], normalized pairs, source projection, target projection, pair count, missing source labels).","answer":{"kind":"expression","value":"([2, 2], [[0, 0], [0, 1]], [0], [0, 1], 2, [1])","explanation":"Project each coordinate of the complete pair set; missing sources are legal for a relation. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([2, 2], [[0, 0], [0, 1]], [0], [0, 1], 2, [1])."},"skills":["Typed Relation Projections and Missing Sources"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r03-relation-0006","batch_id":"batch-20261012-advance-20261017-r03-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8539080e27ac5d4dceb083953982688abee31193157de83161f064ef1ac09bd8","family_id":"qf-density17-relation-full-typed-inventory","seed":202610173006,"semantic_template":"qf-density17-relation-full-typed-inventory/{\"m\":3,\"n\":3,\"pairs\":[[0,0]]}"},"fields":{"title":"Typed Relation Projections and Missing Sources","stem":"The explicitly typed relation R⊆Fin(3)×Fin(3) has ordered pairs=[[0, 0]]. 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. Give ([n,m], normalized pairs, source projection, target projection, pair count, missing source labels).","answer":{"kind":"expression","value":"([3, 3], [[0, 0]], [0], [0], 1, [1, 2])","explanation":"Project each coordinate of the complete pair set; missing sources are legal for a relation. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([3, 3], [[0, 0]], [0], [0], 1, [1, 2])."},"skills":["Typed Relation Projections and Missing Sources"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r03-relation-0007","batch_id":"batch-20261012-advance-20261017-r03-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"079dcc6db17c9c02d2ed24f17673be885e9acb46c9f0a2696917979d6d4d54da","family_id":"qf-density17-relation-full-typed-inventory","seed":202610173007,"semantic_template":"qf-density17-relation-full-typed-inventory/{\"m\":3,\"n\":3,\"pairs\":[[0,1]]}"},"fields":{"title":"Typed Relation Projections and Missing Sources","stem":"The explicitly typed relation R⊆Fin(3)×Fin(3) has ordered pairs=[[0, 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. Give ([n,m], normalized pairs, source projection, target projection, pair count, missing source labels).","answer":{"kind":"expression","value":"([3, 3], [[0, 1]], [0], [1], 1, [1, 2])","explanation":"Project each coordinate of the complete pair set; missing sources are legal for a relation. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([3, 3], [[0, 1]], [0], [1], 1, [1, 2])."},"skills":["Typed Relation Projections and Missing Sources"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r03-relation-0008","batch_id":"batch-20261012-advance-20261017-r03-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f8d1767c805122996dacb55e10f282215b46e7e124dd2a0febfed4c547a8786e","family_id":"qf-density17-relation-full-typed-inventory","seed":202610173008,"semantic_template":"qf-density17-relation-full-typed-inventory/{\"m\":3,\"n\":3,\"pairs\":[[0,0],[0,1]]}"},"fields":{"title":"Typed Relation Projections and Missing Sources","stem":"The explicitly typed relation R⊆Fin(3)×Fin(3) has ordered pairs=[[0, 0], [0, 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. Give ([n,m], normalized pairs, source projection, target projection, pair count, missing source labels).","answer":{"kind":"expression","value":"([3, 3], [[0, 0], [0, 1]], [0], [0, 1], 2, [1, 2])","explanation":"Project each coordinate of the complete pair set; missing sources are legal for a relation. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([3, 3], [[0, 0], [0, 1]], [0], [0, 1], 2, [1, 2])."},"skills":["Typed Relation Projections and Missing Sources"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r03-relation-0009","batch_id":"batch-20261012-advance-20261017-r03-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"75c440213a94cbe089a3cd5783ef181591302a3d259be84927662dfa9b995fcd","family_id":"qf-density17-relation-full-typed-inventory","seed":202610173009,"semantic_template":"qf-density17-relation-full-typed-inventory/{\"m\":3,\"n\":3,\"pairs\":[[0,2]]}"},"fields":{"title":"Typed Relation Projections and Missing Sources","stem":"The explicitly typed relation R⊆Fin(3)×Fin(3) has ordered pairs=[[0, 2]]. 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. Give ([n,m], normalized pairs, source projection, target projection, pair count, missing source labels).","answer":{"kind":"expression","value":"([3, 3], [[0, 2]], [0], [2], 1, [1, 2])","explanation":"Project each coordinate of the complete pair set; missing sources are legal for a relation. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([3, 3], [[0, 2]], [0], [2], 1, [1, 2])."},"skills":["Typed Relation Projections and Missing Sources"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r03-relation-0010","batch_id":"batch-20261012-advance-20261017-r03-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3a6d8de421b4448d144643bf63a217d5382e654e95a1bbf953a0bcb78738ed40","family_id":"qf-density17-relation-full-typed-inventory","seed":202610173010,"semantic_template":"qf-density17-relation-full-typed-inventory/{\"m\":3,\"n\":3,\"pairs\":[[0,0],[0,2]]}"},"fields":{"title":"Typed Relation Projections and Missing Sources","stem":"The explicitly typed relation R⊆Fin(3)×Fin(3) has ordered pairs=[[0, 0], [0, 2]]. 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. Give ([n,m], normalized pairs, source projection, target projection, pair count, missing source labels).","answer":{"kind":"expression","value":"([3, 3], [[0, 0], [0, 2]], [0], [0, 2], 2, [1, 2])","explanation":"Project each coordinate of the complete pair set; missing sources are legal for a relation. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([3, 3], [[0, 0], [0, 2]], [0], [0, 2], 2, [1, 2])."},"skills":["Typed Relation Projections and Missing Sources"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r03-relation-0011","batch_id":"batch-20261012-advance-20261017-r03-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e19fddf838f3f72a59cd5529495c52c1df480f10d6d538ca837aaa272b3c4433","family_id":"qf-density17-relation-full-typed-inventory","seed":202610173011,"semantic_template":"qf-density17-relation-full-typed-inventory/{\"m\":3,\"n\":3,\"pairs\":[[0,1],[0,2]]}"},"fields":{"title":"Typed Relation Projections and Missing Sources","stem":"The explicitly typed relation R⊆Fin(3)×Fin(3) has ordered pairs=[[0, 1], [0, 2]]. 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. Give ([n,m], normalized pairs, source projection, target projection, pair count, missing source labels).","answer":{"kind":"expression","value":"([3, 3], [[0, 1], [0, 2]], [0], [1, 2], 2, [1, 2])","explanation":"Project each coordinate of the complete pair set; missing sources are legal for a relation. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([3, 3], [[0, 1], [0, 2]], [0], [1, 2], 2, [1, 2])."},"skills":["Typed Relation Projections and Missing Sources"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r03-relation-0012","batch_id":"batch-20261012-advance-20261017-r03-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7e494a3f25152845781dd59c67d77e1c0aa3e592e13220947d3def0d3f169e3b","family_id":"qf-density17-relation-full-typed-inventory","seed":202610173012,"semantic_template":"qf-density17-relation-full-typed-inventory/{\"m\":3,\"n\":3,\"pairs\":[[0,0],[0,1],[0,2]]}"},"fields":{"title":"Typed Relation Projections and Missing Sources","stem":"The explicitly typed relation R⊆Fin(3)×Fin(3) has ordered pairs=[[0, 0], [0, 1], [0, 2]]. 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. Give ([n,m], normalized pairs, source projection, target projection, pair count, missing source labels).","answer":{"kind":"expression","value":"([3, 3], [[0, 0], [0, 1], [0, 2]], [0], [0, 1, 2], 3, [1, 2])","explanation":"Project each coordinate of the complete pair set; missing sources are legal for a relation. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([3, 3], [[0, 0], [0, 1], [0, 2]], [0], [0, 1, 2], 3, [1, 2])."},"skills":["Typed Relation Projections and Missing Sources"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r03-relation-0013","batch_id":"batch-20261012-advance-20261017-r03-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1a317f3d8bb24fddba8c38850cf35a05b17c0abce3869fa59f1891cfd9c7a223","family_id":"qf-density17-relation-full-typed-inventory","seed":202610173013,"semantic_template":"qf-density17-relation-full-typed-inventory/{\"m\":3,\"n\":3,\"pairs\":[[1,0]]}"},"fields":{"title":"Typed Relation Projections and Missing Sources","stem":"The explicitly typed relation R⊆Fin(3)×Fin(3) has ordered pairs=[[1, 0]]. 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. Give ([n,m], normalized pairs, source projection, target projection, pair count, missing source labels).","answer":{"kind":"expression","value":"([3, 3], [[1, 0]], [1], [0], 1, [0, 2])","explanation":"Project each coordinate of the complete pair set; missing sources are legal for a relation. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([3, 3], [[1, 0]], [1], [0], 1, [0, 2])."},"skills":["Typed Relation Projections and Missing Sources"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r03-relation-0014","batch_id":"batch-20261012-advance-20261017-r03-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"eddf22f4c2bf98db107a951136ceda1d018a6734345bca59090273341f4ea4bd","family_id":"qf-density17-relation-full-typed-inventory","seed":202610173014,"semantic_template":"qf-density17-relation-full-typed-inventory/{\"m\":3,\"n\":3,\"pairs\":[[0,0],[1,0]]}"},"fields":{"title":"Typed Relation Projections and Missing Sources","stem":"The explicitly typed relation R⊆Fin(3)×Fin(3) has ordered pairs=[[0, 0], [1, 0]]. 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. Give ([n,m], normalized pairs, source projection, target projection, pair count, missing source labels).","answer":{"kind":"expression","value":"([3, 3], [[0, 0], [1, 0]], [0, 1], [0], 2, [2])","explanation":"Project each coordinate of the complete pair set; missing sources are legal for a relation. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([3, 3], [[0, 0], [1, 0]], [0, 1], [0], 2, [2])."},"skills":["Typed Relation Projections and Missing Sources"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r03-relation-0015","batch_id":"batch-20261012-advance-20261017-r03-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b8501aaff1cd45499e3b0f4d66c8f54f9701d9e9704dcd9a1a0862551f2ef8d4","family_id":"qf-density17-relation-full-typed-inventory","seed":202610173015,"semantic_template":"qf-density17-relation-full-typed-inventory/{\"m\":3,\"n\":3,\"pairs\":[[0,1],[1,0]]}"},"fields":{"title":"Typed Relation Projections and Missing Sources","stem":"The explicitly typed relation R⊆Fin(3)×Fin(3) has ordered pairs=[[0, 1], [1, 0]]. 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. Give ([n,m], normalized pairs, source projection, target projection, pair count, missing source labels).","answer":{"kind":"expression","value":"([3, 3], [[0, 1], [1, 0]], [0, 1], [0, 1], 2, [2])","explanation":"Project each coordinate of the complete pair set; missing sources are legal for a relation. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([3, 3], [[0, 1], [1, 0]], [0, 1], [0, 1], 2, [2])."},"skills":["Typed Relation Projections and Missing Sources"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r03-relation-0016","batch_id":"batch-20261012-advance-20261017-r03-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"447dce709be058635a9b880ed7d50197f7293d2305d51e84abed45d932b9ab82","family_id":"qf-density17-relation-full-typed-inventory","seed":202610173016,"semantic_template":"qf-density17-relation-full-typed-inventory/{\"m\":3,\"n\":3,\"pairs\":[[0,0],[0,1],[1,0]]}"},"fields":{"title":"Typed Relation Projections and Missing Sources","stem":"The explicitly typed relation R⊆Fin(3)×Fin(3) has ordered pairs=[[0, 0], [0, 1], [1, 0]]. 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. Give ([n,m], normalized pairs, source projection, target projection, pair count, missing source labels).","answer":{"kind":"expression","value":"([3, 3], [[0, 0], [0, 1], [1, 0]], [0, 1], [0, 1], 3, [2])","explanation":"Project each coordinate of the complete pair set; missing sources are legal for a relation. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([3, 3], [[0, 0], [0, 1], [1, 0]], [0, 1], [0, 1], 3, [2])."},"skills":["Typed Relation Projections and Missing Sources"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r03-relation-0017","batch_id":"batch-20261012-advance-20261017-r03-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d9147a74d66d5b663d6ac5e67a90484db332e1fc61ff0e3fa1b1364d51aa6e4d","family_id":"qf-density17-relation-full-typed-inventory","seed":202610173017,"semantic_template":"qf-density17-relation-full-typed-inventory/{\"m\":3,\"n\":3,\"pairs\":[[0,2],[1,0]]}"},"fields":{"title":"Typed Relation Projections and Missing Sources","stem":"The explicitly typed relation R⊆Fin(3)×Fin(3) has ordered pairs=[[0, 2], [1, 0]]. 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. Give ([n,m], normalized pairs, source projection, target projection, pair count, missing source labels).","answer":{"kind":"expression","value":"([3, 3], [[0, 2], [1, 0]], [0, 1], [0, 2], 2, [2])","explanation":"Project each coordinate of the complete pair set; missing sources are legal for a relation. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([3, 3], [[0, 2], [1, 0]], [0, 1], [0, 2], 2, [2])."},"skills":["Typed Relation Projections and Missing Sources"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r03-relation-0018","batch_id":"batch-20261012-advance-20261017-r03-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6bab930b0b7fb1aa32349a2c35b7e25f092ff55b8b8b3ef54251d1b674bfd08c","family_id":"qf-density17-relation-full-typed-inventory","seed":202610173018,"semantic_template":"qf-density17-relation-full-typed-inventory/{\"m\":3,\"n\":3,\"pairs\":[[0,0],[0,2],[1,0]]}"},"fields":{"title":"Typed Relation Projections and Missing Sources","stem":"The explicitly typed relation R⊆Fin(3)×Fin(3) has ordered pairs=[[0, 0], [0, 2], [1, 0]]. 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. Give ([n,m], normalized pairs, source projection, target projection, pair count, missing source labels).","answer":{"kind":"expression","value":"([3, 3], [[0, 0], [0, 2], [1, 0]], [0, 1], [0, 2], 3, [2])","explanation":"Project each coordinate of the complete pair set; missing sources are legal for a relation. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([3, 3], [[0, 0], [0, 2], [1, 0]], [0, 1], [0, 2], 3, [2])."},"skills":["Typed Relation Projections and Missing Sources"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r03-relation-0019","batch_id":"batch-20261012-advance-20261017-r03-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fee53d6eb2c4f2191e30b10a86da5f9f47f2f74bdbd7a9d5ddf9b93624f248a4","family_id":"qf-density17-relation-full-typed-inventory","seed":202610173019,"semantic_template":"qf-density17-relation-full-typed-inventory/{\"m\":3,\"n\":3,\"pairs\":[[0,1],[0,2],[1,0]]}"},"fields":{"title":"Typed Relation Projections and Missing Sources","stem":"The explicitly typed relation R⊆Fin(3)×Fin(3) has ordered pairs=[[0, 1], [0, 2], [1, 0]]. 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. Give ([n,m], normalized pairs, source projection, target projection, pair count, missing source labels).","answer":{"kind":"expression","value":"([3, 3], [[0, 1], [0, 2], [1, 0]], [0, 1], [0, 1, 2], 3, [2])","explanation":"Project each coordinate of the complete pair set; missing sources are legal for a relation. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([3, 3], [[0, 1], [0, 2], [1, 0]], [0, 1], [0, 1, 2], 3, [2])."},"skills":["Typed Relation Projections and Missing Sources"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r03-relation-0020","batch_id":"batch-20261012-advance-20261017-r03-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ca65d89039abf8b8fe2e10986b16119ac28f43f38155fd51132ddd9e900ff9e2","family_id":"qf-density17-relation-full-typed-inventory","seed":202610173020,"semantic_template":"qf-density17-relation-full-typed-inventory/{\"m\":3,\"n\":3,\"pairs\":[[0,0],[0,1],[0,2],[1,0]]}"},"fields":{"title":"Typed Relation Projections and Missing Sources","stem":"The explicitly typed relation R⊆Fin(3)×Fin(3) has ordered pairs=[[0, 0], [0, 1], [0, 2], [1, 0]]. 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. Give ([n,m], normalized pairs, source projection, target projection, pair count, missing source labels).","answer":{"kind":"expression","value":"([3, 3], [[0, 0], [0, 1], [0, 2], [1, 0]], [0, 1], [0, 1, 2], 4, [2])","explanation":"Project each coordinate of the complete pair set; missing sources are legal for a relation. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([3, 3], [[0, 0], [0, 1], [0, 2], [1, 0]], [0, 1], [0, 1, 2], 4, [2])."},"skills":["Typed Relation Projections and Missing Sources"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r03-relation-0021","batch_id":"batch-20261012-advance-20261017-r03-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2d312da1ce00c2bc33bd260a876bc2162117232764db576fbe2068fa6ee03e69","family_id":"qf-density17-relation-full-typed-inventory","seed":202610173021,"semantic_template":"qf-density17-relation-full-typed-inventory/{\"m\":3,\"n\":3,\"pairs\":[[1,1]]}"},"fields":{"title":"Typed Relation Projections and Missing Sources","stem":"The explicitly typed relation R⊆Fin(3)×Fin(3) has ordered pairs=[[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. Give ([n,m], normalized pairs, source projection, target projection, pair count, missing source labels).","answer":{"kind":"expression","value":"([3, 3], [[1, 1]], [1], [1], 1, [0, 2])","explanation":"Project each coordinate of the complete pair set; missing sources are legal for a relation. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([3, 3], [[1, 1]], [1], [1], 1, [0, 2])."},"skills":["Typed Relation Projections and Missing Sources"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r03-relation-0022","batch_id":"batch-20261012-advance-20261017-r03-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"215d929bbfff6aef1f022797dc030106188f72cb59f511c1b1217f1ba1b2b402","family_id":"qf-density17-relation-full-typed-inventory","seed":202610173022,"semantic_template":"qf-density17-relation-full-typed-inventory/{\"m\":3,\"n\":3,\"pairs\":[[0,0],[1,1]]}"},"fields":{"title":"Typed Relation Projections and Missing Sources","stem":"The explicitly typed relation R⊆Fin(3)×Fin(3) has ordered pairs=[[0, 0], [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. Give ([n,m], normalized pairs, source projection, target projection, pair count, missing source labels).","answer":{"kind":"expression","value":"([3, 3], [[0, 0], [1, 1]], [0, 1], [0, 1], 2, [2])","explanation":"Project each coordinate of the complete pair set; missing sources are legal for a relation. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([3, 3], [[0, 0], [1, 1]], [0, 1], [0, 1], 2, [2])."},"skills":["Typed Relation Projections and Missing Sources"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r03-relation-0023","batch_id":"batch-20261012-advance-20261017-r03-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0707354c1be01608ca58de5ecb412323a6caa5b86c901d1c3afef6274808fd64","family_id":"qf-density17-relation-full-typed-inventory","seed":202610173023,"semantic_template":"qf-density17-relation-full-typed-inventory/{\"m\":3,\"n\":3,\"pairs\":[[0,1],[1,1]]}"},"fields":{"title":"Typed Relation Projections and Missing Sources","stem":"The explicitly typed relation R⊆Fin(3)×Fin(3) has ordered pairs=[[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. Give ([n,m], normalized pairs, source projection, target projection, pair count, missing source labels).","answer":{"kind":"expression","value":"([3, 3], [[0, 1], [1, 1]], [0, 1], [1], 2, [2])","explanation":"Project each coordinate of the complete pair set; missing sources are legal for a relation. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([3, 3], [[0, 1], [1, 1]], [0, 1], [1], 2, [2])."},"skills":["Typed Relation Projections and Missing Sources"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r03-relation-0024","batch_id":"batch-20261012-advance-20261017-r03-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6f5943625933236a838292ff4c375a773eab95d4adf6d262bebd29e26aac606e","family_id":"qf-density17-relation-full-typed-inventory","seed":202610173024,"semantic_template":"qf-density17-relation-full-typed-inventory/{\"m\":3,\"n\":3,\"pairs\":[[0,0],[0,1],[1,1]]}"},"fields":{"title":"Typed Relation Projections and Missing Sources","stem":"The explicitly typed relation R⊆Fin(3)×Fin(3) 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. Give ([n,m], normalized pairs, source projection, target projection, pair count, missing source labels).","answer":{"kind":"expression","value":"([3, 3], [[0, 0], [0, 1], [1, 1]], [0, 1], [0, 1], 3, [2])","explanation":"Project each coordinate of the complete pair set; missing sources are legal for a relation. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([3, 3], [[0, 0], [0, 1], [1, 1]], [0, 1], [0, 1], 3, [2])."},"skills":["Typed Relation Projections and Missing Sources"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r03-relation-0025","batch_id":"batch-20261012-advance-20261017-r03-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f8735e1292b2b05d96fa5d70c15710d5ea19e160fc2decfa5eb6198f4ccca857","family_id":"qf-density17-relation-full-typed-inventory","seed":202610173025,"semantic_template":"qf-density17-relation-full-typed-inventory/{\"m\":3,\"n\":3,\"pairs\":[[0,2],[1,1]]}"},"fields":{"title":"Typed Relation Projections and Missing Sources","stem":"The explicitly typed relation R⊆Fin(3)×Fin(3) has ordered pairs=[[0, 2], [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. Give ([n,m], normalized pairs, source projection, target projection, pair count, missing source labels).","answer":{"kind":"expression","value":"([3, 3], [[0, 2], [1, 1]], [0, 1], [1, 2], 2, [2])","explanation":"Project each coordinate of the complete pair set; missing sources are legal for a relation. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([3, 3], [[0, 2], [1, 1]], [0, 1], [1, 2], 2, [2])."},"skills":["Typed Relation Projections and Missing Sources"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r03-relation-0026","batch_id":"batch-20261012-advance-20261017-r03-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"61ebe70b6b2bc60658ceca22a2c9dbccc0307f1dfcf75db1f491609f4a98cc4b","family_id":"qf-density17-relation-subset-image-fibers","seed":202610173026,"semantic_template":"qf-density17-relation-subset-image-fibers/{\"m\":0,\"n\":0,\"pairs\":[],\"subset\":[]}"},"fields":{"title":"Restricted Relation Image and All Target Fibers","stem":"The explicitly typed relation R⊆Fin(0)×Fin(0) has ordered pairs=[]. 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 []. 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":"([], [], [], [])","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 ([], [], [], [])."},"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r03-relation-0027","batch_id":"batch-20261012-advance-20261017-r03-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"24ebed2481e446346dd5b8762db8a7d640bcb6f71a38d1475f85df330245c404","family_id":"qf-density17-relation-subset-image-fibers","seed":202610173027,"semantic_template":"qf-density17-relation-subset-image-fibers/{\"m\":2,\"n\":2,\"pairs\":[],\"subset\":[1]}"},"fields":{"title":"Restricted Relation Image and All Target Fibers","stem":"The explicitly typed relation R⊆Fin(2)×Fin(2) has ordered pairs=[]. 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 [1]. 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":"([], [1], [], [[], []])","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 ([], [1], [], [[], []])."},"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r03-relation-0028","batch_id":"batch-20261012-advance-20261017-r03-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"207b0d5a8ff8cf3d4000f4b59800bbd401e164dd51186f5e547e1219d7c13540","family_id":"qf-density17-relation-subset-image-fibers","seed":202610173028,"semantic_template":"qf-density17-relation-subset-image-fibers/{\"m\":2,\"n\":2,\"pairs\":[[0,0],[0,1],[1,1]],\"subset\":[]}"},"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 []. 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]], [], [], [[], []])","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]], [], [], [[], []])."},"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."}}
{"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r03-relation-0030","batch_id":"batch-20261012-advance-20261017-r03-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ce9777daf0c423d4965982456ee8a03272f0dd8ec438d80c2d1a494c27a243f6","family_id":"qf-density17-relation-subset-image-fibers","seed":202610173030,"semantic_template":"qf-density17-relation-subset-image-fibers/{\"m\":2,\"n\":2,\"pairs\":[[0,0],[1,1]],\"subset\":[1]}"},"fields":{"title":"Restricted Relation Image and All Target Fibers","stem":"The explicitly typed relation R⊆Fin(2)×Fin(2) has ordered pairs=[[0, 0], [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 [1]. 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], [1, 1]], [1], [1], [[], [1]])","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], [1, 1]], [1], [1], [[], [1]])."},"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r03-relation-0031","batch_id":"batch-20261012-advance-20261017-r03-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9d3b638ee0bb8fd5f69574742e101c44b0e8e18dabd1590c9eba569494da3dbf","family_id":"qf-density17-relation-subset-image-fibers","seed":202610173031,"semantic_template":"qf-density17-relation-subset-image-fibers/{\"m\":3,\"n\":3,\"pairs\":[[0,0]],\"subset\":[0,2]}"},"fields":{"title":"Restricted Relation Image and All Target Fibers","stem":"The explicitly typed relation R⊆Fin(3)×Fin(3) has ordered pairs=[[0, 0]]. 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 [2, 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, 2], [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, 2], [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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r03-relation-0032","batch_id":"batch-20261012-advance-20261017-r03-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"78862f3a6ec796b2d96431ef83c8ffe5ce7b3faf0cde413770a2f0dd5dc3ab14","family_id":"qf-density17-relation-subset-image-fibers","seed":202610173032,"semantic_template":"qf-density17-relation-subset-image-fibers/{\"m\":3,\"n\":3,\"pairs\":[[0,1]],\"subset\":[0,2]}"},"fields":{"title":"Restricted Relation Image and All Target Fibers","stem":"The explicitly typed relation R⊆Fin(3)×Fin(3) has ordered pairs=[[0, 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 [2, 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, 1]], [0, 2], [1], [[], [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, 1]], [0, 2], [1], [[], [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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r03-relation-0033","batch_id":"batch-20261012-advance-20261017-r03-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1946436ed7edfd1e88152d0652f8c723ce4902b515fb9e5d35d5395a586af64d","family_id":"qf-density17-relation-subset-image-fibers","seed":202610173033,"semantic_template":"qf-density17-relation-subset-image-fibers/{\"m\":3,\"n\":3,\"pairs\":[[0,0],[0,1]],\"subset\":[0,2]}"},"fields":{"title":"Restricted Relation Image and All Target Fibers","stem":"The explicitly typed relation R⊆Fin(3)×Fin(3) has ordered pairs=[[0, 0], [0, 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 [2, 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]], [0, 2], [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]], [0, 2], [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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r03-relation-0034","batch_id":"batch-20261012-advance-20261017-r03-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2f696d7916cb15941db14291ff00356ba2978dde5da1346d1fdebe52c2274a1e","family_id":"qf-density17-relation-subset-image-fibers","seed":202610173034,"semantic_template":"qf-density17-relation-subset-image-fibers/{\"m\":3,\"n\":3,\"pairs\":[[0,2]],\"subset\":[0,2]}"},"fields":{"title":"Restricted Relation Image and All Target Fibers","stem":"The explicitly typed relation R⊆Fin(3)×Fin(3) has ordered pairs=[[0, 2]]. 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 [2, 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, 2]], [0, 2], [2], [[], [], [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, 2]], [0, 2], [2], [[], [], [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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r03-relation-0035","batch_id":"batch-20261012-advance-20261017-r03-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9fa05a0dea7a0791d79a10810964d75ffcaa1ac75261f6aff8c787f1aede7177","family_id":"qf-density17-relation-subset-image-fibers","seed":202610173035,"semantic_template":"qf-density17-relation-subset-image-fibers/{\"m\":3,\"n\":3,\"pairs\":[[0,0],[0,2]],\"subset\":[0,2]}"},"fields":{"title":"Restricted Relation Image and All Target Fibers","stem":"The explicitly typed relation R⊆Fin(3)×Fin(3) has ordered pairs=[[0, 0], [0, 2]]. 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 [2, 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, 2]], [0, 2], [0, 2], [[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, 2]], [0, 2], [0, 2], [[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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r03-relation-0036","batch_id":"batch-20261012-advance-20261017-r03-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5f03e02b0ad707c5a424aa9ec64e0776e782480695aa599fbdf46d121168f91f","family_id":"qf-density17-relation-subset-image-fibers","seed":202610173036,"semantic_template":"qf-density17-relation-subset-image-fibers/{\"m\":3,\"n\":3,\"pairs\":[[0,1],[0,2]],\"subset\":[0,2]}"},"fields":{"title":"Restricted Relation Image and All Target Fibers","stem":"The explicitly typed relation R⊆Fin(3)×Fin(3) has ordered pairs=[[0, 1], [0, 2]]. 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 [2, 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, 1], [0, 2]], [0, 2], [1, 2], [[], [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, 1], [0, 2]], [0, 2], [1, 2], [[], [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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r03-relation-0037","batch_id":"batch-20261012-advance-20261017-r03-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e08a8693d8b7709dacfa0da9b5de7981a07c3cd296fe55aae040ba0f768fb292","family_id":"qf-density17-relation-subset-image-fibers","seed":202610173037,"semantic_template":"qf-density17-relation-subset-image-fibers/{\"m\":3,\"n\":3,\"pairs\":[[0,0],[0,1],[0,2]],\"subset\":[0,2]}"},"fields":{"title":"Restricted Relation Image and All Target Fibers","stem":"The explicitly typed relation R⊆Fin(3)×Fin(3) has ordered pairs=[[0, 0], [0, 1], [0, 2]]. 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 [2, 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], [0, 2]], [0, 2], [0, 1, 2], [[0], [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], [0, 2]], [0, 2], [0, 1, 2], [[0], [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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r03-relation-0038","batch_id":"batch-20261012-advance-20261017-r03-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c17ca92d1770c618fcb2f12638505364359917b8d992f6779073ff44021c0e74","family_id":"qf-density17-relation-subset-image-fibers","seed":202610173038,"semantic_template":"qf-density17-relation-subset-image-fibers/{\"m\":3,\"n\":3,\"pairs\":[[1,0]],\"subset\":[0,2]}"},"fields":{"title":"Restricted Relation Image and All Target Fibers","stem":"The explicitly typed relation R⊆Fin(3)×Fin(3) has ordered pairs=[[1, 0]]. 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 [2, 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":"([[1, 0]], [0, 2], [], [[], [], []])","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 ([[1, 0]], [0, 2], [], [[], [], []])."},"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r03-relation-0039","batch_id":"batch-20261012-advance-20261017-r03-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e780cb3327b063587918e389eaaef9ba21271c9d3e393fd84078cd7627f69fde","family_id":"qf-density17-relation-subset-image-fibers","seed":202610173039,"semantic_template":"qf-density17-relation-subset-image-fibers/{\"m\":3,\"n\":3,\"pairs\":[[0,0],[1,0]],\"subset\":[0,2]}"},"fields":{"title":"Restricted Relation Image and All Target Fibers","stem":"The explicitly typed relation R⊆Fin(3)×Fin(3) has ordered pairs=[[0, 0], [1, 0]]. 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 [2, 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], [1, 0]], [0, 2], [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], [1, 0]], [0, 2], [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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r03-relation-0040","batch_id":"batch-20261012-advance-20261017-r03-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"310f243855f1c895344826373a6af27457836ac48d5a1ea3cc59ec78ecc10ee3","family_id":"qf-density17-relation-subset-image-fibers","seed":202610173040,"semantic_template":"qf-density17-relation-subset-image-fibers/{\"m\":3,\"n\":3,\"pairs\":[[0,1],[1,0]],\"subset\":[0,2]}"},"fields":{"title":"Restricted Relation Image and All Target Fibers","stem":"The explicitly typed relation R⊆Fin(3)×Fin(3) has ordered pairs=[[0, 1], [1, 0]]. 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 [2, 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, 1], [1, 0]], [0, 2], [1], [[], [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, 1], [1, 0]], [0, 2], [1], [[], [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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r03-relation-0041","batch_id":"batch-20261012-advance-20261017-r03-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a62bcaae762a4c8272fb12551c236c3d10dd35ce7d81d0bbea44d9a93af76e43","family_id":"qf-density17-relation-subset-image-fibers","seed":202610173041,"semantic_template":"qf-density17-relation-subset-image-fibers/{\"m\":3,\"n\":3,\"pairs\":[[0,0],[0,1],[1,0]],\"subset\":[0,2]}"},"fields":{"title":"Restricted Relation Image and All Target Fibers","stem":"The explicitly typed relation R⊆Fin(3)×Fin(3) has ordered pairs=[[0, 0], [0, 1], [1, 0]]. 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 [2, 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, 0]], [0, 2], [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, 0]], [0, 2], [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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r03-relation-0042","batch_id":"batch-20261012-advance-20261017-r03-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"51da334c65d556770d135a7636b305831c8b66c4072687942458efafd1841159","family_id":"qf-density17-relation-subset-image-fibers","seed":202610173042,"semantic_template":"qf-density17-relation-subset-image-fibers/{\"m\":3,\"n\":3,\"pairs\":[[0,2],[1,0]],\"subset\":[0,2]}"},"fields":{"title":"Restricted Relation Image and All Target Fibers","stem":"The explicitly typed relation R⊆Fin(3)×Fin(3) has ordered pairs=[[0, 2], [1, 0]]. 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 [2, 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, 2], [1, 0]], [0, 2], [2], [[], [], [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, 2], [1, 0]], [0, 2], [2], [[], [], [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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r03-relation-0043","batch_id":"batch-20261012-advance-20261017-r03-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"893a6760e7cd3fc00a0635e2ecddc989ca837913d3dc1c2461e427747aab3e89","family_id":"qf-density17-relation-subset-image-fibers","seed":202610173043,"semantic_template":"qf-density17-relation-subset-image-fibers/{\"m\":3,\"n\":3,\"pairs\":[[0,0],[0,2],[1,0]],\"subset\":[0,2]}"},"fields":{"title":"Restricted Relation Image and All Target Fibers","stem":"The explicitly typed relation R⊆Fin(3)×Fin(3) has ordered pairs=[[0, 0], [0, 2], [1, 0]]. 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 [2, 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, 2], [1, 0]], [0, 2], [0, 2], [[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, 2], [1, 0]], [0, 2], [0, 2], [[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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r03-relation-0044","batch_id":"batch-20261012-advance-20261017-r03-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"00c204442c82fc447c6dbfefdcd8ca0cd5cb3cab55823db16d61cdd64314a9e8","family_id":"qf-density17-relation-subset-image-fibers","seed":202610173044,"semantic_template":"qf-density17-relation-subset-image-fibers/{\"m\":3,\"n\":3,\"pairs\":[[0,1],[0,2],[1,0]],\"subset\":[0,2]}"},"fields":{"title":"Restricted Relation Image and All Target Fibers","stem":"The explicitly typed relation R⊆Fin(3)×Fin(3) has ordered pairs=[[0, 1], [0, 2], [1, 0]]. 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 [2, 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, 1], [0, 2], [1, 0]], [0, 2], [1, 2], [[], [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, 1], [0, 2], [1, 0]], [0, 2], [1, 2], [[], [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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r03-relation-0045","batch_id":"batch-20261012-advance-20261017-r03-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ced1542e77bd4fafc0296ac3dda3a4308231ea59666617e67e2f20592567d868","family_id":"qf-density17-relation-subset-image-fibers","seed":202610173045,"semantic_template":"qf-density17-relation-subset-image-fibers/{\"m\":3,\"n\":3,\"pairs\":[[0,0],[0,1],[0,2],[1,0]],\"subset\":[0,2]}"},"fields":{"title":"Restricted Relation Image and All Target Fibers","stem":"The explicitly typed relation R⊆Fin(3)×Fin(3) has ordered pairs=[[0, 0], [0, 1], [0, 2], [1, 0]]. 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 [2, 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], [0, 2], [1, 0]], [0, 2], [0, 1, 2], [[0], [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], [0, 2], [1, 0]], [0, 2], [0, 1, 2], [[0], [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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r03-relation-0046","batch_id":"batch-20261012-advance-20261017-r03-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"decaa4e43134447286e65aed0f8697aec2bb1ed55696da5e9cf95faa1fa2c0ca","family_id":"qf-density17-relation-subset-image-fibers","seed":202610173046,"semantic_template":"qf-density17-relation-subset-image-fibers/{\"m\":3,\"n\":3,\"pairs\":[[1,1]],\"subset\":[0,2]}"},"fields":{"title":"Restricted Relation Image and All Target Fibers","stem":"The explicitly typed relation R⊆Fin(3)×Fin(3) has ordered pairs=[[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 [2, 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":"([[1, 1]], [0, 2], [], [[], [], []])","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 ([[1, 1]], [0, 2], [], [[], [], []])."},"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r03-relation-0047","batch_id":"batch-20261012-advance-20261017-r03-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"54fd04dae8ceb344cbb341f37750639611da39196b95e9e0567c57d985ae580a","family_id":"qf-density17-relation-subset-image-fibers","seed":202610173047,"semantic_template":"qf-density17-relation-subset-image-fibers/{\"m\":3,\"n\":3,\"pairs\":[[0,0],[1,1]],\"subset\":[0,2]}"},"fields":{"title":"Restricted Relation Image and All Target Fibers","stem":"The explicitly typed relation R⊆Fin(3)×Fin(3) has ordered pairs=[[0, 0], [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 [2, 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], [1, 1]], [0, 2], [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], [1, 1]], [0, 2], [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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r03-relation-0048","batch_id":"batch-20261012-advance-20261017-r03-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c182544be8ede9fe96cf0d356321d55d987d292b2b7731a058fb08f97dc63a8a","family_id":"qf-density17-relation-subset-image-fibers","seed":202610173048,"semantic_template":"qf-density17-relation-subset-image-fibers/{\"m\":3,\"n\":3,\"pairs\":[[0,1],[1,1]],\"subset\":[0,2]}"},"fields":{"title":"Restricted Relation Image and All Target Fibers","stem":"The explicitly typed relation R⊆Fin(3)×Fin(3) has ordered pairs=[[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 [2, 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, 1], [1, 1]], [0, 2], [1], [[], [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, 1], [1, 1]], [0, 2], [1], [[], [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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r03-relation-0049","batch_id":"batch-20261012-advance-20261017-r03-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ff3172ee5cf3ea48c755f6c795c946f174adbcb169c278a2141c7688aa5c4254","family_id":"qf-density17-relation-subset-image-fibers","seed":202610173049,"semantic_template":"qf-density17-relation-subset-image-fibers/{\"m\":3,\"n\":3,\"pairs\":[[0,0],[0,1],[1,1]],\"subset\":[0,2]}"},"fields":{"title":"Restricted Relation Image and All Target Fibers","stem":"The explicitly typed relation R⊆Fin(3)×Fin(3) 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 [2, 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, 2], [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, 2], [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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r03-relation-0050","batch_id":"batch-20261012-advance-20261017-r03-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"63de732fca28d241764d74a42c083a1811dd7042268cdab0ce3622ccacaf1787","family_id":"qf-density17-relation-subset-image-fibers","seed":202610173050,"semantic_template":"qf-density17-relation-subset-image-fibers/{\"m\":3,\"n\":3,\"pairs\":[[0,2],[1,1]],\"subset\":[0,2]}"},"fields":{"title":"Restricted Relation Image and All Target Fibers","stem":"The explicitly typed relation R⊆Fin(3)×Fin(3) has ordered pairs=[[0, 2], [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 [2, 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, 2], [1, 1]], [0, 2], [2], [[], [], [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, 2], [1, 1]], [0, 2], [2], [[], [], [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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r03-relation-0051","batch_id":"batch-20261012-advance-20261017-r03-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5297bbfa8bbe55d46adb8031deb9a37a80898e3157fe466160edf09cb3dfa346","family_id":"qf-density17-relation-inverse-double","seed":202610173051,"semantic_template":"qf-density17-relation-inverse-double/{\"m\":0,\"n\":0,\"pairs\":[]}"},"fields":{"title":"Typed Converse Relation and Double Converse","stem":"The explicitly typed relation R⊆Fin(0)×Fin(0) has ordered pairs=[]. 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. Give ([n,m], swapped type sizes [m,n], original normalized pairs, normalized converse pairs, double-converse pairs, does the double converse equal the original?).","answer":{"kind":"expression","value":"([0, 0], [0, 0], [], [], [], Yes)","explanation":"Reverse both pair coordinates and the source/target types, then reverse again; do not claim that either relation is a function. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 0], [0, 0], [], [], [], Yes)."},"skills":["Typed Converse Relation and Double Converse"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r03-relation-0052","batch_id":"batch-20261012-advance-20261017-r03-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d7cdc3aec08943bd4d46240277b6989f22a498d5bfa89ee3ea73d1328acc802d","family_id":"qf-density17-relation-inverse-double","seed":202610173052,"semantic_template":"qf-density17-relation-inverse-double/{\"m\":0,\"n\":2,\"pairs\":[]}"},"fields":{"title":"Typed Converse Relation and Double Converse","stem":"The explicitly typed relation R⊆Fin(2)×Fin(0) has ordered pairs=[]. 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. Give ([n,m], swapped type sizes [m,n], original normalized pairs, normalized converse pairs, double-converse pairs, does the double converse equal the original?).","answer":{"kind":"expression","value":"([2, 0], [0, 2], [], [], [], Yes)","explanation":"Reverse both pair coordinates and the source/target types, then reverse again; do not claim that either relation is a function. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([2, 0], [0, 2], [], [], [], Yes)."},"skills":["Typed Converse Relation and Double Converse"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r03-relation-0053","batch_id":"batch-20261012-advance-20261017-r03-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f92e3d8c38fc8be320f8213ecbdf46dab789f7ec47f12fa0606269e4be2cbc7d","family_id":"qf-density17-relation-inverse-double","seed":202610173053,"semantic_template":"qf-density17-relation-inverse-double/{\"m\":2,\"n\":0,\"pairs\":[]}"},"fields":{"title":"Typed Converse Relation and Double Converse","stem":"The explicitly typed relation R⊆Fin(0)×Fin(2) has ordered pairs=[]. 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. Give ([n,m], swapped type sizes [m,n], original normalized pairs, normalized converse pairs, double-converse pairs, does the double converse equal the original?).","answer":{"kind":"expression","value":"([0, 2], [2, 0], [], [], [], Yes)","explanation":"Reverse both pair coordinates and the source/target types, then reverse again; do not claim that either relation is a function. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 2], [2, 0], [], [], [], Yes)."},"skills":["Typed Converse Relation and Double Converse"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r03-relation-0054","batch_id":"batch-20261012-advance-20261017-r03-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a2d0e76a659e898f094dbaf9b21639579cfc83a84ce13c4eca834008ee5a6d36","family_id":"qf-density17-relation-inverse-double","seed":202610173054,"semantic_template":"qf-density17-relation-inverse-double/{\"m\":2,\"n\":2,\"pairs\":[[0,0],[1,1]]}"},"fields":{"title":"Typed Converse Relation and Double Converse","stem":"The explicitly typed relation R⊆Fin(2)×Fin(2) has ordered pairs=[[0, 0], [1, 1], [0, 0]]. 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. Give ([n,m], swapped type sizes [m,n], original normalized pairs, normalized converse pairs, double-converse pairs, does the double converse equal the original?).","answer":{"kind":"expression","value":"([2, 2], [2, 2], [[0, 0], [1, 1]], [[0, 0], [1, 1]], [[0, 0], [1, 1]], Yes)","explanation":"Reverse both pair coordinates and the source/target types, then reverse again; do not claim that either relation is a function. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([2, 2], [2, 2], [[0, 0], [1, 1]], [[0, 0], [1, 1]], [[0, 0], [1, 1]], Yes)."},"skills":["Typed Converse Relation and Double Converse"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r03-relation-0055","batch_id":"batch-20261012-advance-20261017-r03-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3a44184c2ed3a131d537137677625d0c46294154b0bce5d57d3afb798a1b187e","family_id":"qf-density17-relation-inverse-double","seed":202610173055,"semantic_template":"qf-density17-relation-inverse-double/{\"m\":2,\"n\":2,\"pairs\":[[0,0],[0,1]]}"},"fields":{"title":"Typed Converse Relation and Double Converse","stem":"The explicitly typed relation R⊆Fin(2)×Fin(2) has ordered pairs=[[0, 0], [0, 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. Give ([n,m], swapped type sizes [m,n], original normalized pairs, normalized converse pairs, double-converse pairs, does the double converse equal the original?).","answer":{"kind":"expression","value":"([2, 2], [2, 2], [[0, 0], [0, 1]], [[0, 0], [1, 0]], [[0, 0], [0, 1]], Yes)","explanation":"Reverse both pair coordinates and the source/target types, then reverse again; do not claim that either relation is a function. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([2, 2], [2, 2], [[0, 0], [0, 1]], [[0, 0], [1, 0]], [[0, 0], [0, 1]], Yes)."},"skills":["Typed Converse Relation and Double Converse"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r03-relation-0056","batch_id":"batch-20261012-advance-20261017-r03-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e61e46587b1a7916d9e52f98d9722c9642b0119c5fb08c60445b352352bc291a","family_id":"qf-density17-relation-inverse-double","seed":202610173056,"semantic_template":"qf-density17-relation-inverse-double/{\"m\":3,\"n\":3,\"pairs\":[[0,0]]}"},"fields":{"title":"Typed Converse Relation and Double Converse","stem":"The explicitly typed relation R⊆Fin(3)×Fin(3) has ordered pairs=[[0, 0]]. 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. Give ([n,m], swapped type sizes [m,n], original normalized pairs, normalized converse pairs, double-converse pairs, does the double converse equal the original?).","answer":{"kind":"expression","value":"([3, 3], [3, 3], [[0, 0]], [[0, 0]], [[0, 0]], Yes)","explanation":"Reverse both pair coordinates and the source/target types, then reverse again; do not claim that either relation is a function. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([3, 3], [3, 3], [[0, 0]], [[0, 0]], [[0, 0]], Yes)."},"skills":["Typed Converse Relation and Double Converse"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r03-relation-0057","batch_id":"batch-20261012-advance-20261017-r03-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1870a17fa600e2254d3c772b775f9867f688501bbc828bcfbf7f1da1b28fc876","family_id":"qf-density17-relation-inverse-double","seed":202610173057,"semantic_template":"qf-density17-relation-inverse-double/{\"m\":3,\"n\":3,\"pairs\":[[0,1]]}"},"fields":{"title":"Typed Converse Relation and Double Converse","stem":"The explicitly typed relation R⊆Fin(3)×Fin(3) has ordered pairs=[[0, 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. Give ([n,m], swapped type sizes [m,n], original normalized pairs, normalized converse pairs, double-converse pairs, does the double converse equal the original?).","answer":{"kind":"expression","value":"([3, 3], [3, 3], [[0, 1]], [[1, 0]], [[0, 1]], Yes)","explanation":"Reverse both pair coordinates and the source/target types, then reverse again; do not claim that either relation is a function. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([3, 3], [3, 3], [[0, 1]], [[1, 0]], [[0, 1]], Yes)."},"skills":["Typed Converse Relation and Double Converse"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r03-relation-0058","batch_id":"batch-20261012-advance-20261017-r03-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e37b794a8f80a01bbf445b568e99be3374441e746640ad15d0b8c6b0a2e09824","family_id":"qf-density17-relation-inverse-double","seed":202610173058,"semantic_template":"qf-density17-relation-inverse-double/{\"m\":3,\"n\":3,\"pairs\":[[0,0],[0,1]]}"},"fields":{"title":"Typed Converse Relation and Double Converse","stem":"The explicitly typed relation R⊆Fin(3)×Fin(3) has ordered pairs=[[0, 0], [0, 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. Give ([n,m], swapped type sizes [m,n], original normalized pairs, normalized converse pairs, double-converse pairs, does the double converse equal the original?).","answer":{"kind":"expression","value":"([3, 3], [3, 3], [[0, 0], [0, 1]], [[0, 0], [1, 0]], [[0, 0], [0, 1]], Yes)","explanation":"Reverse both pair coordinates and the source/target types, then reverse again; do not claim that either relation is a function. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([3, 3], [3, 3], [[0, 0], [0, 1]], [[0, 0], [1, 0]], [[0, 0], [0, 1]], Yes)."},"skills":["Typed Converse Relation and Double Converse"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r03-relation-0059","batch_id":"batch-20261012-advance-20261017-r03-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8acf061bbc182e17233f5c8ac4cbbb229653302f8538cba69daaf078f9b5feeb","family_id":"qf-density17-relation-inverse-double","seed":202610173059,"semantic_template":"qf-density17-relation-inverse-double/{\"m\":3,\"n\":3,\"pairs\":[[0,2]]}"},"fields":{"title":"Typed Converse Relation and Double Converse","stem":"The explicitly typed relation R⊆Fin(3)×Fin(3) has ordered pairs=[[0, 2]]. 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. Give ([n,m], swapped type sizes [m,n], original normalized pairs, normalized converse pairs, double-converse pairs, does the double converse equal the original?).","answer":{"kind":"expression","value":"([3, 3], [3, 3], [[0, 2]], [[2, 0]], [[0, 2]], Yes)","explanation":"Reverse both pair coordinates and the source/target types, then reverse again; do not claim that either relation is a function. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([3, 3], [3, 3], [[0, 2]], [[2, 0]], [[0, 2]], Yes)."},"skills":["Typed Converse Relation and Double Converse"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r03-relation-0060","batch_id":"batch-20261012-advance-20261017-r03-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6aa3a3912a91fdb3d61ec07d6a5dce42d3da9ef9edcf0a4e7e75be53d94e3026","family_id":"qf-density17-relation-inverse-double","seed":202610173060,"semantic_template":"qf-density17-relation-inverse-double/{\"m\":3,\"n\":3,\"pairs\":[[0,0],[0,2]]}"},"fields":{"title":"Typed Converse Relation and Double Converse","stem":"The explicitly typed relation R⊆Fin(3)×Fin(3) has ordered pairs=[[0, 0], [0, 2]]. 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. Give ([n,m], swapped type sizes [m,n], original normalized pairs, normalized converse pairs, double-converse pairs, does the double converse equal the original?).","answer":{"kind":"expression","value":"([3, 3], [3, 3], [[0, 0], [0, 2]], [[0, 0], [2, 0]], [[0, 0], [0, 2]], Yes)","explanation":"Reverse both pair coordinates and the source/target types, then reverse again; do not claim that either relation is a function. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([3, 3], [3, 3], [[0, 0], [0, 2]], [[0, 0], [2, 0]], [[0, 0], [0, 2]], Yes)."},"skills":["Typed Converse Relation and Double Converse"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r03-relation-0061","batch_id":"batch-20261012-advance-20261017-r03-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"90cc2e9a36e439b1f542ee1d48e2524aa3b69701c544685c464396342d5ec446","family_id":"qf-density17-relation-inverse-double","seed":202610173061,"semantic_template":"qf-density17-relation-inverse-double/{\"m\":3,\"n\":3,\"pairs\":[[0,1],[0,2]]}"},"fields":{"title":"Typed Converse Relation and Double Converse","stem":"The explicitly typed relation R⊆Fin(3)×Fin(3) has ordered pairs=[[0, 1], [0, 2]]. 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. Give ([n,m], swapped type sizes [m,n], original normalized pairs, normalized converse pairs, double-converse pairs, does the double converse equal the original?).","answer":{"kind":"expression","value":"([3, 3], [3, 3], [[0, 1], [0, 2]], [[1, 0], [2, 0]], [[0, 1], [0, 2]], Yes)","explanation":"Reverse both pair coordinates and the source/target types, then reverse again; do not claim that either relation is a function. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([3, 3], [3, 3], [[0, 1], [0, 2]], [[1, 0], [2, 0]], [[0, 1], [0, 2]], Yes)."},"skills":["Typed Converse Relation and Double Converse"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r03-relation-0062","batch_id":"batch-20261012-advance-20261017-r03-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9219649b4292425447a05b8e0a0e9b9acab85cae62a2fd238312dd459172a6ee","family_id":"qf-density17-relation-inverse-double","seed":202610173062,"semantic_template":"qf-density17-relation-inverse-double/{\"m\":3,\"n\":3,\"pairs\":[[0,0],[0,1],[0,2]]}"},"fields":{"title":"Typed Converse Relation and Double Converse","stem":"The explicitly typed relation R⊆Fin(3)×Fin(3) has ordered pairs=[[0, 0], [0, 1], [0, 2]]. 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. Give ([n,m], swapped type sizes [m,n], original normalized pairs, normalized converse pairs, double-converse pairs, does the double converse equal the original?).","answer":{"kind":"expression","value":"([3, 3], [3, 3], [[0, 0], [0, 1], [0, 2]], [[0, 0], [1, 0], [2, 0]], [[0, 0], [0, 1], [0, 2]], Yes)","explanation":"Reverse both pair coordinates and the source/target types, then reverse again; do not claim that either relation is a function. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([3, 3], [3, 3], [[0, 0], [0, 1], [0, 2]], [[0, 0], [1, 0], [2, 0]], [[0, 0], [0, 1], [0, 2]], Yes)."},"skills":["Typed Converse Relation and Double Converse"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r03-relation-0063","batch_id":"batch-20261012-advance-20261017-r03-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f25d053a2fd3abcedee65c28881ceaa99303eb62c2ba2bfbd49b7028f5ea4b34","family_id":"qf-density17-relation-inverse-double","seed":202610173063,"semantic_template":"qf-density17-relation-inverse-double/{\"m\":3,\"n\":3,\"pairs\":[[1,0]]}"},"fields":{"title":"Typed Converse Relation and Double Converse","stem":"The explicitly typed relation R⊆Fin(3)×Fin(3) has ordered pairs=[[1, 0]]. 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. Give ([n,m], swapped type sizes [m,n], original normalized pairs, normalized converse pairs, double-converse pairs, does the double converse equal the original?).","answer":{"kind":"expression","value":"([3, 3], [3, 3], [[1, 0]], [[0, 1]], [[1, 0]], Yes)","explanation":"Reverse both pair coordinates and the source/target types, then reverse again; do not claim that either relation is a function. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([3, 3], [3, 3], [[1, 0]], [[0, 1]], [[1, 0]], Yes)."},"skills":["Typed Converse Relation and Double Converse"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r03-relation-0064","batch_id":"batch-20261012-advance-20261017-r03-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5dc607869bea68a0388aab0008967769756c965131d42192dd8ad482924469f1","family_id":"qf-density17-relation-inverse-double","seed":202610173064,"semantic_template":"qf-density17-relation-inverse-double/{\"m\":3,\"n\":3,\"pairs\":[[0,0],[1,0]]}"},"fields":{"title":"Typed Converse Relation and Double Converse","stem":"The explicitly typed relation R⊆Fin(3)×Fin(3) has ordered pairs=[[0, 0], [1, 0]]. 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. Give ([n,m], swapped type sizes [m,n], original normalized pairs, normalized converse pairs, double-converse pairs, does the double converse equal the original?).","answer":{"kind":"expression","value":"([3, 3], [3, 3], [[0, 0], [1, 0]], [[0, 0], [0, 1]], [[0, 0], [1, 0]], Yes)","explanation":"Reverse both pair coordinates and the source/target types, then reverse again; do not claim that either relation is a function. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([3, 3], [3, 3], [[0, 0], [1, 0]], [[0, 0], [0, 1]], [[0, 0], [1, 0]], Yes)."},"skills":["Typed Converse Relation and Double Converse"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r03-relation-0065","batch_id":"batch-20261012-advance-20261017-r03-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d2818ecac7684e8d0cb12306b05b79e254c0eb7dcfdbd003610e7724555fe1eb","family_id":"qf-density17-relation-inverse-double","seed":202610173065,"semantic_template":"qf-density17-relation-inverse-double/{\"m\":3,\"n\":3,\"pairs\":[[0,1],[1,0]]}"},"fields":{"title":"Typed Converse Relation and Double Converse","stem":"The explicitly typed relation R⊆Fin(3)×Fin(3) has ordered pairs=[[0, 1], [1, 0]]. 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. Give ([n,m], swapped type sizes [m,n], original normalized pairs, normalized converse pairs, double-converse pairs, does the double converse equal the original?).","answer":{"kind":"expression","value":"([3, 3], [3, 3], [[0, 1], [1, 0]], [[0, 1], [1, 0]], [[0, 1], [1, 0]], Yes)","explanation":"Reverse both pair coordinates and the source/target types, then reverse again; do not claim that either relation is a function. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([3, 3], [3, 3], [[0, 1], [1, 0]], [[0, 1], [1, 0]], [[0, 1], [1, 0]], Yes)."},"skills":["Typed Converse Relation and Double Converse"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r03-relation-0066","batch_id":"batch-20261012-advance-20261017-r03-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2a9debe2a273e672a3c1ad146eb77939a4a5dce6d5c9b417139ce1cb6253093e","family_id":"qf-density17-relation-inverse-double","seed":202610173066,"semantic_template":"qf-density17-relation-inverse-double/{\"m\":3,\"n\":3,\"pairs\":[[0,0],[0,1],[1,0]]}"},"fields":{"title":"Typed Converse Relation and Double Converse","stem":"The explicitly typed relation R⊆Fin(3)×Fin(3) has ordered pairs=[[0, 0], [0, 1], [1, 0]]. 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. Give ([n,m], swapped type sizes [m,n], original normalized pairs, normalized converse pairs, double-converse pairs, does the double converse equal the original?).","answer":{"kind":"expression","value":"([3, 3], [3, 3], [[0, 0], [0, 1], [1, 0]], [[0, 0], [0, 1], [1, 0]], [[0, 0], [0, 1], [1, 0]], Yes)","explanation":"Reverse both pair coordinates and the source/target types, then reverse again; do not claim that either relation is a function. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([3, 3], [3, 3], [[0, 0], [0, 1], [1, 0]], [[0, 0], [0, 1], [1, 0]], [[0, 0], [0, 1], [1, 0]], Yes)."},"skills":["Typed Converse Relation and Double Converse"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r03-relation-0067","batch_id":"batch-20261012-advance-20261017-r03-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"584e8405207041b3bcff782ef69731230d9cfd532e385293656cd1f826161ecb","family_id":"qf-density17-relation-inverse-double","seed":202610173067,"semantic_template":"qf-density17-relation-inverse-double/{\"m\":3,\"n\":3,\"pairs\":[[0,2],[1,0]]}"},"fields":{"title":"Typed Converse Relation and Double Converse","stem":"The explicitly typed relation R⊆Fin(3)×Fin(3) has ordered pairs=[[0, 2], [1, 0]]. 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. Give ([n,m], swapped type sizes [m,n], original normalized pairs, normalized converse pairs, double-converse pairs, does the double converse equal the original?).","answer":{"kind":"expression","value":"([3, 3], [3, 3], [[0, 2], [1, 0]], [[0, 1], [2, 0]], [[0, 2], [1, 0]], Yes)","explanation":"Reverse both pair coordinates and the source/target types, then reverse again; do not claim that either relation is a function. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([3, 3], [3, 3], [[0, 2], [1, 0]], [[0, 1], [2, 0]], [[0, 2], [1, 0]], Yes)."},"skills":["Typed Converse Relation and Double Converse"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r03-relation-0068","batch_id":"batch-20261012-advance-20261017-r03-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"50a805c6b5d522071b248d5d6e8c465f2e98e457f2e521e5f31275e9e2a35fb9","family_id":"qf-density17-relation-inverse-double","seed":202610173068,"semantic_template":"qf-density17-relation-inverse-double/{\"m\":3,\"n\":3,\"pairs\":[[0,0],[0,2],[1,0]]}"},"fields":{"title":"Typed Converse Relation and Double Converse","stem":"The explicitly typed relation R⊆Fin(3)×Fin(3) has ordered pairs=[[0, 0], [0, 2], [1, 0]]. 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. Give ([n,m], swapped type sizes [m,n], original normalized pairs, normalized converse pairs, double-converse pairs, does the double converse equal the original?).","answer":{"kind":"expression","value":"([3, 3], [3, 3], [[0, 0], [0, 2], [1, 0]], [[0, 0], [0, 1], [2, 0]], [[0, 0], [0, 2], [1, 0]], Yes)","explanation":"Reverse both pair coordinates and the source/target types, then reverse again; do not claim that either relation is a function. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([3, 3], [3, 3], [[0, 0], [0, 2], [1, 0]], [[0, 0], [0, 1], [2, 0]], [[0, 0], [0, 2], [1, 0]], Yes)."},"skills":["Typed Converse Relation and Double Converse"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r03-relation-0069","batch_id":"batch-20261012-advance-20261017-r03-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8f016acafb4cd0b5cd4c75d271bd7d86d0d080d7dd4a8f96e6282faaaf4380cd","family_id":"qf-density17-relation-inverse-double","seed":202610173069,"semantic_template":"qf-density17-relation-inverse-double/{\"m\":3,\"n\":3,\"pairs\":[[0,1],[0,2],[1,0]]}"},"fields":{"title":"Typed Converse Relation and Double Converse","stem":"The explicitly typed relation R⊆Fin(3)×Fin(3) has ordered pairs=[[0, 1], [0, 2], [1, 0]]. 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. Give ([n,m], swapped type sizes [m,n], original normalized pairs, normalized converse pairs, double-converse pairs, does the double converse equal the original?).","answer":{"kind":"expression","value":"([3, 3], [3, 3], [[0, 1], [0, 2], [1, 0]], [[0, 1], [1, 0], [2, 0]], [[0, 1], [0, 2], [1, 0]], Yes)","explanation":"Reverse both pair coordinates and the source/target types, then reverse again; do not claim that either relation is a function. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([3, 3], [3, 3], [[0, 1], [0, 2], [1, 0]], [[0, 1], [1, 0], [2, 0]], [[0, 1], [0, 2], [1, 0]], Yes)."},"skills":["Typed Converse Relation and Double Converse"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r03-relation-0070","batch_id":"batch-20261012-advance-20261017-r03-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e13467059079cc9adb67e7869003a080ad777671903934ff742982083a51d966","family_id":"qf-density17-relation-inverse-double","seed":202610173070,"semantic_template":"qf-density17-relation-inverse-double/{\"m\":3,\"n\":3,\"pairs\":[[0,0],[0,1],[0,2],[1,0]]}"},"fields":{"title":"Typed Converse Relation and Double Converse","stem":"The explicitly typed relation R⊆Fin(3)×Fin(3) has ordered pairs=[[0, 0], [0, 1], [0, 2], [1, 0]]. 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. Give ([n,m], swapped type sizes [m,n], original normalized pairs, normalized converse pairs, double-converse pairs, does the double converse equal the original?).","answer":{"kind":"expression","value":"([3, 3], [3, 3], [[0, 0], [0, 1], [0, 2], [1, 0]], [[0, 0], [0, 1], [1, 0], [2, 0]], [[0, 0], [0, 1], [0, 2], [1, 0]], Yes)","explanation":"Reverse both pair coordinates and the source/target types, then reverse again; do not claim that either relation is a function. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([3, 3], [3, 3], [[0, 0], [0, 1], [0, 2], [1, 0]], [[0, 0], [0, 1], [1, 0], [2, 0]], [[0, 0], [0, 1], [0, 2], [1, 0]], Yes)."},"skills":["Typed Converse Relation and Double Converse"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r03-relation-0071","batch_id":"batch-20261012-advance-20261017-r03-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7a043e2bf4ea66571586adbfb9a239ce9299ae1b6d1f7da02947c9366fa52af2","family_id":"qf-density17-relation-inverse-double","seed":202610173071,"semantic_template":"qf-density17-relation-inverse-double/{\"m\":3,\"n\":3,\"pairs\":[[1,1]]}"},"fields":{"title":"Typed Converse Relation and Double Converse","stem":"The explicitly typed relation R⊆Fin(3)×Fin(3) has ordered pairs=[[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. Give ([n,m], swapped type sizes [m,n], original normalized pairs, normalized converse pairs, double-converse pairs, does the double converse equal the original?).","answer":{"kind":"expression","value":"([3, 3], [3, 3], [[1, 1]], [[1, 1]], [[1, 1]], Yes)","explanation":"Reverse both pair coordinates and the source/target types, then reverse again; do not claim that either relation is a function. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([3, 3], [3, 3], [[1, 1]], [[1, 1]], [[1, 1]], Yes)."},"skills":["Typed Converse Relation and Double Converse"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r03-relation-0072","batch_id":"batch-20261012-advance-20261017-r03-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e0367372d688fbca3755ebaea984653c7237d6aa202b556b39b5e97aa0e1283e","family_id":"qf-density17-relation-inverse-double","seed":202610173072,"semantic_template":"qf-density17-relation-inverse-double/{\"m\":3,\"n\":3,\"pairs\":[[0,0],[1,1]]}"},"fields":{"title":"Typed Converse Relation and Double Converse","stem":"The explicitly typed relation R⊆Fin(3)×Fin(3) has ordered pairs=[[0, 0], [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. Give ([n,m], swapped type sizes [m,n], original normalized pairs, normalized converse pairs, double-converse pairs, does the double converse equal the original?).","answer":{"kind":"expression","value":"([3, 3], [3, 3], [[0, 0], [1, 1]], [[0, 0], [1, 1]], [[0, 0], [1, 1]], Yes)","explanation":"Reverse both pair coordinates and the source/target types, then reverse again; do not claim that either relation is a function. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([3, 3], [3, 3], [[0, 0], [1, 1]], [[0, 0], [1, 1]], [[0, 0], [1, 1]], Yes)."},"skills":["Typed Converse Relation and Double Converse"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r03-relation-0073","batch_id":"batch-20261012-advance-20261017-r03-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cab7c1f3b69ea82bf0fe67f110231afafb4bfe80a2ced1ff066eb2f04568db0a","family_id":"qf-density17-relation-inverse-double","seed":202610173073,"semantic_template":"qf-density17-relation-inverse-double/{\"m\":3,\"n\":3,\"pairs\":[[0,1],[1,1]]}"},"fields":{"title":"Typed Converse Relation and Double Converse","stem":"The explicitly typed relation R⊆Fin(3)×Fin(3) has ordered pairs=[[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. Give ([n,m], swapped type sizes [m,n], original normalized pairs, normalized converse pairs, double-converse pairs, does the double converse equal the original?).","answer":{"kind":"expression","value":"([3, 3], [3, 3], [[0, 1], [1, 1]], [[1, 0], [1, 1]], [[0, 1], [1, 1]], Yes)","explanation":"Reverse both pair coordinates and the source/target types, then reverse again; do not claim that either relation is a function. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([3, 3], [3, 3], [[0, 1], [1, 1]], [[1, 0], [1, 1]], [[0, 1], [1, 1]], Yes)."},"skills":["Typed Converse Relation and Double Converse"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r03-relation-0074","batch_id":"batch-20261012-advance-20261017-r03-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9ebbe35056bcf87e0d2e9b607bbaa7f8cdef6e3adffc21668405ad935f7263d9","family_id":"qf-density17-relation-inverse-double","seed":202610173074,"semantic_template":"qf-density17-relation-inverse-double/{\"m\":3,\"n\":3,\"pairs\":[[0,0],[0,1],[1,1]]}"},"fields":{"title":"Typed Converse Relation and Double Converse","stem":"The explicitly typed relation R⊆Fin(3)×Fin(3) 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. Give ([n,m], swapped type sizes [m,n], original normalized pairs, normalized converse pairs, double-converse pairs, does the double converse equal the original?).","answer":{"kind":"expression","value":"([3, 3], [3, 3], [[0, 0], [0, 1], [1, 1]], [[0, 0], [1, 0], [1, 1]], [[0, 0], [0, 1], [1, 1]], Yes)","explanation":"Reverse both pair coordinates and the source/target types, then reverse again; do not claim that either relation is a function. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([3, 3], [3, 3], [[0, 0], [0, 1], [1, 1]], [[0, 0], [1, 0], [1, 1]], [[0, 0], [0, 1], [1, 1]], Yes)."},"skills":["Typed Converse Relation and Double Converse"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r03-relation-0075","batch_id":"batch-20261012-advance-20261017-r03-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"026507a20adad2f8e57c7633b87d034ac767afff6a22fd7042fa5edff26d54fd","family_id":"qf-density17-relation-inverse-double","seed":202610173075,"semantic_template":"qf-density17-relation-inverse-double/{\"m\":3,\"n\":3,\"pairs\":[[0,2],[1,1]]}"},"fields":{"title":"Typed Converse Relation and Double Converse","stem":"The explicitly typed relation R⊆Fin(3)×Fin(3) has ordered pairs=[[0, 2], [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. Give ([n,m], swapped type sizes [m,n], original normalized pairs, normalized converse pairs, double-converse pairs, does the double converse equal the original?).","answer":{"kind":"expression","value":"([3, 3], [3, 3], [[0, 2], [1, 1]], [[1, 1], [2, 0]], [[0, 2], [1, 1]], Yes)","explanation":"Reverse both pair coordinates and the source/target types, then reverse again; do not claim that either relation is a function. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([3, 3], [3, 3], [[0, 2], [1, 1]], [[1, 1], [2, 0]], [[0, 2], [1, 1]], Yes)."},"skills":["Typed Converse Relation and Double Converse"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r03-relation-0076","batch_id":"batch-20261012-advance-20261017-r03-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"27283c134e99c8b9e9554884cb2e228e66a4204ffcfdc4ac85218723f09eb6d5","family_id":"qf-density17-relation-single-valued-total","seed":202610173076,"semantic_template":"qf-density17-relation-single-valued-total/{\"m\":0,\"n\":0,\"pairs\":[]}"},"fields":{"title":"Relation Single-Valuedness versus Total-Function Qualification","stem":"The explicitly typed relation R⊆Fin(0)×Fin(0) has ordered pairs=[]. 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. Give ([n,m], normalized pairs, each source fiber size, at-most-one value per source?, exactly-one value per source?, total image table or not-a-total-function).","answer":{"kind":"expression","value":"([0, 0], [], [], Yes, Yes, [])","explanation":"At-most-one permits missing sources; totality requires exactly one image at every source. Both universal conditions hold vacuously on an empty source. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 0], [], [], Yes, Yes, [])."},"skills":["Relation Single-Valuedness versus Total-Function Qualification"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r03-relation-0077","batch_id":"batch-20261012-advance-20261017-r03-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"998fbf6604705d5425beb13b29ff2b7c306cee630e0c6127a1d0e3555708618d","family_id":"qf-density17-relation-single-valued-total","seed":202610173077,"semantic_template":"qf-density17-relation-single-valued-total/{\"m\":0,\"n\":2,\"pairs\":[]}"},"fields":{"title":"Relation Single-Valuedness versus Total-Function Qualification","stem":"The explicitly typed relation R⊆Fin(2)×Fin(0) has ordered pairs=[]. 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. Give ([n,m], normalized pairs, each source fiber size, at-most-one value per source?, exactly-one value per source?, total image table or not-a-total-function).","answer":{"kind":"expression","value":"([2, 0], [], [0, 0], Yes, No, not-a-total-function)","explanation":"At-most-one permits missing sources; totality requires exactly one image at every source. Both universal conditions hold vacuously on an empty source. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([2, 0], [], [0, 0], Yes, No, not-a-total-function)."},"skills":["Relation Single-Valuedness versus Total-Function Qualification"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r03-relation-0078","batch_id":"batch-20261012-advance-20261017-r03-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"53881a555e1962380c9b2da1de74e3bc1a5b800d4140612a1e1192b921be8706","family_id":"qf-density17-relation-single-valued-total","seed":202610173078,"semantic_template":"qf-density17-relation-single-valued-total/{\"m\":2,\"n\":0,\"pairs\":[]}"},"fields":{"title":"Relation Single-Valuedness versus Total-Function Qualification","stem":"The explicitly typed relation R⊆Fin(0)×Fin(2) has ordered pairs=[]. 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. Give ([n,m], normalized pairs, each source fiber size, at-most-one value per source?, exactly-one value per source?, total image table or not-a-total-function).","answer":{"kind":"expression","value":"([0, 2], [], [], Yes, Yes, [])","explanation":"At-most-one permits missing sources; totality requires exactly one image at every source. Both universal conditions hold vacuously on an empty source. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([0, 2], [], [], Yes, Yes, [])."},"skills":["Relation Single-Valuedness versus Total-Function Qualification"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r03-relation-0079","batch_id":"batch-20261012-advance-20261017-r03-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c26ff09b0f7b0785a3e96b312a0ad31d53025fe3b4968d509d13edfd23626afd","family_id":"qf-density17-relation-single-valued-total","seed":202610173079,"semantic_template":"qf-density17-relation-single-valued-total/{\"m\":2,\"n\":2,\"pairs\":[[0,0],[1,1]]}"},"fields":{"title":"Relation Single-Valuedness versus Total-Function Qualification","stem":"The explicitly typed relation R⊆Fin(2)×Fin(2) has ordered pairs=[[0, 0], [1, 1], [0, 0]]. 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. Give ([n,m], normalized pairs, each source fiber size, at-most-one value per source?, exactly-one value per source?, total image table or not-a-total-function).","answer":{"kind":"expression","value":"([2, 2], [[0, 0], [1, 1]], [1, 1], Yes, Yes, [0, 1])","explanation":"At-most-one permits missing sources; totality requires exactly one image at every source. Both universal conditions hold vacuously on an empty source. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([2, 2], [[0, 0], [1, 1]], [1, 1], Yes, Yes, [0, 1])."},"skills":["Relation Single-Valuedness versus Total-Function Qualification"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r03-relation-0080","batch_id":"batch-20261012-advance-20261017-r03-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3794402796fd033f962f05bbbb1967ef36a8c4837bdc1350eff68192b800698b","family_id":"qf-density17-relation-single-valued-total","seed":202610173080,"semantic_template":"qf-density17-relation-single-valued-total/{\"m\":2,\"n\":2,\"pairs\":[[0,0],[0,1]]}"},"fields":{"title":"Relation Single-Valuedness versus Total-Function Qualification","stem":"The explicitly typed relation R⊆Fin(2)×Fin(2) has ordered pairs=[[0, 0], [0, 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. Give ([n,m], normalized pairs, each source fiber size, at-most-one value per source?, exactly-one value per source?, total image table or not-a-total-function).","answer":{"kind":"expression","value":"([2, 2], [[0, 0], [0, 1]], [2, 0], No, No, not-a-total-function)","explanation":"At-most-one permits missing sources; totality requires exactly one image at every source. Both universal conditions hold vacuously on an empty source. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([2, 2], [[0, 0], [0, 1]], [2, 0], No, No, not-a-total-function)."},"skills":["Relation Single-Valuedness versus Total-Function Qualification"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r03-relation-0081","batch_id":"batch-20261012-advance-20261017-r03-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6c1978740bbed6bb2bddc619d853585994a33f9eb3ad3d4e29cfada593c7f549","family_id":"qf-density17-relation-single-valued-total","seed":202610173081,"semantic_template":"qf-density17-relation-single-valued-total/{\"m\":3,\"n\":3,\"pairs\":[[0,0]]}"},"fields":{"title":"Relation Single-Valuedness versus Total-Function Qualification","stem":"The explicitly typed relation R⊆Fin(3)×Fin(3) has ordered pairs=[[0, 0]]. 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. Give ([n,m], normalized pairs, each source fiber size, at-most-one value per source?, exactly-one value per source?, total image table or not-a-total-function).","answer":{"kind":"expression","value":"([3, 3], [[0, 0]], [1, 0, 0], Yes, No, not-a-total-function)","explanation":"At-most-one permits missing sources; totality requires exactly one image at every source. Both universal conditions hold vacuously on an empty source. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([3, 3], [[0, 0]], [1, 0, 0], Yes, No, not-a-total-function)."},"skills":["Relation Single-Valuedness versus Total-Function Qualification"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r03-relation-0082","batch_id":"batch-20261012-advance-20261017-r03-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"83a80bc98d529ee299a547397490bf271cbb1dcc68e9c80abedf67696b624b63","family_id":"qf-density17-relation-single-valued-total","seed":202610173082,"semantic_template":"qf-density17-relation-single-valued-total/{\"m\":3,\"n\":3,\"pairs\":[[0,1]]}"},"fields":{"title":"Relation Single-Valuedness versus Total-Function Qualification","stem":"The explicitly typed relation R⊆Fin(3)×Fin(3) has ordered pairs=[[0, 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. Give ([n,m], normalized pairs, each source fiber size, at-most-one value per source?, exactly-one value per source?, total image table or not-a-total-function).","answer":{"kind":"expression","value":"([3, 3], [[0, 1]], [1, 0, 0], Yes, No, not-a-total-function)","explanation":"At-most-one permits missing sources; totality requires exactly one image at every source. Both universal conditions hold vacuously on an empty source. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([3, 3], [[0, 1]], [1, 0, 0], Yes, No, not-a-total-function)."},"skills":["Relation Single-Valuedness versus Total-Function Qualification"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r03-relation-0083","batch_id":"batch-20261012-advance-20261017-r03-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6a78154757599d058fcd4e2ca3f7df9f17be51d8fe76f1b57dbbe5a69b6df5b0","family_id":"qf-density17-relation-single-valued-total","seed":202610173083,"semantic_template":"qf-density17-relation-single-valued-total/{\"m\":3,\"n\":3,\"pairs\":[[0,0],[0,1]]}"},"fields":{"title":"Relation Single-Valuedness versus Total-Function Qualification","stem":"The explicitly typed relation R⊆Fin(3)×Fin(3) has ordered pairs=[[0, 0], [0, 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. Give ([n,m], normalized pairs, each source fiber size, at-most-one value per source?, exactly-one value per source?, total image table or not-a-total-function).","answer":{"kind":"expression","value":"([3, 3], [[0, 0], [0, 1]], [2, 0, 0], No, No, not-a-total-function)","explanation":"At-most-one permits missing sources; totality requires exactly one image at every source. Both universal conditions hold vacuously on an empty source. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([3, 3], [[0, 0], [0, 1]], [2, 0, 0], No, No, not-a-total-function)."},"skills":["Relation Single-Valuedness versus Total-Function Qualification"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r03-relation-0084","batch_id":"batch-20261012-advance-20261017-r03-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"842a27ec4f352527003f03c2065eecd174fbb93ea8320004be1ddb38277efaff","family_id":"qf-density17-relation-single-valued-total","seed":202610173084,"semantic_template":"qf-density17-relation-single-valued-total/{\"m\":3,\"n\":3,\"pairs\":[[0,2]]}"},"fields":{"title":"Relation Single-Valuedness versus Total-Function Qualification","stem":"The explicitly typed relation R⊆Fin(3)×Fin(3) has ordered pairs=[[0, 2]]. 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. Give ([n,m], normalized pairs, each source fiber size, at-most-one value per source?, exactly-one value per source?, total image table or not-a-total-function).","answer":{"kind":"expression","value":"([3, 3], [[0, 2]], [1, 0, 0], Yes, No, not-a-total-function)","explanation":"At-most-one permits missing sources; totality requires exactly one image at every source. Both universal conditions hold vacuously on an empty source. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([3, 3], [[0, 2]], [1, 0, 0], Yes, No, not-a-total-function)."},"skills":["Relation Single-Valuedness versus Total-Function Qualification"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r03-relation-0085","batch_id":"batch-20261012-advance-20261017-r03-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f0ca72d4f8ad4bf1c9c70d593df9c82ca88965251ca623f81fa01b5d3b17cfb2","family_id":"qf-density17-relation-single-valued-total","seed":202610173085,"semantic_template":"qf-density17-relation-single-valued-total/{\"m\":3,\"n\":3,\"pairs\":[[0,0],[0,2]]}"},"fields":{"title":"Relation Single-Valuedness versus Total-Function Qualification","stem":"The explicitly typed relation R⊆Fin(3)×Fin(3) has ordered pairs=[[0, 0], [0, 2]]. 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. Give ([n,m], normalized pairs, each source fiber size, at-most-one value per source?, exactly-one value per source?, total image table or not-a-total-function).","answer":{"kind":"expression","value":"([3, 3], [[0, 0], [0, 2]], [2, 0, 0], No, No, not-a-total-function)","explanation":"At-most-one permits missing sources; totality requires exactly one image at every source. Both universal conditions hold vacuously on an empty source. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([3, 3], [[0, 0], [0, 2]], [2, 0, 0], No, No, not-a-total-function)."},"skills":["Relation Single-Valuedness versus Total-Function Qualification"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r03-relation-0086","batch_id":"batch-20261012-advance-20261017-r03-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b545983a87078e6a6bb35de2b67debeb670843344caeaab0cb7a551eba4ce25e","family_id":"qf-density17-relation-single-valued-total","seed":202610173086,"semantic_template":"qf-density17-relation-single-valued-total/{\"m\":3,\"n\":3,\"pairs\":[[0,1],[0,2]]}"},"fields":{"title":"Relation Single-Valuedness versus Total-Function Qualification","stem":"The explicitly typed relation R⊆Fin(3)×Fin(3) has ordered pairs=[[0, 1], [0, 2]]. 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. Give ([n,m], normalized pairs, each source fiber size, at-most-one value per source?, exactly-one value per source?, total image table or not-a-total-function).","answer":{"kind":"expression","value":"([3, 3], [[0, 1], [0, 2]], [2, 0, 0], No, No, not-a-total-function)","explanation":"At-most-one permits missing sources; totality requires exactly one image at every source. Both universal conditions hold vacuously on an empty source. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([3, 3], [[0, 1], [0, 2]], [2, 0, 0], No, No, not-a-total-function)."},"skills":["Relation Single-Valuedness versus Total-Function Qualification"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r03-relation-0087","batch_id":"batch-20261012-advance-20261017-r03-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"173d247066d44ffc3ff6b0abbdfd440f8099bca5389f0359a73aeaefb34014ca","family_id":"qf-density17-relation-single-valued-total","seed":202610173087,"semantic_template":"qf-density17-relation-single-valued-total/{\"m\":3,\"n\":3,\"pairs\":[[0,0],[0,1],[0,2]]}"},"fields":{"title":"Relation Single-Valuedness versus Total-Function Qualification","stem":"The explicitly typed relation R⊆Fin(3)×Fin(3) has ordered pairs=[[0, 0], [0, 1], [0, 2]]. 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. Give ([n,m], normalized pairs, each source fiber size, at-most-one value per source?, exactly-one value per source?, total image table or not-a-total-function).","answer":{"kind":"expression","value":"([3, 3], [[0, 0], [0, 1], [0, 2]], [3, 0, 0], No, No, not-a-total-function)","explanation":"At-most-one permits missing sources; totality requires exactly one image at every source. Both universal conditions hold vacuously on an empty source. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([3, 3], [[0, 0], [0, 1], [0, 2]], [3, 0, 0], No, No, not-a-total-function)."},"skills":["Relation Single-Valuedness versus Total-Function Qualification"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r03-relation-0088","batch_id":"batch-20261012-advance-20261017-r03-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"af3c565f3909fe0038d23497b219e187fa22c9d78aca1bdc391538ae2c121b53","family_id":"qf-density17-relation-single-valued-total","seed":202610173088,"semantic_template":"qf-density17-relation-single-valued-total/{\"m\":3,\"n\":3,\"pairs\":[[1,0]]}"},"fields":{"title":"Relation Single-Valuedness versus Total-Function Qualification","stem":"The explicitly typed relation R⊆Fin(3)×Fin(3) has ordered pairs=[[1, 0]]. 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. Give ([n,m], normalized pairs, each source fiber size, at-most-one value per source?, exactly-one value per source?, total image table or not-a-total-function).","answer":{"kind":"expression","value":"([3, 3], [[1, 0]], [0, 1, 0], Yes, No, not-a-total-function)","explanation":"At-most-one permits missing sources; totality requires exactly one image at every source. Both universal conditions hold vacuously on an empty source. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([3, 3], [[1, 0]], [0, 1, 0], Yes, No, not-a-total-function)."},"skills":["Relation Single-Valuedness versus Total-Function Qualification"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r03-relation-0089","batch_id":"batch-20261012-advance-20261017-r03-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"81ed2c7ba0b861155fd1ac1271cd9d7f210c6f8db282ded3cb8c82a6fb97f7d7","family_id":"qf-density17-relation-single-valued-total","seed":202610173089,"semantic_template":"qf-density17-relation-single-valued-total/{\"m\":3,\"n\":3,\"pairs\":[[0,0],[1,0]]}"},"fields":{"title":"Relation Single-Valuedness versus Total-Function Qualification","stem":"The explicitly typed relation R⊆Fin(3)×Fin(3) has ordered pairs=[[0, 0], [1, 0]]. 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. Give ([n,m], normalized pairs, each source fiber size, at-most-one value per source?, exactly-one value per source?, total image table or not-a-total-function).","answer":{"kind":"expression","value":"([3, 3], [[0, 0], [1, 0]], [1, 1, 0], Yes, No, not-a-total-function)","explanation":"At-most-one permits missing sources; totality requires exactly one image at every source. Both universal conditions hold vacuously on an empty source. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([3, 3], [[0, 0], [1, 0]], [1, 1, 0], Yes, No, not-a-total-function)."},"skills":["Relation Single-Valuedness versus Total-Function Qualification"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r03-relation-0090","batch_id":"batch-20261012-advance-20261017-r03-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b6822ddcdce5aa0a7f73975f454d9472e191e94fd0e70aa0bcf2e2e9d01fbf9d","family_id":"qf-density17-relation-single-valued-total","seed":202610173090,"semantic_template":"qf-density17-relation-single-valued-total/{\"m\":3,\"n\":3,\"pairs\":[[0,1],[1,0]]}"},"fields":{"title":"Relation Single-Valuedness versus Total-Function Qualification","stem":"The explicitly typed relation R⊆Fin(3)×Fin(3) has ordered pairs=[[0, 1], [1, 0]]. 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. Give ([n,m], normalized pairs, each source fiber size, at-most-one value per source?, exactly-one value per source?, total image table or not-a-total-function).","answer":{"kind":"expression","value":"([3, 3], [[0, 1], [1, 0]], [1, 1, 0], Yes, No, not-a-total-function)","explanation":"At-most-one permits missing sources; totality requires exactly one image at every source. Both universal conditions hold vacuously on an empty source. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([3, 3], [[0, 1], [1, 0]], [1, 1, 0], Yes, No, not-a-total-function)."},"skills":["Relation Single-Valuedness versus Total-Function Qualification"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r03-relation-0091","batch_id":"batch-20261012-advance-20261017-r03-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"052d206efb229fb6d30903a26ff3eb93c285cbcd3b5397599b09b3edae455b26","family_id":"qf-density17-relation-single-valued-total","seed":202610173091,"semantic_template":"qf-density17-relation-single-valued-total/{\"m\":3,\"n\":3,\"pairs\":[[0,0],[0,1],[1,0]]}"},"fields":{"title":"Relation Single-Valuedness versus Total-Function Qualification","stem":"The explicitly typed relation R⊆Fin(3)×Fin(3) has ordered pairs=[[0, 0], [0, 1], [1, 0]]. 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. Give ([n,m], normalized pairs, each source fiber size, at-most-one value per source?, exactly-one value per source?, total image table or not-a-total-function).","answer":{"kind":"expression","value":"([3, 3], [[0, 0], [0, 1], [1, 0]], [2, 1, 0], No, No, not-a-total-function)","explanation":"At-most-one permits missing sources; totality requires exactly one image at every source. Both universal conditions hold vacuously on an empty source. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([3, 3], [[0, 0], [0, 1], [1, 0]], [2, 1, 0], No, No, not-a-total-function)."},"skills":["Relation Single-Valuedness versus Total-Function Qualification"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r03-relation-0092","batch_id":"batch-20261012-advance-20261017-r03-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9ff5bb9740c13a56ed823ad85a7c08c39a138bc5f1e847d8bbe8126884dfced4","family_id":"qf-density17-relation-single-valued-total","seed":202610173092,"semantic_template":"qf-density17-relation-single-valued-total/{\"m\":3,\"n\":3,\"pairs\":[[0,2],[1,0]]}"},"fields":{"title":"Relation Single-Valuedness versus Total-Function Qualification","stem":"The explicitly typed relation R⊆Fin(3)×Fin(3) has ordered pairs=[[0, 2], [1, 0]]. 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. Give ([n,m], normalized pairs, each source fiber size, at-most-one value per source?, exactly-one value per source?, total image table or not-a-total-function).","answer":{"kind":"expression","value":"([3, 3], [[0, 2], [1, 0]], [1, 1, 0], Yes, No, not-a-total-function)","explanation":"At-most-one permits missing sources; totality requires exactly one image at every source. Both universal conditions hold vacuously on an empty source. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([3, 3], [[0, 2], [1, 0]], [1, 1, 0], Yes, No, not-a-total-function)."},"skills":["Relation Single-Valuedness versus Total-Function Qualification"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r03-relation-0093","batch_id":"batch-20261012-advance-20261017-r03-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"70b2077c5c67f782c4e6f935eb3b9137f8b97395822c3276c6d02a599b889ef4","family_id":"qf-density17-relation-single-valued-total","seed":202610173093,"semantic_template":"qf-density17-relation-single-valued-total/{\"m\":3,\"n\":3,\"pairs\":[[0,0],[0,2],[1,0]]}"},"fields":{"title":"Relation Single-Valuedness versus Total-Function Qualification","stem":"The explicitly typed relation R⊆Fin(3)×Fin(3) has ordered pairs=[[0, 0], [0, 2], [1, 0]]. 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. Give ([n,m], normalized pairs, each source fiber size, at-most-one value per source?, exactly-one value per source?, total image table or not-a-total-function).","answer":{"kind":"expression","value":"([3, 3], [[0, 0], [0, 2], [1, 0]], [2, 1, 0], No, No, not-a-total-function)","explanation":"At-most-one permits missing sources; totality requires exactly one image at every source. Both universal conditions hold vacuously on an empty source. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([3, 3], [[0, 0], [0, 2], [1, 0]], [2, 1, 0], No, No, not-a-total-function)."},"skills":["Relation Single-Valuedness versus Total-Function Qualification"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r03-relation-0094","batch_id":"batch-20261012-advance-20261017-r03-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b5f76749e3678abd8458a32c8bef68f35c2018d03367f74b4d3c94fd89fb6000","family_id":"qf-density17-relation-single-valued-total","seed":202610173094,"semantic_template":"qf-density17-relation-single-valued-total/{\"m\":3,\"n\":3,\"pairs\":[[0,1],[0,2],[1,0]]}"},"fields":{"title":"Relation Single-Valuedness versus Total-Function Qualification","stem":"The explicitly typed relation R⊆Fin(3)×Fin(3) has ordered pairs=[[0, 1], [0, 2], [1, 0]]. 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. Give ([n,m], normalized pairs, each source fiber size, at-most-one value per source?, exactly-one value per source?, total image table or not-a-total-function).","answer":{"kind":"expression","value":"([3, 3], [[0, 1], [0, 2], [1, 0]], [2, 1, 0], No, No, not-a-total-function)","explanation":"At-most-one permits missing sources; totality requires exactly one image at every source. Both universal conditions hold vacuously on an empty source. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([3, 3], [[0, 1], [0, 2], [1, 0]], [2, 1, 0], No, No, not-a-total-function)."},"skills":["Relation Single-Valuedness versus Total-Function Qualification"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r03-relation-0095","batch_id":"batch-20261012-advance-20261017-r03-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"059158ed119591110dde3cc968532a1672371d9e0816c76214af336feed4f109","family_id":"qf-density17-relation-single-valued-total","seed":202610173095,"semantic_template":"qf-density17-relation-single-valued-total/{\"m\":3,\"n\":3,\"pairs\":[[0,0],[0,1],[0,2],[1,0]]}"},"fields":{"title":"Relation Single-Valuedness versus Total-Function Qualification","stem":"The explicitly typed relation R⊆Fin(3)×Fin(3) has ordered pairs=[[0, 0], [0, 1], [0, 2], [1, 0]]. 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. Give ([n,m], normalized pairs, each source fiber size, at-most-one value per source?, exactly-one value per source?, total image table or not-a-total-function).","answer":{"kind":"expression","value":"([3, 3], [[0, 0], [0, 1], [0, 2], [1, 0]], [3, 1, 0], No, No, not-a-total-function)","explanation":"At-most-one permits missing sources; totality requires exactly one image at every source. Both universal conditions hold vacuously on an empty source. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([3, 3], [[0, 0], [0, 1], [0, 2], [1, 0]], [3, 1, 0], No, No, not-a-total-function)."},"skills":["Relation Single-Valuedness versus Total-Function Qualification"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r03-relation-0096","batch_id":"batch-20261012-advance-20261017-r03-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a705f6f4929f14acf86fba82266a740e4bebae8a6a00f236bc5c7121ad007c96","family_id":"qf-density17-relation-single-valued-total","seed":202610173096,"semantic_template":"qf-density17-relation-single-valued-total/{\"m\":3,\"n\":3,\"pairs\":[[1,1]]}"},"fields":{"title":"Relation Single-Valuedness versus Total-Function Qualification","stem":"The explicitly typed relation R⊆Fin(3)×Fin(3) has ordered pairs=[[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. Give ([n,m], normalized pairs, each source fiber size, at-most-one value per source?, exactly-one value per source?, total image table or not-a-total-function).","answer":{"kind":"expression","value":"([3, 3], [[1, 1]], [0, 1, 0], Yes, No, not-a-total-function)","explanation":"At-most-one permits missing sources; totality requires exactly one image at every source. Both universal conditions hold vacuously on an empty source. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([3, 3], [[1, 1]], [0, 1, 0], Yes, No, not-a-total-function)."},"skills":["Relation Single-Valuedness versus Total-Function Qualification"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r03-relation-0097","batch_id":"batch-20261012-advance-20261017-r03-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7d43ed40c6ccf3e11ec7d38314fcd2b5f26a2cc19b41d243365ccd8998ff0afe","family_id":"qf-density17-relation-single-valued-total","seed":202610173097,"semantic_template":"qf-density17-relation-single-valued-total/{\"m\":3,\"n\":3,\"pairs\":[[0,0],[1,1]]}"},"fields":{"title":"Relation Single-Valuedness versus Total-Function Qualification","stem":"The explicitly typed relation R⊆Fin(3)×Fin(3) has ordered pairs=[[0, 0], [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. Give ([n,m], normalized pairs, each source fiber size, at-most-one value per source?, exactly-one value per source?, total image table or not-a-total-function).","answer":{"kind":"expression","value":"([3, 3], [[0, 0], [1, 1]], [1, 1, 0], Yes, No, not-a-total-function)","explanation":"At-most-one permits missing sources; totality requires exactly one image at every source. Both universal conditions hold vacuously on an empty source. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([3, 3], [[0, 0], [1, 1]], [1, 1, 0], Yes, No, not-a-total-function)."},"skills":["Relation Single-Valuedness versus Total-Function Qualification"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r03-relation-0098","batch_id":"batch-20261012-advance-20261017-r03-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c4074236c2f57fcbb6178fa62b6f24e98bc58abcf2d806e88c484c4d95b784d6","family_id":"qf-density17-relation-single-valued-total","seed":202610173098,"semantic_template":"qf-density17-relation-single-valued-total/{\"m\":3,\"n\":3,\"pairs\":[[0,1],[1,1]]}"},"fields":{"title":"Relation Single-Valuedness versus Total-Function Qualification","stem":"The explicitly typed relation R⊆Fin(3)×Fin(3) has ordered pairs=[[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. Give ([n,m], normalized pairs, each source fiber size, at-most-one value per source?, exactly-one value per source?, total image table or not-a-total-function).","answer":{"kind":"expression","value":"([3, 3], [[0, 1], [1, 1]], [1, 1, 0], Yes, No, not-a-total-function)","explanation":"At-most-one permits missing sources; totality requires exactly one image at every source. Both universal conditions hold vacuously on an empty source. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([3, 3], [[0, 1], [1, 1]], [1, 1, 0], Yes, No, not-a-total-function)."},"skills":["Relation Single-Valuedness versus Total-Function Qualification"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r03-relation-0099","batch_id":"batch-20261012-advance-20261017-r03-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7757b9296c705a540b4798644240a5c020a46b4a1a85a064228e03bd1da41a23","family_id":"qf-density17-relation-single-valued-total","seed":202610173099,"semantic_template":"qf-density17-relation-single-valued-total/{\"m\":3,\"n\":3,\"pairs\":[[0,0],[0,1],[1,1]]}"},"fields":{"title":"Relation Single-Valuedness versus Total-Function Qualification","stem":"The explicitly typed relation R⊆Fin(3)×Fin(3) 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. Give ([n,m], normalized pairs, each source fiber size, at-most-one value per source?, exactly-one value per source?, total image table or not-a-total-function).","answer":{"kind":"expression","value":"([3, 3], [[0, 0], [0, 1], [1, 1]], [2, 1, 0], No, No, not-a-total-function)","explanation":"At-most-one permits missing sources; totality requires exactly one image at every source. Both universal conditions hold vacuously on an empty source. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([3, 3], [[0, 0], [0, 1], [1, 1]], [2, 1, 0], No, No, not-a-total-function)."},"skills":["Relation Single-Valuedness versus Total-Function Qualification"],"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."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261012-advance-20261017-r03-relation-0100","batch_id":"batch-20261012-advance-20261017-r03-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9a55be5792112e16d8fcd68f292ead91b69cf1567b811c79284cb4e3777f3d1c","family_id":"qf-density17-relation-single-valued-total","seed":202610173100,"semantic_template":"qf-density17-relation-single-valued-total/{\"m\":3,\"n\":3,\"pairs\":[[0,2],[1,1]]}"},"fields":{"title":"Relation Single-Valuedness versus Total-Function Qualification","stem":"The explicitly typed relation R⊆Fin(3)×Fin(3) has ordered pairs=[[0, 2], [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. Give ([n,m], normalized pairs, each source fiber size, at-most-one value per source?, exactly-one value per source?, total image table or not-a-total-function).","answer":{"kind":"expression","value":"([3, 3], [[0, 2], [1, 1]], [1, 1, 0], Yes, No, not-a-total-function)","explanation":"At-most-one permits missing sources; totality requires exactly one image at every source. Both universal conditions hold vacuously on an empty source. These exact finite checks are not new general formal Evidence. The exact result, retaining the requested order, is ([3, 3], [[0, 2], [1, 1]], [1, 1, 0], Yes, No, not-a-total-function)."},"skills":["Relation Single-Valuedness versus Total-Function Qualification"],"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."}}