{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261004-advance-20261005-r02-surjective-0001","batch_id":"batch-20261004-advance-20261005-r02-surjective-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2341022ef6c6dbcc4d9f6703962985825a3fd9acadb36c531fdf828297167273","family_id":"qf-surjective-table-property","seed":202610052001,"semantic_template":"qf-surjective-table-property/{\"A\":[],\"B\":[],\"operation\":\"surjective\",\"pairs\":[]}"},"fields":{"title":"Surjectivity of a Finite Total Function","stem":"The declared total function f:A→B has A=∅, B=∅, and table ∅. Is f surjective onto the declared codomain B?","answer":{"kind":"short_text","value":"Yes","explanation":"Check a preimage for every target in B, giving Yes. Multiple preimages are allowed, and the attained range is not automatically the declared codomain."},"skills":["Surjectivity of a Finite Total Function"],"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-20261004-advance-20261005-r02-surjective-0002","batch_id":"batch-20261004-advance-20261005-r02-surjective-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1a24e94992c99f7df47a9466d30ec1955bc40ca9bf16ef8658f1e68b50d5b1fa","family_id":"qf-surjective-table-property","seed":202610052002,"semantic_template":"qf-surjective-table-property/{\"A\":[-1,0],\"B\":[0,1],\"operation\":\"surjective\",\"pairs\":[[-1,1],[0,0]]}"},"fields":{"title":"Surjectivity of a Finite Total Function","stem":"The declared total function f:A→B has A={-1, 0}, B={0, 1}, and table {(-1, 1), (0, 0)}. Is f surjective onto the declared codomain B?","answer":{"kind":"short_text","value":"Yes","explanation":"Check a preimage for every target in B, giving Yes. Multiple preimages are allowed, and the attained range is not automatically the declared codomain."},"skills":["Surjectivity of a Finite Total Function"],"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-20261004-advance-20261005-r02-surjective-0003","batch_id":"batch-20261004-advance-20261005-r02-surjective-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d5f04ff8b180d3d1c80650057ad3e3bc97c1f7ba0fd49cec825c32b984f915c1","family_id":"qf-surjective-table-property","seed":202610052003,"semantic_template":"qf-surjective-table-property/{\"A\":[-1,0],\"B\":[1,3,5],\"operation\":\"surjective\",\"pairs\":[[-1,1],[0,3]]}"},"fields":{"title":"Surjectivity of a Finite Total Function","stem":"The declared total function f:A→B has A={-1, 0}, B={1, 3, 5}, and table {(-1, 1), (0, 3)}. Is f surjective onto the declared codomain B?","answer":{"kind":"short_text","value":"No","explanation":"Check a preimage for every target in B, giving No. Multiple preimages are allowed, and the attained range is not automatically the declared codomain."},"skills":["Surjectivity of a Finite Total Function"],"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-20261004-advance-20261005-r02-surjective-0004","batch_id":"batch-20261004-advance-20261005-r02-surjective-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d8ff5e7fa1e6f2e69b4a260a67f574613d48f3cd4da15963d1461b7d1b3da21d","family_id":"qf-surjective-table-property","seed":202610052004,"semantic_template":"qf-surjective-table-property/{\"A\":[-1,0],\"B\":[0],\"operation\":\"surjective\",\"pairs\":[[-1,0],[0,0]]}"},"fields":{"title":"Surjectivity of a Finite Total Function","stem":"The declared total function f:A→B has A={-1, 0}, B={0}, and table {(-1, 0), (0, 0)}. Is f surjective onto the declared codomain B?","answer":{"kind":"short_text","value":"Yes","explanation":"Check a preimage for every target in B, giving Yes. Multiple preimages are allowed, and the attained range is not automatically the declared codomain."},"skills":["Surjectivity of a Finite Total Function"],"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-20261004-advance-20261005-r02-surjective-0005","batch_id":"batch-20261004-advance-20261005-r02-surjective-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a194546400c36cd11de82043f24e1f0bc2ce2351af52ab2220227d0ba9aa9225","family_id":"qf-surjective-table-property","seed":202610052005,"semantic_template":"qf-surjective-table-property/{\"A\":[-1,0],\"B\":[-3,-2,-1],\"operation\":\"surjective\",\"pairs\":[[-1,-1],[0,-1]]}"},"fields":{"title":"Surjectivity of a Finite Total Function","stem":"The declared total function f:A→B has A={-1, 0}, B={-3, -2, -1}, and table {(-1, -1), (0, -1)}. Is f surjective onto the declared codomain B?","answer":{"kind":"short_text","value":"No","explanation":"Check a preimage for every target in B, giving No. Multiple preimages are allowed, and the attained range is not automatically the declared codomain."},"skills":["Surjectivity of a Finite Total Function"],"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-20261004-advance-20261005-r02-surjective-0006","batch_id":"batch-20261004-advance-20261005-r02-surjective-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e2ed9306cb18a57ea67507b055e4ebb7820733373257741a7d40c2f950bb965f","family_id":"qf-surjective-table-property","seed":202610052006,"semantic_template":"qf-surjective-table-property/{\"A\":[-2,-1,0],\"B\":[0,1,2],\"operation\":\"surjective\",\"pairs\":[[-2,1],[-1,2],[0,0]]}"},"fields":{"title":"Surjectivity of a Finite Total Function","stem":"The declared total function f:A→B has A={-2, -1, 0}, B={0, 1, 2}, and table {(-2, 1), (-1, 2), (0, 0)}. Is f surjective onto the declared codomain B?","answer":{"kind":"short_text","value":"Yes","explanation":"Check a preimage for every target in B, giving Yes. Multiple preimages are allowed, and the attained range is not automatically the declared codomain."},"skills":["Surjectivity of a Finite Total Function"],"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-20261004-advance-20261005-r02-surjective-0007","batch_id":"batch-20261004-advance-20261005-r02-surjective-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"60362af9c87ba4b1d90d663f62a9f9356676c8d5b35ba3278567c93eb99198be","family_id":"qf-surjective-table-property","seed":202610052007,"semantic_template":"qf-surjective-table-property/{\"A\":[-2,-1,0],\"B\":[1,3,5,7],\"operation\":\"surjective\",\"pairs\":[[-2,1],[-1,3],[0,5]]}"},"fields":{"title":"Surjectivity of a Finite Total Function","stem":"The declared total function f:A→B has A={-2, -1, 0}, B={1, 3, 5, 7}, and table {(-2, 1), (-1, 3), (0, 5)}. Is f surjective onto the declared codomain B?","answer":{"kind":"short_text","value":"No","explanation":"Check a preimage for every target in B, giving No. Multiple preimages are allowed, and the attained range is not automatically the declared codomain."},"skills":["Surjectivity of a Finite Total Function"],"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-20261004-advance-20261005-r02-surjective-0008","batch_id":"batch-20261004-advance-20261005-r02-surjective-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"dab0d4eb7fb400e6a4d253b698eaf3a43a4bc3450022e5c68552188b59dc8920","family_id":"qf-surjective-table-property","seed":202610052008,"semantic_template":"qf-surjective-table-property/{\"A\":[-2,-1,0],\"B\":[0,1],\"operation\":\"surjective\",\"pairs\":[[-2,0],[-1,1],[0,0]]}"},"fields":{"title":"Surjectivity of a Finite Total Function","stem":"The declared total function f:A→B has A={-2, -1, 0}, B={0, 1}, and table {(-2, 0), (-1, 1), (0, 0)}. Is f surjective onto the declared codomain B?","answer":{"kind":"short_text","value":"Yes","explanation":"Check a preimage for every target in B, giving Yes. Multiple preimages are allowed, and the attained range is not automatically the declared codomain."},"skills":["Surjectivity of a Finite Total Function"],"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-20261004-advance-20261005-r02-surjective-0009","batch_id":"batch-20261004-advance-20261005-r02-surjective-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b3c3a088a66cf994340eb655cbeb1385c52c08e6ce4506981eb789b68bf6970c","family_id":"qf-surjective-table-property","seed":202610052009,"semantic_template":"qf-surjective-table-property/{\"A\":[-2,-1,0],\"B\":[-4,-3,-2,-1],\"operation\":\"surjective\",\"pairs\":[[-2,-1],[-1,-1],[0,-2]]}"},"fields":{"title":"Surjectivity of a Finite Total Function","stem":"The declared total function f:A→B has A={-2, -1, 0}, B={-4, -3, -2, -1}, and table {(-2, -1), (-1, -1), (0, -2)}. Is f surjective onto the declared codomain B?","answer":{"kind":"short_text","value":"No","explanation":"Check a preimage for every target in B, giving No. Multiple preimages are allowed, and the attained range is not automatically the declared codomain."},"skills":["Surjectivity of a Finite Total Function"],"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-20261004-advance-20261005-r02-surjective-0010","batch_id":"batch-20261004-advance-20261005-r02-surjective-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9063eaadfc064f7c7887d68e19f18d6715463976b12da71ffe416fe43eb64623","family_id":"qf-surjective-table-property","seed":202610052010,"semantic_template":"qf-surjective-table-property/{\"A\":[-3,-2,-1,0],\"B\":[0,1,2,3],\"operation\":\"surjective\",\"pairs\":[[-3,1],[-2,2],[-1,3],[0,0]]}"},"fields":{"title":"Surjectivity of a Finite Total Function","stem":"The declared total function f:A→B has A={-3, -2, -1, 0}, B={0, 1, 2, 3}, and table {(-3, 1), (-2, 2), (-1, 3), (0, 0)}. Is f surjective onto the declared codomain B?","answer":{"kind":"short_text","value":"Yes","explanation":"Check a preimage for every target in B, giving Yes. Multiple preimages are allowed, and the attained range is not automatically the declared codomain."},"skills":["Surjectivity of a Finite Total Function"],"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-20261004-advance-20261005-r02-surjective-0011","batch_id":"batch-20261004-advance-20261005-r02-surjective-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1899659fcc370de968bd3ee3938905556a76204a99a202db363e8ae04eb12639","family_id":"qf-surjective-table-property","seed":202610052011,"semantic_template":"qf-surjective-table-property/{\"A\":[-3,-2,-1,0],\"B\":[1,3,5,7,9],\"operation\":\"surjective\",\"pairs\":[[-3,1],[-2,3],[-1,5],[0,7]]}"},"fields":{"title":"Surjectivity of a Finite Total Function","stem":"The declared total function f:A→B has A={-3, -2, -1, 0}, B={1, 3, 5, 7, 9}, and table {(-3, 1), (-2, 3), (-1, 5), (0, 7)}. Is f surjective onto the declared codomain B?","answer":{"kind":"short_text","value":"No","explanation":"Check a preimage for every target in B, giving No. Multiple preimages are allowed, and the attained range is not automatically the declared codomain."},"skills":["Surjectivity of a Finite Total Function"],"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-20261004-advance-20261005-r02-surjective-0012","batch_id":"batch-20261004-advance-20261005-r02-surjective-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e7a178d9dfb97d3c17f4cb9bfc78c6c632ee9b7e13147b3cf2b7521f20a01eea","family_id":"qf-surjective-table-property","seed":202610052012,"semantic_template":"qf-surjective-table-property/{\"A\":[-3,-2,-1,0],\"B\":[0,1,2],\"operation\":\"surjective\",\"pairs\":[[-3,0],[-2,1],[-1,2],[0,0]]}"},"fields":{"title":"Surjectivity of a Finite Total Function","stem":"The declared total function f:A→B has A={-3, -2, -1, 0}, B={0, 1, 2}, and table {(-3, 0), (-2, 1), (-1, 2), (0, 0)}. Is f surjective onto the declared codomain B?","answer":{"kind":"short_text","value":"Yes","explanation":"Check a preimage for every target in B, giving Yes. Multiple preimages are allowed, and the attained range is not automatically the declared codomain."},"skills":["Surjectivity of a Finite Total Function"],"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-20261004-advance-20261005-r02-surjective-0013","batch_id":"batch-20261004-advance-20261005-r02-surjective-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7d989a1645ea937cefc900de7d7a406fdb914bff090783b5caba60e7e614e3a6","family_id":"qf-surjective-table-property","seed":202610052013,"semantic_template":"qf-surjective-table-property/{\"A\":[-3,-2,-1,0],\"B\":[-5,-4,-3,-2,-1],\"operation\":\"surjective\",\"pairs\":[[-3,-1],[-2,-1],[-1,-2],[0,-2]]}"},"fields":{"title":"Surjectivity of a Finite Total Function","stem":"The declared total function f:A→B has A={-3, -2, -1, 0}, B={-5, -4, -3, -2, -1}, and table {(-3, -1), (-2, -1), (-1, -2), (0, -2)}. Is f surjective onto the declared codomain B?","answer":{"kind":"short_text","value":"No","explanation":"Check a preimage for every target in B, giving No. Multiple preimages are allowed, and the attained range is not automatically the declared codomain."},"skills":["Surjectivity of a Finite Total Function"],"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-20261004-advance-20261005-r02-surjective-0014","batch_id":"batch-20261004-advance-20261005-r02-surjective-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e93ef840a1fb37b7608be08cd2adb8a12b5ed3c016e09baf87ea8d60483e1f0e","family_id":"qf-surjective-table-property","seed":202610052014,"semantic_template":"qf-surjective-table-property/{\"A\":[-4,-3,-2,-1,0],\"B\":[0,1,2,3,4],\"operation\":\"surjective\",\"pairs\":[[-4,1],[-3,2],[-2,3],[-1,4],[0,0]]}"},"fields":{"title":"Surjectivity of a Finite Total Function","stem":"The declared total function f:A→B has A={-4, -3, -2, -1, 0}, B={0, 1, 2, 3, 4}, and table {(-4, 1), (-3, 2), (-2, 3), (-1, 4), (0, 0)}. Is f surjective onto the declared codomain B?","answer":{"kind":"short_text","value":"Yes","explanation":"Check a preimage for every target in B, giving Yes. Multiple preimages are allowed, and the attained range is not automatically the declared codomain."},"skills":["Surjectivity of a Finite Total Function"],"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-20261004-advance-20261005-r02-surjective-0015","batch_id":"batch-20261004-advance-20261005-r02-surjective-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fc4cac07aa200f391ee74aa1d23186fa5971e1aaf8302d40cade784ffe3fccf6","family_id":"qf-surjective-table-property","seed":202610052015,"semantic_template":"qf-surjective-table-property/{\"A\":[-4,-3,-2,-1,0],\"B\":[1,3,5,7,9,11],\"operation\":\"surjective\",\"pairs\":[[-4,1],[-3,3],[-2,5],[-1,7],[0,9]]}"},"fields":{"title":"Surjectivity of a Finite Total Function","stem":"The declared total function f:A→B has A={-4, -3, -2, -1, 0}, B={1, 3, 5, 7, 9, 11}, and table {(-4, 1), (-3, 3), (-2, 5), (-1, 7), (0, 9)}. Is f surjective onto the declared codomain B?","answer":{"kind":"short_text","value":"No","explanation":"Check a preimage for every target in B, giving No. Multiple preimages are allowed, and the attained range is not automatically the declared codomain."},"skills":["Surjectivity of a Finite Total Function"],"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-20261004-advance-20261005-r02-surjective-0016","batch_id":"batch-20261004-advance-20261005-r02-surjective-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6da0ac5d22a985540f9d354ae188cb9aef071c23b6e973f0e7e24c38fc1992c7","family_id":"qf-surjective-table-property","seed":202610052016,"semantic_template":"qf-surjective-table-property/{\"A\":[-4,-3,-2,-1,0],\"B\":[0,1,2,3],\"operation\":\"surjective\",\"pairs\":[[-4,0],[-3,1],[-2,2],[-1,3],[0,0]]}"},"fields":{"title":"Surjectivity of a Finite Total Function","stem":"The declared total function f:A→B has A={-4, -3, -2, -1, 0}, B={0, 1, 2, 3}, and table {(-4, 0), (-3, 1), (-2, 2), (-1, 3), (0, 0)}. Is f surjective onto the declared codomain B?","answer":{"kind":"short_text","value":"Yes","explanation":"Check a preimage for every target in B, giving Yes. Multiple preimages are allowed, and the attained range is not automatically the declared codomain."},"skills":["Surjectivity of a Finite Total Function"],"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-20261004-advance-20261005-r02-surjective-0017","batch_id":"batch-20261004-advance-20261005-r02-surjective-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2e47a791d99059d17e2feb5ff43cfee90e99a363fc517060d50883d7ed269a84","family_id":"qf-surjective-table-property","seed":202610052017,"semantic_template":"qf-surjective-table-property/{\"A\":[-4,-3,-2,-1,0],\"B\":[-6,-5,-4,-3,-2,-1],\"operation\":\"surjective\",\"pairs\":[[-4,-1],[-3,-1],[-2,-2],[-1,-2],[0,-3]]}"},"fields":{"title":"Surjectivity of a Finite Total Function","stem":"The declared total function f:A→B has A={-4, -3, -2, -1, 0}, B={-6, -5, -4, -3, -2, -1}, and table {(-4, -1), (-3, -1), (-2, -2), (-1, -2), (0, -3)}. Is f surjective onto the declared codomain B?","answer":{"kind":"short_text","value":"No","explanation":"Check a preimage for every target in B, giving No. Multiple preimages are allowed, and the attained range is not automatically the declared codomain."},"skills":["Surjectivity of a Finite Total Function"],"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-20261004-advance-20261005-r02-surjective-0018","batch_id":"batch-20261004-advance-20261005-r02-surjective-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bf3db0106c23a60cf2eb314565987dd483eeab733caffe6760a4548894692edc","family_id":"qf-surjective-table-property","seed":202610052018,"semantic_template":"qf-surjective-table-property/{\"A\":[-5,-4,-3,-2,-1,0],\"B\":[0,1,2,3,4,5],\"operation\":\"surjective\",\"pairs\":[[-5,1],[-4,2],[-3,3],[-2,4],[-1,5],[0,0]]}"},"fields":{"title":"Surjectivity of a Finite Total Function","stem":"The declared total function f:A→B has A={-5, -4, -3, -2, -1, 0}, B={0, 1, 2, 3, 4, 5}, and table {(-5, 1), (-4, 2), (-3, 3), (-2, 4), (-1, 5), (0, 0)}. Is f surjective onto the declared codomain B?","answer":{"kind":"short_text","value":"Yes","explanation":"Check a preimage for every target in B, giving Yes. Multiple preimages are allowed, and the attained range is not automatically the declared codomain."},"skills":["Surjectivity of a Finite Total Function"],"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-20261004-advance-20261005-r02-surjective-0019","batch_id":"batch-20261004-advance-20261005-r02-surjective-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a8bb2816508edfbae1fc4803edbb1524af30b74c2a94f6eb592f83be3b2cf546","family_id":"qf-surjective-table-property","seed":202610052019,"semantic_template":"qf-surjective-table-property/{\"A\":[-5,-4,-3,-2,-1,0],\"B\":[1,3,5,7,9,11,13],\"operation\":\"surjective\",\"pairs\":[[-5,1],[-4,3],[-3,5],[-2,7],[-1,9],[0,11]]}"},"fields":{"title":"Surjectivity of a Finite Total Function","stem":"The declared total function f:A→B has A={-5, -4, -3, -2, -1, 0}, B={1, 3, 5, 7, 9, 11, 13}, and table {(-5, 1), (-4, 3), (-3, 5), (-2, 7), (-1, 9), (0, 11)}. Is f surjective onto the declared codomain B?","answer":{"kind":"short_text","value":"No","explanation":"Check a preimage for every target in B, giving No. Multiple preimages are allowed, and the attained range is not automatically the declared codomain."},"skills":["Surjectivity of a Finite Total Function"],"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-20261004-advance-20261005-r02-surjective-0020","batch_id":"batch-20261004-advance-20261005-r02-surjective-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"22059964934642cdc8f1278c1da35f42d26837d2f7e3e27f160063ae444515bd","family_id":"qf-surjective-table-property","seed":202610052020,"semantic_template":"qf-surjective-table-property/{\"A\":[-5,-4,-3,-2,-1,0],\"B\":[0,1,2,3,4],\"operation\":\"surjective\",\"pairs\":[[-5,0],[-4,1],[-3,2],[-2,3],[-1,4],[0,0]]}"},"fields":{"title":"Surjectivity of a Finite Total Function","stem":"The declared total function f:A→B has A={-5, -4, -3, -2, -1, 0}, B={0, 1, 2, 3, 4}, and table {(-5, 0), (-4, 1), (-3, 2), (-2, 3), (-1, 4), (0, 0)}. Is f surjective onto the declared codomain B?","answer":{"kind":"short_text","value":"Yes","explanation":"Check a preimage for every target in B, giving Yes. Multiple preimages are allowed, and the attained range is not automatically the declared codomain."},"skills":["Surjectivity of a Finite Total Function"],"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-20261004-advance-20261005-r02-surjective-0021","batch_id":"batch-20261004-advance-20261005-r02-surjective-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"eca086f487f1f7f263aa9719752cd72e7f9652b2a87e1c93709224871291957f","family_id":"qf-surjective-table-property","seed":202610052021,"semantic_template":"qf-surjective-table-property/{\"A\":[-5,-4,-3,-2,-1,0],\"B\":[-7,-6,-5,-4,-3,-2,-1],\"operation\":\"surjective\",\"pairs\":[[-5,-1],[-4,-1],[-3,-2],[-2,-2],[-1,-3],[0,-3]]}"},"fields":{"title":"Surjectivity of a Finite Total Function","stem":"The declared total function f:A→B has A={-5, -4, -3, -2, -1, 0}, B={-7, -6, -5, -4, -3, -2, -1}, and table {(-5, -1), (-4, -1), (-3, -2), (-2, -2), (-1, -3), (0, -3)}. Is f surjective onto the declared codomain B?","answer":{"kind":"short_text","value":"No","explanation":"Check a preimage for every target in B, giving No. Multiple preimages are allowed, and the attained range is not automatically the declared codomain."},"skills":["Surjectivity of a Finite Total Function"],"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-20261004-advance-20261005-r02-surjective-0022","batch_id":"batch-20261004-advance-20261005-r02-surjective-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0af13a626017b54a5deda4ebcbfea63049e32e2f9278a7e43f35f868aca5d310","family_id":"qf-surjective-table-property","seed":202610052022,"semantic_template":"qf-surjective-table-property/{\"A\":[-6,-5,-4,-3,-2,-1,0],\"B\":[0,1,2,3,4,5,6],\"operation\":\"surjective\",\"pairs\":[[-6,1],[-5,2],[-4,3],[-3,4],[-2,5],[-1,6],[0,0]]}"},"fields":{"title":"Surjectivity of a Finite Total Function","stem":"The declared total function f:A→B has A={-6, -5, -4, -3, -2, -1, 0}, B={0, 1, 2, 3, 4, 5, 6}, and table {(-6, 1), (-5, 2), (-4, 3), (-3, 4), (-2, 5), (-1, 6), (0, 0)}. Is f surjective onto the declared codomain B?","answer":{"kind":"short_text","value":"Yes","explanation":"Check a preimage for every target in B, giving Yes. Multiple preimages are allowed, and the attained range is not automatically the declared codomain."},"skills":["Surjectivity of a Finite Total Function"],"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-20261004-advance-20261005-r02-surjective-0023","batch_id":"batch-20261004-advance-20261005-r02-surjective-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"58a7388d44a373e540286c333474a1c94eca570d9c3a6b27f165c35bf5174b97","family_id":"qf-surjective-table-property","seed":202610052023,"semantic_template":"qf-surjective-table-property/{\"A\":[-6,-5,-4,-3,-2,-1,0],\"B\":[1,3,5,7,9,11,13,15],\"operation\":\"surjective\",\"pairs\":[[-6,1],[-5,3],[-4,5],[-3,7],[-2,9],[-1,11],[0,13]]}"},"fields":{"title":"Surjectivity of a Finite Total Function","stem":"The declared total function f:A→B has A={-6, -5, -4, -3, -2, -1, 0}, B={1, 3, 5, 7, 9, 11, 13, 15}, and table {(-6, 1), (-5, 3), (-4, 5), (-3, 7), (-2, 9), (-1, 11), (0, 13)}. Is f surjective onto the declared codomain B?","answer":{"kind":"short_text","value":"No","explanation":"Check a preimage for every target in B, giving No. Multiple preimages are allowed, and the attained range is not automatically the declared codomain."},"skills":["Surjectivity of a Finite Total Function"],"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-20261004-advance-20261005-r02-surjective-0024","batch_id":"batch-20261004-advance-20261005-r02-surjective-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7abc6c93526213e0642c60c7c779ebbd9775727c94dd8a36241d60f3b7048922","family_id":"qf-surjective-table-property","seed":202610052024,"semantic_template":"qf-surjective-table-property/{\"A\":[-6,-5,-4,-3,-2,-1,0],\"B\":[0,1,2,3,4,5],\"operation\":\"surjective\",\"pairs\":[[-6,0],[-5,1],[-4,2],[-3,3],[-2,4],[-1,5],[0,0]]}"},"fields":{"title":"Surjectivity of a Finite Total Function","stem":"The declared total function f:A→B has A={-6, -5, -4, -3, -2, -1, 0}, B={0, 1, 2, 3, 4, 5}, and table {(-6, 0), (-5, 1), (-4, 2), (-3, 3), (-2, 4), (-1, 5), (0, 0)}. Is f surjective onto the declared codomain B?","answer":{"kind":"short_text","value":"Yes","explanation":"Check a preimage for every target in B, giving Yes. Multiple preimages are allowed, and the attained range is not automatically the declared codomain."},"skills":["Surjectivity of a Finite Total Function"],"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-20261004-advance-20261005-r02-surjective-0025","batch_id":"batch-20261004-advance-20261005-r02-surjective-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"919a0aea627c35ce7fe6887c1b47ea3e14f9fd2b94027554b2e911fb00461a3d","family_id":"qf-surjective-table-property","seed":202610052025,"semantic_template":"qf-surjective-table-property/{\"A\":[-6,-5,-4,-3,-2,-1,0],\"B\":[-8,-7,-6,-5,-4,-3,-2,-1],\"operation\":\"surjective\",\"pairs\":[[-6,-1],[-5,-1],[-4,-2],[-3,-2],[-2,-3],[-1,-3],[0,-4]]}"},"fields":{"title":"Surjectivity of a Finite Total Function","stem":"The declared total function f:A→B has A={-6, -5, -4, -3, -2, -1, 0}, B={-8, -7, -6, -5, -4, -3, -2, -1}, and table {(-6, -1), (-5, -1), (-4, -2), (-3, -2), (-2, -3), (-1, -3), (0, -4)}. Is f surjective onto the declared codomain B?","answer":{"kind":"short_text","value":"No","explanation":"Check a preimage for every target in B, giving No. Multiple preimages are allowed, and the attained range is not automatically the declared codomain."},"skills":["Surjectivity of a Finite Total Function"],"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-20261004-advance-20261005-r02-surjective-0026","batch_id":"batch-20261004-advance-20261005-r02-surjective-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"233dd7bd168c61e1a9bc127f61a60338e5606e88a400107aeec671a0181b74e8","family_id":"qf-surjective-missing-targets","seed":202610052026,"semantic_template":"qf-surjective-missing-targets/{\"A\":[],\"B\":[],\"operation\":\"missing_targets\",\"pairs\":[]}"},"fields":{"title":"Uncovered Codomain Targets","stem":"The declared total function f:A→B has A=∅, B=∅, and table ∅. List the uncovered targets B∖f[A] in increasing order, using ∅ for the empty set.","answer":{"kind":"short_text","value":"∅","explanation":"Collect the attained outputs and remove them from the declared codomain. The uncovered set is ∅; any missing target defeats surjectivity."},"skills":["Uncovered Codomain Targets"],"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-20261004-advance-20261005-r02-surjective-0027","batch_id":"batch-20261004-advance-20261005-r02-surjective-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2f310c77bf77129c7b01c601cec420eb868449be018ff8ead1bf7c08af86b0b3","family_id":"qf-surjective-missing-targets","seed":202610052027,"semantic_template":"qf-surjective-missing-targets/{\"A\":[-1,0],\"B\":[0,1],\"operation\":\"missing_targets\",\"pairs\":[[-1,1],[0,0]]}"},"fields":{"title":"Uncovered Codomain Targets","stem":"The declared total function f:A→B has A={-1, 0}, B={0, 1}, and table {(-1, 1), (0, 0)}. List the uncovered targets B∖f[A] in increasing order, using ∅ for the empty set.","answer":{"kind":"short_text","value":"∅","explanation":"Collect the attained outputs and remove them from the declared codomain. The uncovered set is ∅; any missing target defeats surjectivity."},"skills":["Uncovered Codomain Targets"],"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-20261004-advance-20261005-r02-surjective-0028","batch_id":"batch-20261004-advance-20261005-r02-surjective-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a5c8b0ad330315857b834e5713723447a8d6e26e7e82580c80127c855abf6501","family_id":"qf-surjective-missing-targets","seed":202610052028,"semantic_template":"qf-surjective-missing-targets/{\"A\":[-1,0],\"B\":[1,3,5],\"operation\":\"missing_targets\",\"pairs\":[[-1,1],[0,3]]}"},"fields":{"title":"Uncovered Codomain Targets","stem":"The declared total function f:A→B has A={-1, 0}, B={1, 3, 5}, and table {(-1, 1), (0, 3)}. List the uncovered targets B∖f[A] in increasing order, using ∅ for the empty set.","answer":{"kind":"short_text","value":"{5}","explanation":"Collect the attained outputs and remove them from the declared codomain. The uncovered set is {5}; any missing target defeats surjectivity."},"skills":["Uncovered Codomain Targets"],"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-20261004-advance-20261005-r02-surjective-0029","batch_id":"batch-20261004-advance-20261005-r02-surjective-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ec458ff922650eddc1a9325cc6f82800758713e5807f3c7022163da35a506568","family_id":"qf-surjective-missing-targets","seed":202610052029,"semantic_template":"qf-surjective-missing-targets/{\"A\":[-1,0],\"B\":[0],\"operation\":\"missing_targets\",\"pairs\":[[-1,0],[0,0]]}"},"fields":{"title":"Uncovered Codomain Targets","stem":"The declared total function f:A→B has A={-1, 0}, B={0}, and table {(-1, 0), (0, 0)}. List the uncovered targets B∖f[A] in increasing order, using ∅ for the empty set.","answer":{"kind":"short_text","value":"∅","explanation":"Collect the attained outputs and remove them from the declared codomain. The uncovered set is ∅; any missing target defeats surjectivity."},"skills":["Uncovered Codomain Targets"],"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-20261004-advance-20261005-r02-surjective-0030","batch_id":"batch-20261004-advance-20261005-r02-surjective-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"69d924e0442637449d4ccc78dc8a8e0e9d1d923ae752db14751d1a5750fbf651","family_id":"qf-surjective-missing-targets","seed":202610052030,"semantic_template":"qf-surjective-missing-targets/{\"A\":[-1,0],\"B\":[-3,-2,-1],\"operation\":\"missing_targets\",\"pairs\":[[-1,-1],[0,-1]]}"},"fields":{"title":"Uncovered Codomain Targets","stem":"The declared total function f:A→B has A={-1, 0}, B={-3, -2, -1}, and table {(-1, -1), (0, -1)}. List the uncovered targets B∖f[A] in increasing order, using ∅ for the empty set.","answer":{"kind":"short_text","value":"{-3, -2}","explanation":"Collect the attained outputs and remove them from the declared codomain. The uncovered set is {-3, -2}; any missing target defeats surjectivity."},"skills":["Uncovered Codomain Targets"],"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-20261004-advance-20261005-r02-surjective-0031","batch_id":"batch-20261004-advance-20261005-r02-surjective-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bdbea419339f347b87642d39a26cca57505b5a7cb6b720730e5722e1d503f90b","family_id":"qf-surjective-missing-targets","seed":202610052031,"semantic_template":"qf-surjective-missing-targets/{\"A\":[-2,-1,0],\"B\":[0,1,2],\"operation\":\"missing_targets\",\"pairs\":[[-2,1],[-1,2],[0,0]]}"},"fields":{"title":"Uncovered Codomain Targets","stem":"The declared total function f:A→B has A={-2, -1, 0}, B={0, 1, 2}, and table {(-2, 1), (-1, 2), (0, 0)}. List the uncovered targets B∖f[A] in increasing order, using ∅ for the empty set.","answer":{"kind":"short_text","value":"∅","explanation":"Collect the attained outputs and remove them from the declared codomain. The uncovered set is ∅; any missing target defeats surjectivity."},"skills":["Uncovered Codomain Targets"],"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-20261004-advance-20261005-r02-surjective-0032","batch_id":"batch-20261004-advance-20261005-r02-surjective-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"745ad9e2228ce6adfdc9eb20607eab13fb7b74391a6a64960a52fa40bb289463","family_id":"qf-surjective-missing-targets","seed":202610052032,"semantic_template":"qf-surjective-missing-targets/{\"A\":[-2,-1,0],\"B\":[1,3,5,7],\"operation\":\"missing_targets\",\"pairs\":[[-2,1],[-1,3],[0,5]]}"},"fields":{"title":"Uncovered Codomain Targets","stem":"The declared total function f:A→B has A={-2, -1, 0}, B={1, 3, 5, 7}, and table {(-2, 1), (-1, 3), (0, 5)}. List the uncovered targets B∖f[A] in increasing order, using ∅ for the empty set.","answer":{"kind":"short_text","value":"{7}","explanation":"Collect the attained outputs and remove them from the declared codomain. The uncovered set is {7}; any missing target defeats surjectivity."},"skills":["Uncovered Codomain Targets"],"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-20261004-advance-20261005-r02-surjective-0033","batch_id":"batch-20261004-advance-20261005-r02-surjective-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cd08d8b38c19cf2b2cec6e8e74b3c7e814e466a31020eea955d9768a651b17c7","family_id":"qf-surjective-missing-targets","seed":202610052033,"semantic_template":"qf-surjective-missing-targets/{\"A\":[-2,-1,0],\"B\":[0,1],\"operation\":\"missing_targets\",\"pairs\":[[-2,0],[-1,1],[0,0]]}"},"fields":{"title":"Uncovered Codomain Targets","stem":"The declared total function f:A→B has A={-2, -1, 0}, B={0, 1}, and table {(-2, 0), (-1, 1), (0, 0)}. List the uncovered targets B∖f[A] in increasing order, using ∅ for the empty set.","answer":{"kind":"short_text","value":"∅","explanation":"Collect the attained outputs and remove them from the declared codomain. The uncovered set is ∅; any missing target defeats surjectivity."},"skills":["Uncovered Codomain Targets"],"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-20261004-advance-20261005-r02-surjective-0034","batch_id":"batch-20261004-advance-20261005-r02-surjective-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3412e673ba752430457917838719b8ef5d63df2d6873c98655ee256ebe63f5c6","family_id":"qf-surjective-missing-targets","seed":202610052034,"semantic_template":"qf-surjective-missing-targets/{\"A\":[-2,-1,0],\"B\":[-4,-3,-2,-1],\"operation\":\"missing_targets\",\"pairs\":[[-2,-1],[-1,-1],[0,-2]]}"},"fields":{"title":"Uncovered Codomain Targets","stem":"The declared total function f:A→B has A={-2, -1, 0}, B={-4, -3, -2, -1}, and table {(-2, -1), (-1, -1), (0, -2)}. List the uncovered targets B∖f[A] in increasing order, using ∅ for the empty set.","answer":{"kind":"short_text","value":"{-4, -3}","explanation":"Collect the attained outputs and remove them from the declared codomain. The uncovered set is {-4, -3}; any missing target defeats surjectivity."},"skills":["Uncovered Codomain Targets"],"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-20261004-advance-20261005-r02-surjective-0035","batch_id":"batch-20261004-advance-20261005-r02-surjective-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f7f1cda6b585b4530684e392361d058ac20f40de5991f71ef9d483d2617a5198","family_id":"qf-surjective-missing-targets","seed":202610052035,"semantic_template":"qf-surjective-missing-targets/{\"A\":[-3,-2,-1,0],\"B\":[0,1,2,3],\"operation\":\"missing_targets\",\"pairs\":[[-3,1],[-2,2],[-1,3],[0,0]]}"},"fields":{"title":"Uncovered Codomain Targets","stem":"The declared total function f:A→B has A={-3, -2, -1, 0}, B={0, 1, 2, 3}, and table {(-3, 1), (-2, 2), (-1, 3), (0, 0)}. List the uncovered targets B∖f[A] in increasing order, using ∅ for the empty set.","answer":{"kind":"short_text","value":"∅","explanation":"Collect the attained outputs and remove them from the declared codomain. The uncovered set is ∅; any missing target defeats surjectivity."},"skills":["Uncovered Codomain Targets"],"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-20261004-advance-20261005-r02-surjective-0036","batch_id":"batch-20261004-advance-20261005-r02-surjective-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7771782577d07bfd2b98d8a2acf8a9a502203d68447e866d62aa003f3d7ef8d6","family_id":"qf-surjective-missing-targets","seed":202610052036,"semantic_template":"qf-surjective-missing-targets/{\"A\":[-3,-2,-1,0],\"B\":[1,3,5,7,9],\"operation\":\"missing_targets\",\"pairs\":[[-3,1],[-2,3],[-1,5],[0,7]]}"},"fields":{"title":"Uncovered Codomain Targets","stem":"The declared total function f:A→B has A={-3, -2, -1, 0}, B={1, 3, 5, 7, 9}, and table {(-3, 1), (-2, 3), (-1, 5), (0, 7)}. List the uncovered targets B∖f[A] in increasing order, using ∅ for the empty set.","answer":{"kind":"short_text","value":"{9}","explanation":"Collect the attained outputs and remove them from the declared codomain. The uncovered set is {9}; any missing target defeats surjectivity."},"skills":["Uncovered Codomain Targets"],"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-20261004-advance-20261005-r02-surjective-0037","batch_id":"batch-20261004-advance-20261005-r02-surjective-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2cb9930653541be0fd8235ab7cc3ddbb2e33df209de13eea4c8355cf7666aded","family_id":"qf-surjective-missing-targets","seed":202610052037,"semantic_template":"qf-surjective-missing-targets/{\"A\":[-3,-2,-1,0],\"B\":[0,1,2],\"operation\":\"missing_targets\",\"pairs\":[[-3,0],[-2,1],[-1,2],[0,0]]}"},"fields":{"title":"Uncovered Codomain Targets","stem":"The declared total function f:A→B has A={-3, -2, -1, 0}, B={0, 1, 2}, and table {(-3, 0), (-2, 1), (-1, 2), (0, 0)}. List the uncovered targets B∖f[A] in increasing order, using ∅ for the empty set.","answer":{"kind":"short_text","value":"∅","explanation":"Collect the attained outputs and remove them from the declared codomain. The uncovered set is ∅; any missing target defeats surjectivity."},"skills":["Uncovered Codomain Targets"],"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-20261004-advance-20261005-r02-surjective-0038","batch_id":"batch-20261004-advance-20261005-r02-surjective-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1d2c940e9c6783a8b5c1d8f0b5d09e2908e2994a8e9f4deaac2c02a37b382b40","family_id":"qf-surjective-missing-targets","seed":202610052038,"semantic_template":"qf-surjective-missing-targets/{\"A\":[-3,-2,-1,0],\"B\":[-5,-4,-3,-2,-1],\"operation\":\"missing_targets\",\"pairs\":[[-3,-1],[-2,-1],[-1,-2],[0,-2]]}"},"fields":{"title":"Uncovered Codomain Targets","stem":"The declared total function f:A→B has A={-3, -2, -1, 0}, B={-5, -4, -3, -2, -1}, and table {(-3, -1), (-2, -1), (-1, -2), (0, -2)}. List the uncovered targets B∖f[A] in increasing order, using ∅ for the empty set.","answer":{"kind":"short_text","value":"{-5, -4, -3}","explanation":"Collect the attained outputs and remove them from the declared codomain. The uncovered set is {-5, -4, -3}; any missing target defeats surjectivity."},"skills":["Uncovered Codomain Targets"],"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-20261004-advance-20261005-r02-surjective-0039","batch_id":"batch-20261004-advance-20261005-r02-surjective-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f45d867ea8093eabd1fd8824e54da0191f39903f3552276657afab302689d11b","family_id":"qf-surjective-missing-targets","seed":202610052039,"semantic_template":"qf-surjective-missing-targets/{\"A\":[-4,-3,-2,-1,0],\"B\":[0,1,2,3,4],\"operation\":\"missing_targets\",\"pairs\":[[-4,1],[-3,2],[-2,3],[-1,4],[0,0]]}"},"fields":{"title":"Uncovered Codomain Targets","stem":"The declared total function f:A→B has A={-4, -3, -2, -1, 0}, B={0, 1, 2, 3, 4}, and table {(-4, 1), (-3, 2), (-2, 3), (-1, 4), (0, 0)}. List the uncovered targets B∖f[A] in increasing order, using ∅ for the empty set.","answer":{"kind":"short_text","value":"∅","explanation":"Collect the attained outputs and remove them from the declared codomain. The uncovered set is ∅; any missing target defeats surjectivity."},"skills":["Uncovered Codomain Targets"],"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-20261004-advance-20261005-r02-surjective-0040","batch_id":"batch-20261004-advance-20261005-r02-surjective-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"44a09aa2291fb2594218e148c10984fba57b8bab041e9278c48e5928d20e9670","family_id":"qf-surjective-missing-targets","seed":202610052040,"semantic_template":"qf-surjective-missing-targets/{\"A\":[-4,-3,-2,-1,0],\"B\":[1,3,5,7,9,11],\"operation\":\"missing_targets\",\"pairs\":[[-4,1],[-3,3],[-2,5],[-1,7],[0,9]]}"},"fields":{"title":"Uncovered Codomain Targets","stem":"The declared total function f:A→B has A={-4, -3, -2, -1, 0}, B={1, 3, 5, 7, 9, 11}, and table {(-4, 1), (-3, 3), (-2, 5), (-1, 7), (0, 9)}. List the uncovered targets B∖f[A] in increasing order, using ∅ for the empty set.","answer":{"kind":"short_text","value":"{11}","explanation":"Collect the attained outputs and remove them from the declared codomain. The uncovered set is {11}; any missing target defeats surjectivity."},"skills":["Uncovered Codomain Targets"],"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-20261004-advance-20261005-r02-surjective-0041","batch_id":"batch-20261004-advance-20261005-r02-surjective-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4f8a36878fef5b8a86f3fcf6ee5aa04d06af83a4e5fcd37fcf75d6ef60e71545","family_id":"qf-surjective-missing-targets","seed":202610052041,"semantic_template":"qf-surjective-missing-targets/{\"A\":[-4,-3,-2,-1,0],\"B\":[0,1,2,3],\"operation\":\"missing_targets\",\"pairs\":[[-4,0],[-3,1],[-2,2],[-1,3],[0,0]]}"},"fields":{"title":"Uncovered Codomain Targets","stem":"The declared total function f:A→B has A={-4, -3, -2, -1, 0}, B={0, 1, 2, 3}, and table {(-4, 0), (-3, 1), (-2, 2), (-1, 3), (0, 0)}. List the uncovered targets B∖f[A] in increasing order, using ∅ for the empty set.","answer":{"kind":"short_text","value":"∅","explanation":"Collect the attained outputs and remove them from the declared codomain. The uncovered set is ∅; any missing target defeats surjectivity."},"skills":["Uncovered Codomain Targets"],"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-20261004-advance-20261005-r02-surjective-0042","batch_id":"batch-20261004-advance-20261005-r02-surjective-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b0113d23b8c6631e785c65491e12e33f046387995cba6e84c137f2d79f7c120c","family_id":"qf-surjective-missing-targets","seed":202610052042,"semantic_template":"qf-surjective-missing-targets/{\"A\":[-4,-3,-2,-1,0],\"B\":[-6,-5,-4,-3,-2,-1],\"operation\":\"missing_targets\",\"pairs\":[[-4,-1],[-3,-1],[-2,-2],[-1,-2],[0,-3]]}"},"fields":{"title":"Uncovered Codomain Targets","stem":"The declared total function f:A→B has A={-4, -3, -2, -1, 0}, B={-6, -5, -4, -3, -2, -1}, and table {(-4, -1), (-3, -1), (-2, -2), (-1, -2), (0, -3)}. List the uncovered targets B∖f[A] in increasing order, using ∅ for the empty set.","answer":{"kind":"short_text","value":"{-6, -5, -4}","explanation":"Collect the attained outputs and remove them from the declared codomain. The uncovered set is {-6, -5, -4}; any missing target defeats surjectivity."},"skills":["Uncovered Codomain Targets"],"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-20261004-advance-20261005-r02-surjective-0043","batch_id":"batch-20261004-advance-20261005-r02-surjective-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f76068a33a7c353cdb8eea5a8a91200c8edbfe1713dc716bbf85287a98783d4b","family_id":"qf-surjective-missing-targets","seed":202610052043,"semantic_template":"qf-surjective-missing-targets/{\"A\":[-5,-4,-3,-2,-1,0],\"B\":[0,1,2,3,4,5],\"operation\":\"missing_targets\",\"pairs\":[[-5,1],[-4,2],[-3,3],[-2,4],[-1,5],[0,0]]}"},"fields":{"title":"Uncovered Codomain Targets","stem":"The declared total function f:A→B has A={-5, -4, -3, -2, -1, 0}, B={0, 1, 2, 3, 4, 5}, and table {(-5, 1), (-4, 2), (-3, 3), (-2, 4), (-1, 5), (0, 0)}. List the uncovered targets B∖f[A] in increasing order, using ∅ for the empty set.","answer":{"kind":"short_text","value":"∅","explanation":"Collect the attained outputs and remove them from the declared codomain. The uncovered set is ∅; any missing target defeats surjectivity."},"skills":["Uncovered Codomain Targets"],"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-20261004-advance-20261005-r02-surjective-0044","batch_id":"batch-20261004-advance-20261005-r02-surjective-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"133ec2820f24a6627f4dae39dd407b6bf9aaa2358317664361a0839efa406a68","family_id":"qf-surjective-missing-targets","seed":202610052044,"semantic_template":"qf-surjective-missing-targets/{\"A\":[-5,-4,-3,-2,-1,0],\"B\":[1,3,5,7,9,11,13],\"operation\":\"missing_targets\",\"pairs\":[[-5,1],[-4,3],[-3,5],[-2,7],[-1,9],[0,11]]}"},"fields":{"title":"Uncovered Codomain Targets","stem":"The declared total function f:A→B has A={-5, -4, -3, -2, -1, 0}, B={1, 3, 5, 7, 9, 11, 13}, and table {(-5, 1), (-4, 3), (-3, 5), (-2, 7), (-1, 9), (0, 11)}. List the uncovered targets B∖f[A] in increasing order, using ∅ for the empty set.","answer":{"kind":"short_text","value":"{13}","explanation":"Collect the attained outputs and remove them from the declared codomain. The uncovered set is {13}; any missing target defeats surjectivity."},"skills":["Uncovered Codomain Targets"],"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-20261004-advance-20261005-r02-surjective-0045","batch_id":"batch-20261004-advance-20261005-r02-surjective-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ebb29971b919a8bf6f153d3a40388c876edcb438f2c34369754b838798efe4e1","family_id":"qf-surjective-missing-targets","seed":202610052045,"semantic_template":"qf-surjective-missing-targets/{\"A\":[-5,-4,-3,-2,-1,0],\"B\":[0,1,2,3,4],\"operation\":\"missing_targets\",\"pairs\":[[-5,0],[-4,1],[-3,2],[-2,3],[-1,4],[0,0]]}"},"fields":{"title":"Uncovered Codomain Targets","stem":"The declared total function f:A→B has A={-5, -4, -3, -2, -1, 0}, B={0, 1, 2, 3, 4}, and table {(-5, 0), (-4, 1), (-3, 2), (-2, 3), (-1, 4), (0, 0)}. List the uncovered targets B∖f[A] in increasing order, using ∅ for the empty set.","answer":{"kind":"short_text","value":"∅","explanation":"Collect the attained outputs and remove them from the declared codomain. The uncovered set is ∅; any missing target defeats surjectivity."},"skills":["Uncovered Codomain Targets"],"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-20261004-advance-20261005-r02-surjective-0046","batch_id":"batch-20261004-advance-20261005-r02-surjective-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0e4619e5842f03f88c7b2c818fb06f64ee803fd1db920875a2a7dcfb8f04aeaa","family_id":"qf-surjective-missing-targets","seed":202610052046,"semantic_template":"qf-surjective-missing-targets/{\"A\":[-5,-4,-3,-2,-1,0],\"B\":[-7,-6,-5,-4,-3,-2,-1],\"operation\":\"missing_targets\",\"pairs\":[[-5,-1],[-4,-1],[-3,-2],[-2,-2],[-1,-3],[0,-3]]}"},"fields":{"title":"Uncovered Codomain Targets","stem":"The declared total function f:A→B has A={-5, -4, -3, -2, -1, 0}, B={-7, -6, -5, -4, -3, -2, -1}, and table {(-5, -1), (-4, -1), (-3, -2), (-2, -2), (-1, -3), (0, -3)}. List the uncovered targets B∖f[A] in increasing order, using ∅ for the empty set.","answer":{"kind":"short_text","value":"{-7, -6, -5, -4}","explanation":"Collect the attained outputs and remove them from the declared codomain. The uncovered set is {-7, -6, -5, -4}; any missing target defeats surjectivity."},"skills":["Uncovered Codomain Targets"],"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-20261004-advance-20261005-r02-surjective-0047","batch_id":"batch-20261004-advance-20261005-r02-surjective-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ff752c9f097d8494d9790ea1ce41c34fbf1e66109bbe790a8f1c554f9177d203","family_id":"qf-surjective-missing-targets","seed":202610052047,"semantic_template":"qf-surjective-missing-targets/{\"A\":[-6,-5,-4,-3,-2,-1,0],\"B\":[0,1,2,3,4,5,6],\"operation\":\"missing_targets\",\"pairs\":[[-6,1],[-5,2],[-4,3],[-3,4],[-2,5],[-1,6],[0,0]]}"},"fields":{"title":"Uncovered Codomain Targets","stem":"The declared total function f:A→B has A={-6, -5, -4, -3, -2, -1, 0}, B={0, 1, 2, 3, 4, 5, 6}, and table {(-6, 1), (-5, 2), (-4, 3), (-3, 4), (-2, 5), (-1, 6), (0, 0)}. List the uncovered targets B∖f[A] in increasing order, using ∅ for the empty set.","answer":{"kind":"short_text","value":"∅","explanation":"Collect the attained outputs and remove them from the declared codomain. The uncovered set is ∅; any missing target defeats surjectivity."},"skills":["Uncovered Codomain Targets"],"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-20261004-advance-20261005-r02-surjective-0048","batch_id":"batch-20261004-advance-20261005-r02-surjective-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"de2e3b0cc88eb04c280fd6405e057639168768ddf80e44ea06cee1d871c85109","family_id":"qf-surjective-missing-targets","seed":202610052048,"semantic_template":"qf-surjective-missing-targets/{\"A\":[-6,-5,-4,-3,-2,-1,0],\"B\":[1,3,5,7,9,11,13,15],\"operation\":\"missing_targets\",\"pairs\":[[-6,1],[-5,3],[-4,5],[-3,7],[-2,9],[-1,11],[0,13]]}"},"fields":{"title":"Uncovered Codomain Targets","stem":"The declared total function f:A→B has A={-6, -5, -4, -3, -2, -1, 0}, B={1, 3, 5, 7, 9, 11, 13, 15}, and table {(-6, 1), (-5, 3), (-4, 5), (-3, 7), (-2, 9), (-1, 11), (0, 13)}. List the uncovered targets B∖f[A] in increasing order, using ∅ for the empty set.","answer":{"kind":"short_text","value":"{15}","explanation":"Collect the attained outputs and remove them from the declared codomain. The uncovered set is {15}; any missing target defeats surjectivity."},"skills":["Uncovered Codomain Targets"],"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-20261004-advance-20261005-r02-surjective-0049","batch_id":"batch-20261004-advance-20261005-r02-surjective-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bd68b749377d862d644a5e4fcfde2566c417080073bfac694d607c7aa1d66033","family_id":"qf-surjective-missing-targets","seed":202610052049,"semantic_template":"qf-surjective-missing-targets/{\"A\":[-6,-5,-4,-3,-2,-1,0],\"B\":[0,1,2,3,4,5],\"operation\":\"missing_targets\",\"pairs\":[[-6,0],[-5,1],[-4,2],[-3,3],[-2,4],[-1,5],[0,0]]}"},"fields":{"title":"Uncovered Codomain Targets","stem":"The declared total function f:A→B has A={-6, -5, -4, -3, -2, -1, 0}, B={0, 1, 2, 3, 4, 5}, and table {(-6, 0), (-5, 1), (-4, 2), (-3, 3), (-2, 4), (-1, 5), (0, 0)}. List the uncovered targets B∖f[A] in increasing order, using ∅ for the empty set.","answer":{"kind":"short_text","value":"∅","explanation":"Collect the attained outputs and remove them from the declared codomain. The uncovered set is ∅; any missing target defeats surjectivity."},"skills":["Uncovered Codomain Targets"],"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-20261004-advance-20261005-r02-surjective-0050","batch_id":"batch-20261004-advance-20261005-r02-surjective-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e0043429fd04b8d603052504a203752495687721d9236e019667cca071c19c20","family_id":"qf-surjective-missing-targets","seed":202610052050,"semantic_template":"qf-surjective-missing-targets/{\"A\":[-6,-5,-4,-3,-2,-1,0],\"B\":[-8,-7,-6,-5,-4,-3,-2,-1],\"operation\":\"missing_targets\",\"pairs\":[[-6,-1],[-5,-1],[-4,-2],[-3,-2],[-2,-3],[-1,-3],[0,-4]]}"},"fields":{"title":"Uncovered Codomain Targets","stem":"The declared total function f:A→B has A={-6, -5, -4, -3, -2, -1, 0}, B={-8, -7, -6, -5, -4, -3, -2, -1}, and table {(-6, -1), (-5, -1), (-4, -2), (-3, -2), (-2, -3), (-1, -3), (0, -4)}. List the uncovered targets B∖f[A] in increasing order, using ∅ for the empty set.","answer":{"kind":"short_text","value":"{-8, -7, -6, -5}","explanation":"Collect the attained outputs and remove them from the declared codomain. The uncovered set is {-8, -7, -6, -5}; any missing target defeats surjectivity."},"skills":["Uncovered Codomain Targets"],"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-20261004-advance-20261005-r02-surjective-0051","batch_id":"batch-20261004-advance-20261005-r02-surjective-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"51867c0db133504661aeb165fcd08618317d63130cab9938926cde96a485c435","family_id":"qf-surjective-first-preimage","seed":202610052051,"semantic_template":"qf-surjective-first-preimage/{\"A\":[],\"B\":[0],\"operation\":\"first_preimage\",\"pairs\":[],\"t\":0}"},"fields":{"title":"Least Permitted Preimage of a Target","stem":"The declared total function f:A→B has A=∅, B={0}, and table ∅. For t=0∈B, find its numerically least permitted preimage, answering none when there is no preimage.","answer":{"kind":"short_text","value":"none","explanation":"Enumerate the full fiber and select its least input if it is nonempty. The answer is none. Existence of a preimage does not imply uniqueness."},"skills":["Least Permitted Preimage of a Target"],"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-20261004-advance-20261005-r02-surjective-0052","batch_id":"batch-20261004-advance-20261005-r02-surjective-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e461e36be759ec3aeac00ffe9c22f2b04926f7fd8f749dadadb183bf402b88ba","family_id":"qf-surjective-first-preimage","seed":202610052052,"semantic_template":"qf-surjective-first-preimage/{\"A\":[-1,0],\"B\":[0,1],\"operation\":\"first_preimage\",\"pairs\":[[-1,1],[0,0]],\"t\":1}"},"fields":{"title":"Least Permitted Preimage of a Target","stem":"The declared total function f:A→B has A={-1, 0}, B={0, 1}, and table {(-1, 1), (0, 0)}. For t=1∈B, find its numerically least permitted preimage, answering none when there is no preimage.","answer":{"kind":"short_text","value":"-1","explanation":"Enumerate the full fiber and select its least input if it is nonempty. The answer is -1. Existence of a preimage does not imply uniqueness."},"skills":["Least Permitted Preimage of a Target"],"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-20261004-advance-20261005-r02-surjective-0053","batch_id":"batch-20261004-advance-20261005-r02-surjective-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fdd8717d898580d10f3ce9b931fec95f824536cdf538925e6fb3a74789ee3c91","family_id":"qf-surjective-first-preimage","seed":202610052053,"semantic_template":"qf-surjective-first-preimage/{\"A\":[-1,0],\"B\":[1,3,5],\"operation\":\"first_preimage\",\"pairs\":[[-1,1],[0,3]],\"t\":5}"},"fields":{"title":"Least Permitted Preimage of a Target","stem":"The declared total function f:A→B has A={-1, 0}, B={1, 3, 5}, and table {(-1, 1), (0, 3)}. For t=5∈B, find its numerically least permitted preimage, answering none when there is no preimage.","answer":{"kind":"short_text","value":"none","explanation":"Enumerate the full fiber and select its least input if it is nonempty. The answer is none. Existence of a preimage does not imply uniqueness."},"skills":["Least Permitted Preimage of a Target"],"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-20261004-advance-20261005-r02-surjective-0054","batch_id":"batch-20261004-advance-20261005-r02-surjective-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0a38f8fe7981869c46839a5f554cc988f5589da587208378b489d10205325015","family_id":"qf-surjective-first-preimage","seed":202610052054,"semantic_template":"qf-surjective-first-preimage/{\"A\":[-1,0],\"B\":[0],\"operation\":\"first_preimage\",\"pairs\":[[-1,0],[0,0]],\"t\":0}"},"fields":{"title":"Least Permitted Preimage of a Target","stem":"The declared total function f:A→B has A={-1, 0}, B={0}, and table {(-1, 0), (0, 0)}. For t=0∈B, find its numerically least permitted preimage, answering none when there is no preimage.","answer":{"kind":"short_text","value":"-1","explanation":"Enumerate the full fiber and select its least input if it is nonempty. The answer is -1. Existence of a preimage does not imply uniqueness."},"skills":["Least Permitted Preimage of a Target"],"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-20261004-advance-20261005-r02-surjective-0055","batch_id":"batch-20261004-advance-20261005-r02-surjective-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f789ce6b57ddc42c9c9c8f62e7a0f9b0fddf78d432b87fe69ad7f6c654f1262e","family_id":"qf-surjective-first-preimage","seed":202610052055,"semantic_template":"qf-surjective-first-preimage/{\"A\":[-1,0],\"B\":[-3,-2,-1],\"operation\":\"first_preimage\",\"pairs\":[[-1,-1],[0,-1]],\"t\":-2}"},"fields":{"title":"Least Permitted Preimage of a Target","stem":"The declared total function f:A→B has A={-1, 0}, B={-3, -2, -1}, and table {(-1, -1), (0, -1)}. For t=-2∈B, find its numerically least permitted preimage, answering none when there is no preimage.","answer":{"kind":"short_text","value":"none","explanation":"Enumerate the full fiber and select its least input if it is nonempty. The answer is none. Existence of a preimage does not imply uniqueness."},"skills":["Least Permitted Preimage of a Target"],"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-20261004-advance-20261005-r02-surjective-0056","batch_id":"batch-20261004-advance-20261005-r02-surjective-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9f90bca157c0df0cd82785d4403d477471772e233b1b930297974585872c68bd","family_id":"qf-surjective-first-preimage","seed":202610052056,"semantic_template":"qf-surjective-first-preimage/{\"A\":[-2,-1,0],\"B\":[0,1,2],\"operation\":\"first_preimage\",\"pairs\":[[-2,1],[-1,2],[0,0]],\"t\":2}"},"fields":{"title":"Least Permitted Preimage of a Target","stem":"The declared total function f:A→B has A={-2, -1, 0}, B={0, 1, 2}, and table {(-2, 1), (-1, 2), (0, 0)}. For t=2∈B, find its numerically least permitted preimage, answering none when there is no preimage.","answer":{"kind":"short_text","value":"-1","explanation":"Enumerate the full fiber and select its least input if it is nonempty. The answer is -1. Existence of a preimage does not imply uniqueness."},"skills":["Least Permitted Preimage of a Target"],"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-20261004-advance-20261005-r02-surjective-0057","batch_id":"batch-20261004-advance-20261005-r02-surjective-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4530af343f21f2b80b38037c8e751b7acd8b57ea32f25dad49255d24432065e6","family_id":"qf-surjective-first-preimage","seed":202610052057,"semantic_template":"qf-surjective-first-preimage/{\"A\":[-2,-1,0],\"B\":[1,3,5,7],\"operation\":\"first_preimage\",\"pairs\":[[-2,1],[-1,3],[0,5]],\"t\":7}"},"fields":{"title":"Least Permitted Preimage of a Target","stem":"The declared total function f:A→B has A={-2, -1, 0}, B={1, 3, 5, 7}, and table {(-2, 1), (-1, 3), (0, 5)}. For t=7∈B, find its numerically least permitted preimage, answering none when there is no preimage.","answer":{"kind":"short_text","value":"none","explanation":"Enumerate the full fiber and select its least input if it is nonempty. The answer is none. Existence of a preimage does not imply uniqueness."},"skills":["Least Permitted Preimage of a Target"],"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-20261004-advance-20261005-r02-surjective-0058","batch_id":"batch-20261004-advance-20261005-r02-surjective-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b71290710a6d0a8bb26dc8811819b8c6983f2ea92a37b8c4a2646d31fb32ba3d","family_id":"qf-surjective-first-preimage","seed":202610052058,"semantic_template":"qf-surjective-first-preimage/{\"A\":[-2,-1,0],\"B\":[0,1],\"operation\":\"first_preimage\",\"pairs\":[[-2,0],[-1,1],[0,0]],\"t\":1}"},"fields":{"title":"Least Permitted Preimage of a Target","stem":"The declared total function f:A→B has A={-2, -1, 0}, B={0, 1}, and table {(-2, 0), (-1, 1), (0, 0)}. For t=1∈B, find its numerically least permitted preimage, answering none when there is no preimage.","answer":{"kind":"short_text","value":"-1","explanation":"Enumerate the full fiber and select its least input if it is nonempty. The answer is -1. Existence of a preimage does not imply uniqueness."},"skills":["Least Permitted Preimage of a Target"],"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-20261004-advance-20261005-r02-surjective-0059","batch_id":"batch-20261004-advance-20261005-r02-surjective-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8a5dfd8c6b4742bb8bc7fae00e6296a530b39d09d15b695439001fa3490340fa","family_id":"qf-surjective-first-preimage","seed":202610052059,"semantic_template":"qf-surjective-first-preimage/{\"A\":[-2,-1,0],\"B\":[-4,-3,-2,-1],\"operation\":\"first_preimage\",\"pairs\":[[-2,-1],[-1,-1],[0,-2]],\"t\":-4}"},"fields":{"title":"Least Permitted Preimage of a Target","stem":"The declared total function f:A→B has A={-2, -1, 0}, B={-4, -3, -2, -1}, and table {(-2, -1), (-1, -1), (0, -2)}. For t=-4∈B, find its numerically least permitted preimage, answering none when there is no preimage.","answer":{"kind":"short_text","value":"none","explanation":"Enumerate the full fiber and select its least input if it is nonempty. The answer is none. Existence of a preimage does not imply uniqueness."},"skills":["Least Permitted Preimage of a Target"],"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-20261004-advance-20261005-r02-surjective-0060","batch_id":"batch-20261004-advance-20261005-r02-surjective-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f5ddaaa378c3fe51fa9bec2ef9f19b10180ba19af6582ed96f2ecdc10fc133a2","family_id":"qf-surjective-first-preimage","seed":202610052060,"semantic_template":"qf-surjective-first-preimage/{\"A\":[-3,-2,-1,0],\"B\":[0,1,2,3],\"operation\":\"first_preimage\",\"pairs\":[[-3,1],[-2,2],[-1,3],[0,0]],\"t\":1}"},"fields":{"title":"Least Permitted Preimage of a Target","stem":"The declared total function f:A→B has A={-3, -2, -1, 0}, B={0, 1, 2, 3}, and table {(-3, 1), (-2, 2), (-1, 3), (0, 0)}. For t=1∈B, find its numerically least permitted preimage, answering none when there is no preimage.","answer":{"kind":"short_text","value":"-3","explanation":"Enumerate the full fiber and select its least input if it is nonempty. The answer is -3. Existence of a preimage does not imply uniqueness."},"skills":["Least Permitted Preimage of a Target"],"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-20261004-advance-20261005-r02-surjective-0061","batch_id":"batch-20261004-advance-20261005-r02-surjective-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"28a49448b4190dae586b0af42a832fff418d306ce059099016cc45ac2d3c31fb","family_id":"qf-surjective-first-preimage","seed":202610052061,"semantic_template":"qf-surjective-first-preimage/{\"A\":[-3,-2,-1,0],\"B\":[1,3,5,7,9],\"operation\":\"first_preimage\",\"pairs\":[[-3,1],[-2,3],[-1,5],[0,7]],\"t\":9}"},"fields":{"title":"Least Permitted Preimage of a Target","stem":"The declared total function f:A→B has A={-3, -2, -1, 0}, B={1, 3, 5, 7, 9}, and table {(-3, 1), (-2, 3), (-1, 5), (0, 7)}. For t=9∈B, find its numerically least permitted preimage, answering none when there is no preimage.","answer":{"kind":"short_text","value":"none","explanation":"Enumerate the full fiber and select its least input if it is nonempty. The answer is none. Existence of a preimage does not imply uniqueness."},"skills":["Least Permitted Preimage of a Target"],"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-20261004-advance-20261005-r02-surjective-0062","batch_id":"batch-20261004-advance-20261005-r02-surjective-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2c1d22408065a25fc940a271a24bec6f8795641fe94d7335b42be9f0f91dbdd1","family_id":"qf-surjective-first-preimage","seed":202610052062,"semantic_template":"qf-surjective-first-preimage/{\"A\":[-3,-2,-1,0],\"B\":[0,1,2],\"operation\":\"first_preimage\",\"pairs\":[[-3,0],[-2,1],[-1,2],[0,0]],\"t\":2}"},"fields":{"title":"Least Permitted Preimage of a Target","stem":"The declared total function f:A→B has A={-3, -2, -1, 0}, B={0, 1, 2}, and table {(-3, 0), (-2, 1), (-1, 2), (0, 0)}. For t=2∈B, find its numerically least permitted preimage, answering none when there is no preimage.","answer":{"kind":"short_text","value":"-1","explanation":"Enumerate the full fiber and select its least input if it is nonempty. The answer is -1. Existence of a preimage does not imply uniqueness."},"skills":["Least Permitted Preimage of a Target"],"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-20261004-advance-20261005-r02-surjective-0063","batch_id":"batch-20261004-advance-20261005-r02-surjective-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7ae9d17b4f416b6ae8c72fbc7f84ff7abc810d42944eb2660892fd23a36e4c54","family_id":"qf-surjective-first-preimage","seed":202610052063,"semantic_template":"qf-surjective-first-preimage/{\"A\":[-3,-2,-1,0],\"B\":[-5,-4,-3,-2,-1],\"operation\":\"first_preimage\",\"pairs\":[[-3,-1],[-2,-1],[-1,-2],[0,-2]],\"t\":-3}"},"fields":{"title":"Least Permitted Preimage of a Target","stem":"The declared total function f:A→B has A={-3, -2, -1, 0}, B={-5, -4, -3, -2, -1}, and table {(-3, -1), (-2, -1), (-1, -2), (0, -2)}. For t=-3∈B, find its numerically least permitted preimage, answering none when there is no preimage.","answer":{"kind":"short_text","value":"none","explanation":"Enumerate the full fiber and select its least input if it is nonempty. The answer is none. Existence of a preimage does not imply uniqueness."},"skills":["Least Permitted Preimage of a Target"],"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-20261004-advance-20261005-r02-surjective-0064","batch_id":"batch-20261004-advance-20261005-r02-surjective-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"60e739f598aaf131bceffb1dd657f9f21c7c6ea446dbededd532546644f6e957","family_id":"qf-surjective-first-preimage","seed":202610052064,"semantic_template":"qf-surjective-first-preimage/{\"A\":[-4,-3,-2,-1,0],\"B\":[0,1,2,3,4],\"operation\":\"first_preimage\",\"pairs\":[[-4,1],[-3,2],[-2,3],[-1,4],[0,0]],\"t\":3}"},"fields":{"title":"Least Permitted Preimage of a Target","stem":"The declared total function f:A→B has A={-4, -3, -2, -1, 0}, B={0, 1, 2, 3, 4}, and table {(-4, 1), (-3, 2), (-2, 3), (-1, 4), (0, 0)}. For t=3∈B, find its numerically least permitted preimage, answering none when there is no preimage.","answer":{"kind":"short_text","value":"-2","explanation":"Enumerate the full fiber and select its least input if it is nonempty. The answer is -2. Existence of a preimage does not imply uniqueness."},"skills":["Least Permitted Preimage of a Target"],"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-20261004-advance-20261005-r02-surjective-0065","batch_id":"batch-20261004-advance-20261005-r02-surjective-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"025d562e87292c3ad7c78835789d904ba2c4e59d8d0af00c5d565ec1357721e4","family_id":"qf-surjective-first-preimage","seed":202610052065,"semantic_template":"qf-surjective-first-preimage/{\"A\":[-4,-3,-2,-1,0],\"B\":[1,3,5,7,9,11],\"operation\":\"first_preimage\",\"pairs\":[[-4,1],[-3,3],[-2,5],[-1,7],[0,9]],\"t\":11}"},"fields":{"title":"Least Permitted Preimage of a Target","stem":"The declared total function f:A→B has A={-4, -3, -2, -1, 0}, B={1, 3, 5, 7, 9, 11}, and table {(-4, 1), (-3, 3), (-2, 5), (-1, 7), (0, 9)}. For t=11∈B, find its numerically least permitted preimage, answering none when there is no preimage.","answer":{"kind":"short_text","value":"none","explanation":"Enumerate the full fiber and select its least input if it is nonempty. The answer is none. Existence of a preimage does not imply uniqueness."},"skills":["Least Permitted Preimage of a Target"],"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-20261004-advance-20261005-r02-surjective-0066","batch_id":"batch-20261004-advance-20261005-r02-surjective-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"38c48a34107512e77b2a426e66e6548a4d33a49407067290d3cc54c548ea7e64","family_id":"qf-surjective-first-preimage","seed":202610052066,"semantic_template":"qf-surjective-first-preimage/{\"A\":[-4,-3,-2,-1,0],\"B\":[0,1,2,3],\"operation\":\"first_preimage\",\"pairs\":[[-4,0],[-3,1],[-2,2],[-1,3],[0,0]],\"t\":3}"},"fields":{"title":"Least Permitted Preimage of a Target","stem":"The declared total function f:A→B has A={-4, -3, -2, -1, 0}, B={0, 1, 2, 3}, and table {(-4, 0), (-3, 1), (-2, 2), (-1, 3), (0, 0)}. For t=3∈B, find its numerically least permitted preimage, answering none when there is no preimage.","answer":{"kind":"short_text","value":"-1","explanation":"Enumerate the full fiber and select its least input if it is nonempty. The answer is -1. Existence of a preimage does not imply uniqueness."},"skills":["Least Permitted Preimage of a Target"],"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-20261004-advance-20261005-r02-surjective-0067","batch_id":"batch-20261004-advance-20261005-r02-surjective-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f6d95055ff4eade9290a61aeeca5f044dc3b8b8ec9cde728383e1428a3b41372","family_id":"qf-surjective-first-preimage","seed":202610052067,"semantic_template":"qf-surjective-first-preimage/{\"A\":[-4,-3,-2,-1,0],\"B\":[-6,-5,-4,-3,-2,-1],\"operation\":\"first_preimage\",\"pairs\":[[-4,-1],[-3,-1],[-2,-2],[-1,-2],[0,-3]],\"t\":-2}"},"fields":{"title":"Least Permitted Preimage of a Target","stem":"The declared total function f:A→B has A={-4, -3, -2, -1, 0}, B={-6, -5, -4, -3, -2, -1}, and table {(-4, -1), (-3, -1), (-2, -2), (-1, -2), (0, -3)}. For t=-2∈B, find its numerically least permitted preimage, answering none when there is no preimage.","answer":{"kind":"short_text","value":"-2","explanation":"Enumerate the full fiber and select its least input if it is nonempty. The answer is -2. Existence of a preimage does not imply uniqueness."},"skills":["Least Permitted Preimage of a Target"],"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-20261004-advance-20261005-r02-surjective-0068","batch_id":"batch-20261004-advance-20261005-r02-surjective-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1375cc90c80a054c44c087afec73f440faa5be203934e019167d6129a8680e74","family_id":"qf-surjective-first-preimage","seed":202610052068,"semantic_template":"qf-surjective-first-preimage/{\"A\":[-5,-4,-3,-2,-1,0],\"B\":[0,1,2,3,4,5],\"operation\":\"first_preimage\",\"pairs\":[[-5,1],[-4,2],[-3,3],[-2,4],[-1,5],[0,0]],\"t\":5}"},"fields":{"title":"Least Permitted Preimage of a Target","stem":"The declared total function f:A→B has A={-5, -4, -3, -2, -1, 0}, B={0, 1, 2, 3, 4, 5}, and table {(-5, 1), (-4, 2), (-3, 3), (-2, 4), (-1, 5), (0, 0)}. For t=5∈B, find its numerically least permitted preimage, answering none when there is no preimage.","answer":{"kind":"short_text","value":"-1","explanation":"Enumerate the full fiber and select its least input if it is nonempty. The answer is -1. Existence of a preimage does not imply uniqueness."},"skills":["Least Permitted Preimage of a Target"],"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-20261004-advance-20261005-r02-surjective-0069","batch_id":"batch-20261004-advance-20261005-r02-surjective-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"378efe5ecf552cdd45c5b69e54b322f5e5c22fdf55eaf78a9bdfba18a8556337","family_id":"qf-surjective-first-preimage","seed":202610052069,"semantic_template":"qf-surjective-first-preimage/{\"A\":[-5,-4,-3,-2,-1,0],\"B\":[1,3,5,7,9,11,13],\"operation\":\"first_preimage\",\"pairs\":[[-5,1],[-4,3],[-3,5],[-2,7],[-1,9],[0,11]],\"t\":13}"},"fields":{"title":"Least Permitted Preimage of a Target","stem":"The declared total function f:A→B has A={-5, -4, -3, -2, -1, 0}, B={1, 3, 5, 7, 9, 11, 13}, and table {(-5, 1), (-4, 3), (-3, 5), (-2, 7), (-1, 9), (0, 11)}. For t=13∈B, find its numerically least permitted preimage, answering none when there is no preimage.","answer":{"kind":"short_text","value":"none","explanation":"Enumerate the full fiber and select its least input if it is nonempty. The answer is none. Existence of a preimage does not imply uniqueness."},"skills":["Least Permitted Preimage of a Target"],"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-20261004-advance-20261005-r02-surjective-0070","batch_id":"batch-20261004-advance-20261005-r02-surjective-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cd702409f256b5109bfe380c953371457076742b3a2f78903117968c85031f0f","family_id":"qf-surjective-first-preimage","seed":202610052070,"semantic_template":"qf-surjective-first-preimage/{\"A\":[-5,-4,-3,-2,-1,0],\"B\":[0,1,2,3,4],\"operation\":\"first_preimage\",\"pairs\":[[-5,0],[-4,1],[-3,2],[-2,3],[-1,4],[0,0]],\"t\":4}"},"fields":{"title":"Least Permitted Preimage of a Target","stem":"The declared total function f:A→B has A={-5, -4, -3, -2, -1, 0}, B={0, 1, 2, 3, 4}, and table {(-5, 0), (-4, 1), (-3, 2), (-2, 3), (-1, 4), (0, 0)}. For t=4∈B, find its numerically least permitted preimage, answering none when there is no preimage.","answer":{"kind":"short_text","value":"-1","explanation":"Enumerate the full fiber and select its least input if it is nonempty. The answer is -1. Existence of a preimage does not imply uniqueness."},"skills":["Least Permitted Preimage of a Target"],"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-20261004-advance-20261005-r02-surjective-0071","batch_id":"batch-20261004-advance-20261005-r02-surjective-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"747d4bfce3a0938fdcc89ddc1efdd4b0536bf14ec5485e964bbd48018c078151","family_id":"qf-surjective-first-preimage","seed":202610052071,"semantic_template":"qf-surjective-first-preimage/{\"A\":[-5,-4,-3,-2,-1,0],\"B\":[-7,-6,-5,-4,-3,-2,-1],\"operation\":\"first_preimage\",\"pairs\":[[-5,-1],[-4,-1],[-3,-2],[-2,-2],[-1,-3],[0,-3]],\"t\":-1}"},"fields":{"title":"Least Permitted Preimage of a Target","stem":"The declared total function f:A→B has A={-5, -4, -3, -2, -1, 0}, B={-7, -6, -5, -4, -3, -2, -1}, and table {(-5, -1), (-4, -1), (-3, -2), (-2, -2), (-1, -3), (0, -3)}. For t=-1∈B, find its numerically least permitted preimage, answering none when there is no preimage.","answer":{"kind":"short_text","value":"-5","explanation":"Enumerate the full fiber and select its least input if it is nonempty. The answer is -5. Existence of a preimage does not imply uniqueness."},"skills":["Least Permitted Preimage of a Target"],"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-20261004-advance-20261005-r02-surjective-0072","batch_id":"batch-20261004-advance-20261005-r02-surjective-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"75353519a718e711c09c924b3236a8f3701a5f1b19089ab81e5f3eec620638ea","family_id":"qf-surjective-first-preimage","seed":202610052072,"semantic_template":"qf-surjective-first-preimage/{\"A\":[-6,-5,-4,-3,-2,-1,0],\"B\":[0,1,2,3,4,5,6],\"operation\":\"first_preimage\",\"pairs\":[[-6,1],[-5,2],[-4,3],[-3,4],[-2,5],[-1,6],[0,0]],\"t\":0}"},"fields":{"title":"Least Permitted Preimage of a Target","stem":"The declared total function f:A→B has A={-6, -5, -4, -3, -2, -1, 0}, B={0, 1, 2, 3, 4, 5, 6}, and table {(-6, 1), (-5, 2), (-4, 3), (-3, 4), (-2, 5), (-1, 6), (0, 0)}. For t=0∈B, find its numerically least permitted preimage, answering none when there is no preimage.","answer":{"kind":"short_text","value":"0","explanation":"Enumerate the full fiber and select its least input if it is nonempty. The answer is 0. Existence of a preimage does not imply uniqueness."},"skills":["Least Permitted Preimage of a Target"],"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-20261004-advance-20261005-r02-surjective-0073","batch_id":"batch-20261004-advance-20261005-r02-surjective-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1664ed13d160f743a9bbcc1e59e356ed54e3b19d32f9644dd22ad233bfa3f9f5","family_id":"qf-surjective-first-preimage","seed":202610052073,"semantic_template":"qf-surjective-first-preimage/{\"A\":[-6,-5,-4,-3,-2,-1,0],\"B\":[1,3,5,7,9,11,13,15],\"operation\":\"first_preimage\",\"pairs\":[[-6,1],[-5,3],[-4,5],[-3,7],[-2,9],[-1,11],[0,13]],\"t\":15}"},"fields":{"title":"Least Permitted Preimage of a Target","stem":"The declared total function f:A→B has A={-6, -5, -4, -3, -2, -1, 0}, B={1, 3, 5, 7, 9, 11, 13, 15}, and table {(-6, 1), (-5, 3), (-4, 5), (-3, 7), (-2, 9), (-1, 11), (0, 13)}. For t=15∈B, find its numerically least permitted preimage, answering none when there is no preimage.","answer":{"kind":"short_text","value":"none","explanation":"Enumerate the full fiber and select its least input if it is nonempty. The answer is none. Existence of a preimage does not imply uniqueness."},"skills":["Least Permitted Preimage of a Target"],"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-20261004-advance-20261005-r02-surjective-0074","batch_id":"batch-20261004-advance-20261005-r02-surjective-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d029654c748bada28aa5cd8bcb6e8df96baa168d76600de5b44fdb39e2fe54b5","family_id":"qf-surjective-first-preimage","seed":202610052074,"semantic_template":"qf-surjective-first-preimage/{\"A\":[-6,-5,-4,-3,-2,-1,0],\"B\":[0,1,2,3,4,5],\"operation\":\"first_preimage\",\"pairs\":[[-6,0],[-5,1],[-4,2],[-3,3],[-2,4],[-1,5],[0,0]],\"t\":5}"},"fields":{"title":"Least Permitted Preimage of a Target","stem":"The declared total function f:A→B has A={-6, -5, -4, -3, -2, -1, 0}, B={0, 1, 2, 3, 4, 5}, and table {(-6, 0), (-5, 1), (-4, 2), (-3, 3), (-2, 4), (-1, 5), (0, 0)}. For t=5∈B, find its numerically least permitted preimage, answering none when there is no preimage.","answer":{"kind":"short_text","value":"-1","explanation":"Enumerate the full fiber and select its least input if it is nonempty. The answer is -1. Existence of a preimage does not imply uniqueness."},"skills":["Least Permitted Preimage of a Target"],"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-20261004-advance-20261005-r02-surjective-0075","batch_id":"batch-20261004-advance-20261005-r02-surjective-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"33d54bed8970040c87eef512ae52738fd9191f6ae5326dc80d095bc80f19c0e9","family_id":"qf-surjective-first-preimage","seed":202610052075,"semantic_template":"qf-surjective-first-preimage/{\"A\":[-6,-5,-4,-3,-2,-1,0],\"B\":[-8,-7,-6,-5,-4,-3,-2,-1],\"operation\":\"first_preimage\",\"pairs\":[[-6,-1],[-5,-1],[-4,-2],[-3,-2],[-2,-3],[-1,-3],[0,-4]],\"t\":-8}"},"fields":{"title":"Least Permitted Preimage of a Target","stem":"The declared total function f:A→B has A={-6, -5, -4, -3, -2, -1, 0}, B={-8, -7, -6, -5, -4, -3, -2, -1}, and table {(-6, -1), (-5, -1), (-4, -2), (-3, -2), (-2, -3), (-1, -3), (0, -4)}. For t=-8∈B, find its numerically least permitted preimage, answering none when there is no preimage.","answer":{"kind":"short_text","value":"none","explanation":"Enumerate the full fiber and select its least input if it is nonempty. The answer is none. Existence of a preimage does not imply uniqueness."},"skills":["Least Permitted Preimage of a Target"],"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-20261004-advance-20261005-r02-surjective-0076","batch_id":"batch-20261004-advance-20261005-r02-surjective-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d472846fc9c5066d130c07a0d0dd4dc7a2ef50d45717983447147756c5470261","family_id":"qf-surjective-codomain-change","seed":202610052076,"semantic_template":"qf-surjective-codomain-change/{\"A\":[],\"B2\":[],\"operation\":\"changed_codomain\",\"pairs\":[]}"},"fields":{"title":"Coverage after a Valid Codomain Change","stem":"The declared total function f:A→B has A=∅, B=∅, and table ∅. The new codomain B2=∅ contains every attained output. Is the same output table surjective as A→B2?","answer":{"kind":"short_text","value":"Yes","explanation":"Containing the range first makes the new total-function declaration valid. Compare that range with B2 to obtain Yes; no existing output may be discarded to force surjectivity."},"skills":["Coverage after a Valid Codomain Change"],"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-20261004-advance-20261005-r02-surjective-0077","batch_id":"batch-20261004-advance-20261005-r02-surjective-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"14a9e3bd06b05a14e0d27c3c53cae9b50f5d73a7aa8a18380fc78f6e0177d8ee","family_id":"qf-surjective-codomain-change","seed":202610052077,"semantic_template":"qf-surjective-codomain-change/{\"A\":[-1,0],\"B2\":[0,1,101],\"operation\":\"changed_codomain\",\"pairs\":[[-1,1],[0,0]]}"},"fields":{"title":"Coverage after a Valid Codomain Change","stem":"The declared total function f:A→B has A={-1, 0}, B={0, 1}, and table {(-1, 1), (0, 0)}. The new codomain B2={1, 0, 101} contains every attained output. Is the same output table surjective as A→B2?","answer":{"kind":"short_text","value":"No","explanation":"Containing the range first makes the new total-function declaration valid. Compare that range with B2 to obtain No; no existing output may be discarded to force surjectivity."},"skills":["Coverage after a Valid Codomain Change"],"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-20261004-advance-20261005-r02-surjective-0078","batch_id":"batch-20261004-advance-20261005-r02-surjective-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"37c51a3601d553bdb7de5c97f1bfad6f5e759eda147c754279568dd6d8671963","family_id":"qf-surjective-codomain-change","seed":202610052078,"semantic_template":"qf-surjective-codomain-change/{\"A\":[-1,0],\"B2\":[1,3],\"operation\":\"changed_codomain\",\"pairs\":[[-1,1],[0,3]]}"},"fields":{"title":"Coverage after a Valid Codomain Change","stem":"The declared total function f:A→B has A={-1, 0}, B={1, 3, 5}, and table {(-1, 1), (0, 3)}. The new codomain B2={1, 3} contains every attained output. Is the same output table surjective as A→B2?","answer":{"kind":"short_text","value":"Yes","explanation":"Containing the range first makes the new total-function declaration valid. Compare that range with B2 to obtain Yes; no existing output may be discarded to force surjectivity."},"skills":["Coverage after a Valid Codomain Change"],"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-20261004-advance-20261005-r02-surjective-0079","batch_id":"batch-20261004-advance-20261005-r02-surjective-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9a88f35ffdc931db9d0a54dc97c39b3da4bc02b8290134ebed23a0023d8521fb","family_id":"qf-surjective-codomain-change","seed":202610052079,"semantic_template":"qf-surjective-codomain-change/{\"A\":[-1,0],\"B2\":[0,103],\"operation\":\"changed_codomain\",\"pairs\":[[-1,0],[0,0]]}"},"fields":{"title":"Coverage after a Valid Codomain Change","stem":"The declared total function f:A→B has A={-1, 0}, B={0}, and table {(-1, 0), (0, 0)}. The new codomain B2={0, 103} contains every attained output. Is the same output table surjective as A→B2?","answer":{"kind":"short_text","value":"No","explanation":"Containing the range first makes the new total-function declaration valid. Compare that range with B2 to obtain No; no existing output may be discarded to force surjectivity."},"skills":["Coverage after a Valid Codomain Change"],"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-20261004-advance-20261005-r02-surjective-0080","batch_id":"batch-20261004-advance-20261005-r02-surjective-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"90cb2985d751afe34dcb2a3ebae6ade5e6588da8351ed9ace467055ec8e018c2","family_id":"qf-surjective-codomain-change","seed":202610052080,"semantic_template":"qf-surjective-codomain-change/{\"A\":[-1,0],\"B2\":[-1],\"operation\":\"changed_codomain\",\"pairs\":[[-1,-1],[0,-1]]}"},"fields":{"title":"Coverage after a Valid Codomain Change","stem":"The declared total function f:A→B has A={-1, 0}, B={-3, -2, -1}, and table {(-1, -1), (0, -1)}. The new codomain B2={-1} contains every attained output. Is the same output table surjective as A→B2?","answer":{"kind":"short_text","value":"Yes","explanation":"Containing the range first makes the new total-function declaration valid. Compare that range with B2 to obtain Yes; no existing output may be discarded to force surjectivity."},"skills":["Coverage after a Valid Codomain Change"],"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-20261004-advance-20261005-r02-surjective-0081","batch_id":"batch-20261004-advance-20261005-r02-surjective-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"11fd42bd171f0a644b47dbc8d3cfa82986eaddf017a7fe35261deb25996a9f29","family_id":"qf-surjective-codomain-change","seed":202610052081,"semantic_template":"qf-surjective-codomain-change/{\"A\":[-2,-1,0],\"B2\":[0,1,2,105],\"operation\":\"changed_codomain\",\"pairs\":[[-2,1],[-1,2],[0,0]]}"},"fields":{"title":"Coverage after a Valid Codomain Change","stem":"The declared total function f:A→B has A={-2, -1, 0}, B={0, 1, 2}, and table {(-2, 1), (-1, 2), (0, 0)}. The new codomain B2={1, 2, 0, 105} contains every attained output. Is the same output table surjective as A→B2?","answer":{"kind":"short_text","value":"No","explanation":"Containing the range first makes the new total-function declaration valid. Compare that range with B2 to obtain No; no existing output may be discarded to force surjectivity."},"skills":["Coverage after a Valid Codomain Change"],"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-20261004-advance-20261005-r02-surjective-0082","batch_id":"batch-20261004-advance-20261005-r02-surjective-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fa24577646d04c819b7c3c46abe767b4c7e46ea39306e397db748117ea355a28","family_id":"qf-surjective-codomain-change","seed":202610052082,"semantic_template":"qf-surjective-codomain-change/{\"A\":[-2,-1,0],\"B2\":[1,3,5],\"operation\":\"changed_codomain\",\"pairs\":[[-2,1],[-1,3],[0,5]]}"},"fields":{"title":"Coverage after a Valid Codomain Change","stem":"The declared total function f:A→B has A={-2, -1, 0}, B={1, 3, 5, 7}, and table {(-2, 1), (-1, 3), (0, 5)}. The new codomain B2={1, 3, 5} contains every attained output. Is the same output table surjective as A→B2?","answer":{"kind":"short_text","value":"Yes","explanation":"Containing the range first makes the new total-function declaration valid. Compare that range with B2 to obtain Yes; no existing output may be discarded to force surjectivity."},"skills":["Coverage after a Valid Codomain Change"],"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-20261004-advance-20261005-r02-surjective-0083","batch_id":"batch-20261004-advance-20261005-r02-surjective-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"133e81079b32f83bf620ed0019bd1773a1d7ed760b3c3ee4e0ac8b6b92d8c059","family_id":"qf-surjective-codomain-change","seed":202610052083,"semantic_template":"qf-surjective-codomain-change/{\"A\":[-2,-1,0],\"B2\":[0,1,107],\"operation\":\"changed_codomain\",\"pairs\":[[-2,0],[-1,1],[0,0]]}"},"fields":{"title":"Coverage after a Valid Codomain Change","stem":"The declared total function f:A→B has A={-2, -1, 0}, B={0, 1}, and table {(-2, 0), (-1, 1), (0, 0)}. The new codomain B2={0, 1, 107} contains every attained output. Is the same output table surjective as A→B2?","answer":{"kind":"short_text","value":"No","explanation":"Containing the range first makes the new total-function declaration valid. Compare that range with B2 to obtain No; no existing output may be discarded to force surjectivity."},"skills":["Coverage after a Valid Codomain Change"],"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-20261004-advance-20261005-r02-surjective-0084","batch_id":"batch-20261004-advance-20261005-r02-surjective-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e15bf8a544e12d9519782ce36e8c3e0c4cdc20fe21e3f1ccd95cea32471b5646","family_id":"qf-surjective-codomain-change","seed":202610052084,"semantic_template":"qf-surjective-codomain-change/{\"A\":[-2,-1,0],\"B2\":[-2,-1],\"operation\":\"changed_codomain\",\"pairs\":[[-2,-1],[-1,-1],[0,-2]]}"},"fields":{"title":"Coverage after a Valid Codomain Change","stem":"The declared total function f:A→B has A={-2, -1, 0}, B={-4, -3, -2, -1}, and table {(-2, -1), (-1, -1), (0, -2)}. The new codomain B2={-1, -2} contains every attained output. Is the same output table surjective as A→B2?","answer":{"kind":"short_text","value":"Yes","explanation":"Containing the range first makes the new total-function declaration valid. Compare that range with B2 to obtain Yes; no existing output may be discarded to force surjectivity."},"skills":["Coverage after a Valid Codomain Change"],"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-20261004-advance-20261005-r02-surjective-0085","batch_id":"batch-20261004-advance-20261005-r02-surjective-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a3e2b463e12a1999c1c75f39a7054653e7fea50db374fd0815ecfe6340d308e9","family_id":"qf-surjective-codomain-change","seed":202610052085,"semantic_template":"qf-surjective-codomain-change/{\"A\":[-3,-2,-1,0],\"B2\":[0,1,2,3,109],\"operation\":\"changed_codomain\",\"pairs\":[[-3,1],[-2,2],[-1,3],[0,0]]}"},"fields":{"title":"Coverage after a Valid Codomain Change","stem":"The declared total function f:A→B has A={-3, -2, -1, 0}, B={0, 1, 2, 3}, and table {(-3, 1), (-2, 2), (-1, 3), (0, 0)}. The new codomain B2={1, 2, 3, 0, 109} contains every attained output. Is the same output table surjective as A→B2?","answer":{"kind":"short_text","value":"No","explanation":"Containing the range first makes the new total-function declaration valid. Compare that range with B2 to obtain No; no existing output may be discarded to force surjectivity."},"skills":["Coverage after a Valid Codomain Change"],"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-20261004-advance-20261005-r02-surjective-0086","batch_id":"batch-20261004-advance-20261005-r02-surjective-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"200890cde6615b4680b297d5a4dfcc561dbe5850ea7829bd2266d5c7ba57a3cd","family_id":"qf-surjective-codomain-change","seed":202610052086,"semantic_template":"qf-surjective-codomain-change/{\"A\":[-3,-2,-1,0],\"B2\":[1,3,5,7],\"operation\":\"changed_codomain\",\"pairs\":[[-3,1],[-2,3],[-1,5],[0,7]]}"},"fields":{"title":"Coverage after a Valid Codomain Change","stem":"The declared total function f:A→B has A={-3, -2, -1, 0}, B={1, 3, 5, 7, 9}, and table {(-3, 1), (-2, 3), (-1, 5), (0, 7)}. The new codomain B2={1, 3, 5, 7} contains every attained output. Is the same output table surjective as A→B2?","answer":{"kind":"short_text","value":"Yes","explanation":"Containing the range first makes the new total-function declaration valid. Compare that range with B2 to obtain Yes; no existing output may be discarded to force surjectivity."},"skills":["Coverage after a Valid Codomain Change"],"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-20261004-advance-20261005-r02-surjective-0087","batch_id":"batch-20261004-advance-20261005-r02-surjective-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d17f401fbeb2b959e96a0aac11c7b1c550bd14a5c0857d1fce21b0fe11bc58ee","family_id":"qf-surjective-codomain-change","seed":202610052087,"semantic_template":"qf-surjective-codomain-change/{\"A\":[-3,-2,-1,0],\"B2\":[0,1,2,111],\"operation\":\"changed_codomain\",\"pairs\":[[-3,0],[-2,1],[-1,2],[0,0]]}"},"fields":{"title":"Coverage after a Valid Codomain Change","stem":"The declared total function f:A→B has A={-3, -2, -1, 0}, B={0, 1, 2}, and table {(-3, 0), (-2, 1), (-1, 2), (0, 0)}. The new codomain B2={0, 1, 2, 111} contains every attained output. Is the same output table surjective as A→B2?","answer":{"kind":"short_text","value":"No","explanation":"Containing the range first makes the new total-function declaration valid. Compare that range with B2 to obtain No; no existing output may be discarded to force surjectivity."},"skills":["Coverage after a Valid Codomain Change"],"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-20261004-advance-20261005-r02-surjective-0088","batch_id":"batch-20261004-advance-20261005-r02-surjective-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"eee8d16eab5d1a2f7f790e0bfed0b95fd1aecfc9dfc7c1734ac959492834155a","family_id":"qf-surjective-codomain-change","seed":202610052088,"semantic_template":"qf-surjective-codomain-change/{\"A\":[-3,-2,-1,0],\"B2\":[-2,-1],\"operation\":\"changed_codomain\",\"pairs\":[[-3,-1],[-2,-1],[-1,-2],[0,-2]]}"},"fields":{"title":"Coverage after a Valid Codomain Change","stem":"The declared total function f:A→B has A={-3, -2, -1, 0}, B={-5, -4, -3, -2, -1}, and table {(-3, -1), (-2, -1), (-1, -2), (0, -2)}. The new codomain B2={-1, -2} contains every attained output. Is the same output table surjective as A→B2?","answer":{"kind":"short_text","value":"Yes","explanation":"Containing the range first makes the new total-function declaration valid. Compare that range with B2 to obtain Yes; no existing output may be discarded to force surjectivity."},"skills":["Coverage after a Valid Codomain Change"],"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-20261004-advance-20261005-r02-surjective-0089","batch_id":"batch-20261004-advance-20261005-r02-surjective-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5a257576662d0d520c2be95d2c5063480b3127bf9ff6d5b9d7bea0d511a2007d","family_id":"qf-surjective-codomain-change","seed":202610052089,"semantic_template":"qf-surjective-codomain-change/{\"A\":[-4,-3,-2,-1,0],\"B2\":[0,1,2,3,4,113],\"operation\":\"changed_codomain\",\"pairs\":[[-4,1],[-3,2],[-2,3],[-1,4],[0,0]]}"},"fields":{"title":"Coverage after a Valid Codomain Change","stem":"The declared total function f:A→B has A={-4, -3, -2, -1, 0}, B={0, 1, 2, 3, 4}, and table {(-4, 1), (-3, 2), (-2, 3), (-1, 4), (0, 0)}. The new codomain B2={1, 2, 3, 4, 0, 113} contains every attained output. Is the same output table surjective as A→B2?","answer":{"kind":"short_text","value":"No","explanation":"Containing the range first makes the new total-function declaration valid. Compare that range with B2 to obtain No; no existing output may be discarded to force surjectivity."},"skills":["Coverage after a Valid Codomain Change"],"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-20261004-advance-20261005-r02-surjective-0090","batch_id":"batch-20261004-advance-20261005-r02-surjective-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3cc15093c1c7f61275ffdd25372bdabcdb59f3b5144da750f12371a67771d3bb","family_id":"qf-surjective-codomain-change","seed":202610052090,"semantic_template":"qf-surjective-codomain-change/{\"A\":[-4,-3,-2,-1,0],\"B2\":[1,3,5,7,9],\"operation\":\"changed_codomain\",\"pairs\":[[-4,1],[-3,3],[-2,5],[-1,7],[0,9]]}"},"fields":{"title":"Coverage after a Valid Codomain Change","stem":"The declared total function f:A→B has A={-4, -3, -2, -1, 0}, B={1, 3, 5, 7, 9, 11}, and table {(-4, 1), (-3, 3), (-2, 5), (-1, 7), (0, 9)}. The new codomain B2={1, 3, 5, 7, 9} contains every attained output. Is the same output table surjective as A→B2?","answer":{"kind":"short_text","value":"Yes","explanation":"Containing the range first makes the new total-function declaration valid. Compare that range with B2 to obtain Yes; no existing output may be discarded to force surjectivity."},"skills":["Coverage after a Valid Codomain Change"],"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-20261004-advance-20261005-r02-surjective-0091","batch_id":"batch-20261004-advance-20261005-r02-surjective-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a27050f817e404386393210c4813858f69bcd3940422bff8110b2e925f1a69b3","family_id":"qf-surjective-codomain-change","seed":202610052091,"semantic_template":"qf-surjective-codomain-change/{\"A\":[-4,-3,-2,-1,0],\"B2\":[0,1,2,3,115],\"operation\":\"changed_codomain\",\"pairs\":[[-4,0],[-3,1],[-2,2],[-1,3],[0,0]]}"},"fields":{"title":"Coverage after a Valid Codomain Change","stem":"The declared total function f:A→B has A={-4, -3, -2, -1, 0}, B={0, 1, 2, 3}, and table {(-4, 0), (-3, 1), (-2, 2), (-1, 3), (0, 0)}. The new codomain B2={0, 1, 2, 3, 115} contains every attained output. Is the same output table surjective as A→B2?","answer":{"kind":"short_text","value":"No","explanation":"Containing the range first makes the new total-function declaration valid. Compare that range with B2 to obtain No; no existing output may be discarded to force surjectivity."},"skills":["Coverage after a Valid Codomain Change"],"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-20261004-advance-20261005-r02-surjective-0092","batch_id":"batch-20261004-advance-20261005-r02-surjective-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"85e6eef6b46cd76aefd42518d0ee16f9a70737aa45d865bf8fad520c618199e6","family_id":"qf-surjective-codomain-change","seed":202610052092,"semantic_template":"qf-surjective-codomain-change/{\"A\":[-4,-3,-2,-1,0],\"B2\":[-3,-2,-1],\"operation\":\"changed_codomain\",\"pairs\":[[-4,-1],[-3,-1],[-2,-2],[-1,-2],[0,-3]]}"},"fields":{"title":"Coverage after a Valid Codomain Change","stem":"The declared total function f:A→B has A={-4, -3, -2, -1, 0}, B={-6, -5, -4, -3, -2, -1}, and table {(-4, -1), (-3, -1), (-2, -2), (-1, -2), (0, -3)}. The new codomain B2={-1, -2, -3} contains every attained output. Is the same output table surjective as A→B2?","answer":{"kind":"short_text","value":"Yes","explanation":"Containing the range first makes the new total-function declaration valid. Compare that range with B2 to obtain Yes; no existing output may be discarded to force surjectivity."},"skills":["Coverage after a Valid Codomain Change"],"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-20261004-advance-20261005-r02-surjective-0093","batch_id":"batch-20261004-advance-20261005-r02-surjective-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f2678c6dcd6946c2eb40e7eeceb959b6956a19bf038cc58f54930eee074f62f2","family_id":"qf-surjective-codomain-change","seed":202610052093,"semantic_template":"qf-surjective-codomain-change/{\"A\":[-5,-4,-3,-2,-1,0],\"B2\":[0,1,2,3,4,5,117],\"operation\":\"changed_codomain\",\"pairs\":[[-5,1],[-4,2],[-3,3],[-2,4],[-1,5],[0,0]]}"},"fields":{"title":"Coverage after a Valid Codomain Change","stem":"The declared total function f:A→B has A={-5, -4, -3, -2, -1, 0}, B={0, 1, 2, 3, 4, 5}, and table {(-5, 1), (-4, 2), (-3, 3), (-2, 4), (-1, 5), (0, 0)}. The new codomain B2={1, 2, 3, 4, 5, 0, 117} contains every attained output. Is the same output table surjective as A→B2?","answer":{"kind":"short_text","value":"No","explanation":"Containing the range first makes the new total-function declaration valid. Compare that range with B2 to obtain No; no existing output may be discarded to force surjectivity."},"skills":["Coverage after a Valid Codomain Change"],"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-20261004-advance-20261005-r02-surjective-0094","batch_id":"batch-20261004-advance-20261005-r02-surjective-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cf0d21fe8f233322653e58e94c41b114bf1d5f46e61251a4c1620d52d5ad4483","family_id":"qf-surjective-codomain-change","seed":202610052094,"semantic_template":"qf-surjective-codomain-change/{\"A\":[-5,-4,-3,-2,-1,0],\"B2\":[1,3,5,7,9,11],\"operation\":\"changed_codomain\",\"pairs\":[[-5,1],[-4,3],[-3,5],[-2,7],[-1,9],[0,11]]}"},"fields":{"title":"Coverage after a Valid Codomain Change","stem":"The declared total function f:A→B has A={-5, -4, -3, -2, -1, 0}, B={1, 3, 5, 7, 9, 11, 13}, and table {(-5, 1), (-4, 3), (-3, 5), (-2, 7), (-1, 9), (0, 11)}. The new codomain B2={1, 3, 5, 7, 9, 11} contains every attained output. Is the same output table surjective as A→B2?","answer":{"kind":"short_text","value":"Yes","explanation":"Containing the range first makes the new total-function declaration valid. Compare that range with B2 to obtain Yes; no existing output may be discarded to force surjectivity."},"skills":["Coverage after a Valid Codomain Change"],"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-20261004-advance-20261005-r02-surjective-0095","batch_id":"batch-20261004-advance-20261005-r02-surjective-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2bb2c10665a3c83807350a69f7cb13ac54eac453adc162874ba931b1db990d62","family_id":"qf-surjective-codomain-change","seed":202610052095,"semantic_template":"qf-surjective-codomain-change/{\"A\":[-5,-4,-3,-2,-1,0],\"B2\":[0,1,2,3,4,119],\"operation\":\"changed_codomain\",\"pairs\":[[-5,0],[-4,1],[-3,2],[-2,3],[-1,4],[0,0]]}"},"fields":{"title":"Coverage after a Valid Codomain Change","stem":"The declared total function f:A→B has A={-5, -4, -3, -2, -1, 0}, B={0, 1, 2, 3, 4}, and table {(-5, 0), (-4, 1), (-3, 2), (-2, 3), (-1, 4), (0, 0)}. The new codomain B2={0, 1, 2, 3, 4, 119} contains every attained output. Is the same output table surjective as A→B2?","answer":{"kind":"short_text","value":"No","explanation":"Containing the range first makes the new total-function declaration valid. Compare that range with B2 to obtain No; no existing output may be discarded to force surjectivity."},"skills":["Coverage after a Valid Codomain Change"],"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-20261004-advance-20261005-r02-surjective-0096","batch_id":"batch-20261004-advance-20261005-r02-surjective-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4c65dc06ef54eeaef74bab6fb9440d0c9762e9060d402c91f9c56380a5aba7dc","family_id":"qf-surjective-codomain-change","seed":202610052096,"semantic_template":"qf-surjective-codomain-change/{\"A\":[-5,-4,-3,-2,-1,0],\"B2\":[-3,-2,-1],\"operation\":\"changed_codomain\",\"pairs\":[[-5,-1],[-4,-1],[-3,-2],[-2,-2],[-1,-3],[0,-3]]}"},"fields":{"title":"Coverage after a Valid Codomain Change","stem":"The declared total function f:A→B has A={-5, -4, -3, -2, -1, 0}, B={-7, -6, -5, -4, -3, -2, -1}, and table {(-5, -1), (-4, -1), (-3, -2), (-2, -2), (-1, -3), (0, -3)}. The new codomain B2={-1, -2, -3} contains every attained output. Is the same output table surjective as A→B2?","answer":{"kind":"short_text","value":"Yes","explanation":"Containing the range first makes the new total-function declaration valid. Compare that range with B2 to obtain Yes; no existing output may be discarded to force surjectivity."},"skills":["Coverage after a Valid Codomain Change"],"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-20261004-advance-20261005-r02-surjective-0097","batch_id":"batch-20261004-advance-20261005-r02-surjective-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3bb54d029c19e45e0061dbad0dcd641aadf1b2c447b9a50ed4b59c1a5aaabb29","family_id":"qf-surjective-codomain-change","seed":202610052097,"semantic_template":"qf-surjective-codomain-change/{\"A\":[-6,-5,-4,-3,-2,-1,0],\"B2\":[0,1,2,3,4,5,6,121],\"operation\":\"changed_codomain\",\"pairs\":[[-6,1],[-5,2],[-4,3],[-3,4],[-2,5],[-1,6],[0,0]]}"},"fields":{"title":"Coverage after a Valid Codomain Change","stem":"The declared total function f:A→B has A={-6, -5, -4, -3, -2, -1, 0}, B={0, 1, 2, 3, 4, 5, 6}, and table {(-6, 1), (-5, 2), (-4, 3), (-3, 4), (-2, 5), (-1, 6), (0, 0)}. The new codomain B2={1, 2, 3, 4, 5, 6, 0, 121} contains every attained output. Is the same output table surjective as A→B2?","answer":{"kind":"short_text","value":"No","explanation":"Containing the range first makes the new total-function declaration valid. Compare that range with B2 to obtain No; no existing output may be discarded to force surjectivity."},"skills":["Coverage after a Valid Codomain Change"],"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-20261004-advance-20261005-r02-surjective-0098","batch_id":"batch-20261004-advance-20261005-r02-surjective-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"38590ee82c42fded0004a4e21fd268350227c47ad61aaa08d0331a5b9ca43421","family_id":"qf-surjective-codomain-change","seed":202610052098,"semantic_template":"qf-surjective-codomain-change/{\"A\":[-6,-5,-4,-3,-2,-1,0],\"B2\":[1,3,5,7,9,11,13],\"operation\":\"changed_codomain\",\"pairs\":[[-6,1],[-5,3],[-4,5],[-3,7],[-2,9],[-1,11],[0,13]]}"},"fields":{"title":"Coverage after a Valid Codomain Change","stem":"The declared total function f:A→B has A={-6, -5, -4, -3, -2, -1, 0}, B={1, 3, 5, 7, 9, 11, 13, 15}, and table {(-6, 1), (-5, 3), (-4, 5), (-3, 7), (-2, 9), (-1, 11), (0, 13)}. The new codomain B2={1, 3, 5, 7, 9, 11, 13} contains every attained output. Is the same output table surjective as A→B2?","answer":{"kind":"short_text","value":"Yes","explanation":"Containing the range first makes the new total-function declaration valid. Compare that range with B2 to obtain Yes; no existing output may be discarded to force surjectivity."},"skills":["Coverage after a Valid Codomain Change"],"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-20261004-advance-20261005-r02-surjective-0099","batch_id":"batch-20261004-advance-20261005-r02-surjective-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"677e9b8aaf02fdf1be16d9d0cf116473135dc16b0df26f0cef5a1437bc44347e","family_id":"qf-surjective-codomain-change","seed":202610052099,"semantic_template":"qf-surjective-codomain-change/{\"A\":[-6,-5,-4,-3,-2,-1,0],\"B2\":[0,1,2,3,4,5,123],\"operation\":\"changed_codomain\",\"pairs\":[[-6,0],[-5,1],[-4,2],[-3,3],[-2,4],[-1,5],[0,0]]}"},"fields":{"title":"Coverage after a Valid Codomain Change","stem":"The declared total function f:A→B has A={-6, -5, -4, -3, -2, -1, 0}, B={0, 1, 2, 3, 4, 5}, and table {(-6, 0), (-5, 1), (-4, 2), (-3, 3), (-2, 4), (-1, 5), (0, 0)}. The new codomain B2={0, 1, 2, 3, 4, 5, 123} contains every attained output. Is the same output table surjective as A→B2?","answer":{"kind":"short_text","value":"No","explanation":"Containing the range first makes the new total-function declaration valid. Compare that range with B2 to obtain No; no existing output may be discarded to force surjectivity."},"skills":["Coverage after a Valid Codomain Change"],"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-20261004-advance-20261005-r02-surjective-0100","batch_id":"batch-20261004-advance-20261005-r02-surjective-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ee1a8450086c3dbe7cd409d8a751093bbdb478e4cdf196e2aa9130def262506e","family_id":"qf-surjective-codomain-change","seed":202610052100,"semantic_template":"qf-surjective-codomain-change/{\"A\":[-6,-5,-4,-3,-2,-1,0],\"B2\":[-4,-3,-2,-1],\"operation\":\"changed_codomain\",\"pairs\":[[-6,-1],[-5,-1],[-4,-2],[-3,-2],[-2,-3],[-1,-3],[0,-4]]}"},"fields":{"title":"Coverage after a Valid Codomain Change","stem":"The declared total function f:A→B has A={-6, -5, -4, -3, -2, -1, 0}, B={-8, -7, -6, -5, -4, -3, -2, -1}, and table {(-6, -1), (-5, -1), (-4, -2), (-3, -2), (-2, -3), (-1, -3), (0, -4)}. The new codomain B2={-1, -2, -3, -4} contains every attained output. Is the same output table surjective as A→B2?","answer":{"kind":"short_text","value":"Yes","explanation":"Containing the range first makes the new total-function declaration valid. Compare that range with B2 to obtain Yes; no existing output may be discarded to force surjectivity."},"skills":["Coverage after a Valid Codomain Change"],"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."}}