{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r03-relation-0001","batch_id":"batch-20261003-advance-20261004-r03-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9821e73e8fd44c7fb0e706b97875bb6c2394236fead632dab0b13adcd7e4b45f","family_id":"qf-relation-pair-membership","seed":202610043001,"semantic_template":"qf-relation-pair-membership/{\"A\":[],\"B\":[],\"R\":[],\"operation\":\"pair_membership\",\"x\":0,\"y\":1}"},"fields":{"title":"Ordered-Pair Membership in a Binary Relation","stem":"The binary relation has source A=∅, target B=∅, and R=∅⊆A×B. Does the ordered pair (0, 1) belong to R?","answer":{"kind":"short_text","value":"No","explanation":"Compare the entire ordered pair with the declared relation, keeping component order. Membership gives No."},"skills":["Ordered-Pair Membership in a Binary Relation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r03-relation-0002","batch_id":"batch-20261003-advance-20261004-r03-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"07d3e383b796fb8ad594077d145d812319992301214ca8a924c80e5e7a8da988","family_id":"qf-relation-pair-membership","seed":202610043002,"semantic_template":"qf-relation-pair-membership/{\"A\":[0],\"B\":[1],\"R\":[],\"operation\":\"pair_membership\",\"x\":1,\"y\":0}"},"fields":{"title":"Ordered-Pair Membership in a Binary Relation","stem":"The binary relation has source A={0}, target B={1}, and R=∅⊆A×B. Does the ordered pair (1, 0) belong to R?","answer":{"kind":"short_text","value":"No","explanation":"Compare the entire ordered pair with the declared relation, keeping component order. Membership gives No."},"skills":["Ordered-Pair Membership in a Binary Relation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r03-relation-0003","batch_id":"batch-20261003-advance-20261004-r03-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"985c81765ce38b49b6a8967cc06a764b3d34e35c2e5e0763bffac3e5bf9b3436","family_id":"qf-relation-pair-membership","seed":202610043003,"semantic_template":"qf-relation-pair-membership/{\"A\":[-10,-9,-8],\"B\":[-6,-5,-4],\"R\":[[-10,-6],[-10,-5]],\"operation\":\"pair_membership\",\"x\":-8,\"y\":-6}"},"fields":{"title":"Ordered-Pair Membership in a Binary Relation","stem":"The binary relation has source A={-10, -9, -8}, target B={-6, -5, -4}, and R={(-10, -6), (-10, -5)}⊆A×B. Does the ordered pair (-8, -6) belong to R?","answer":{"kind":"short_text","value":"No","explanation":"Compare the entire ordered pair with the declared relation, keeping component order. Membership gives No."},"skills":["Ordered-Pair Membership in a Binary Relation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r03-relation-0004","batch_id":"batch-20261003-advance-20261004-r03-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0875dc1ef6a73a105d898f26d87cba059c209b94a4a8f3396e2e15784339d171","family_id":"qf-relation-pair-membership","seed":202610043004,"semantic_template":"qf-relation-pair-membership/{\"A\":[-9,-8,-7],\"B\":[-5,-4,-3],\"R\":[[-9,-4],[-8,-4]],\"operation\":\"pair_membership\",\"x\":-9,\"y\":-4}"},"fields":{"title":"Ordered-Pair Membership in a Binary Relation","stem":"The binary relation has source A={-9, -8, -7}, target B={-5, -4, -3}, and R={(-9, -4), (-8, -4)}⊆A×B. Does the ordered pair (-9, -4) belong to R?","answer":{"kind":"short_text","value":"Yes","explanation":"Compare the entire ordered pair with the declared relation, keeping component order. Membership gives Yes."},"skills":["Ordered-Pair Membership in a Binary Relation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r03-relation-0005","batch_id":"batch-20261003-advance-20261004-r03-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c1ca5c950036b9c2eef7489bd1adcf0e37ce7b113926ed6cd0e694ca3f6b8f92","family_id":"qf-relation-pair-membership","seed":202610043005,"semantic_template":"qf-relation-pair-membership/{\"A\":[-8,-7,-6],\"B\":[-4,-3,-2],\"R\":[[-8,-4],[-7,-3],[-6,-2]],\"operation\":\"pair_membership\",\"x\":-4,\"y\":-8}"},"fields":{"title":"Ordered-Pair Membership in a Binary Relation","stem":"The binary relation has source A={-8, -7, -6}, target B={-4, -3, -2}, and R={(-8, -4), (-7, -3), (-6, -2)}⊆A×B. Does the ordered pair (-4, -8) belong to R?","answer":{"kind":"short_text","value":"No","explanation":"Compare the entire ordered pair with the declared relation, keeping component order. Membership gives No."},"skills":["Ordered-Pair Membership in a Binary Relation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r03-relation-0006","batch_id":"batch-20261003-advance-20261004-r03-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"073df47243d81ef25e922b9379f34495193abe4179c9efe54a954cfecaa67fbc","family_id":"qf-relation-pair-membership","seed":202610043006,"semantic_template":"qf-relation-pair-membership/{\"A\":[-7,-6,-5],\"B\":[-3,-2,-1],\"R\":[[-6,-3],[-5,-2]],\"operation\":\"pair_membership\",\"x\":-5,\"y\":-3}"},"fields":{"title":"Ordered-Pair Membership in a Binary Relation","stem":"The binary relation has source A={-7, -6, -5}, target B={-3, -2, -1}, and R={(-6, -3), (-5, -2)}⊆A×B. Does the ordered pair (-5, -3) belong to R?","answer":{"kind":"short_text","value":"No","explanation":"Compare the entire ordered pair with the declared relation, keeping component order. Membership gives No."},"skills":["Ordered-Pair Membership in a Binary Relation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r03-relation-0007","batch_id":"batch-20261003-advance-20261004-r03-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d042fa14cf16853e361a415acc9a653419e1e5d6ef987aa7dd1f9d4cd591bdc9","family_id":"qf-relation-pair-membership","seed":202610043007,"semantic_template":"qf-relation-pair-membership/{\"A\":[-6,-5,-4],\"B\":[-2,-1,0],\"R\":[],\"operation\":\"pair_membership\",\"x\":-6,\"y\":-2}"},"fields":{"title":"Ordered-Pair Membership in a Binary Relation","stem":"The binary relation has source A={-6, -5, -4}, target B={-2, -1, 0}, and R=∅⊆A×B. Does the ordered pair (-6, -2) belong to R?","answer":{"kind":"short_text","value":"No","explanation":"Compare the entire ordered pair with the declared relation, keeping component order. Membership gives No."},"skills":["Ordered-Pair Membership in a Binary Relation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r03-relation-0008","batch_id":"batch-20261003-advance-20261004-r03-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"82e8c9a10244d50a6f422ce6d452431129adfc2f4c86afd7256b64258e8c1823","family_id":"qf-relation-pair-membership","seed":202610043008,"semantic_template":"qf-relation-pair-membership/{\"A\":[-5,-4,-3],\"B\":[-1,0,1],\"R\":[[-5,-1]],\"operation\":\"pair_membership\",\"x\":-1,\"y\":-5}"},"fields":{"title":"Ordered-Pair Membership in a Binary Relation","stem":"The binary relation has source A={-5, -4, -3}, target B={-1, 0, 1}, and R={(-5, -1)}⊆A×B. Does the ordered pair (-1, -5) belong to R?","answer":{"kind":"short_text","value":"No","explanation":"Compare the entire ordered pair with the declared relation, keeping component order. Membership gives No."},"skills":["Ordered-Pair Membership in a Binary Relation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r03-relation-0009","batch_id":"batch-20261003-advance-20261004-r03-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"77cdf06e48bff53a1e784036e6e16746f05ea287412f117f00791c562acf72d1","family_id":"qf-relation-pair-membership","seed":202610043009,"semantic_template":"qf-relation-pair-membership/{\"A\":[-4,-3,-2],\"B\":[0,1,2],\"R\":[[-4,0],[-4,1]],\"operation\":\"pair_membership\",\"x\":-2,\"y\":0}"},"fields":{"title":"Ordered-Pair Membership in a Binary Relation","stem":"The binary relation has source A={-4, -3, -2}, target B={0, 1, 2}, and R={(-4, 0), (-4, 1)}⊆A×B. Does the ordered pair (-2, 0) belong to R?","answer":{"kind":"short_text","value":"No","explanation":"Compare the entire ordered pair with the declared relation, keeping component order. Membership gives No."},"skills":["Ordered-Pair Membership in a Binary Relation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r03-relation-0010","batch_id":"batch-20261003-advance-20261004-r03-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"04c5cb94556fe4b2b159de220308768b1da9f47bae880be5ddac294dd8f901b0","family_id":"qf-relation-pair-membership","seed":202610043010,"semantic_template":"qf-relation-pair-membership/{\"A\":[-3,-2,-1],\"B\":[1,2,3],\"R\":[[-3,2],[-2,2]],\"operation\":\"pair_membership\",\"x\":-3,\"y\":2}"},"fields":{"title":"Ordered-Pair Membership in a Binary Relation","stem":"The binary relation has source A={-3, -2, -1}, target B={1, 2, 3}, and R={(-3, 2), (-2, 2)}⊆A×B. Does the ordered pair (-3, 2) belong to R?","answer":{"kind":"short_text","value":"Yes","explanation":"Compare the entire ordered pair with the declared relation, keeping component order. Membership gives Yes."},"skills":["Ordered-Pair Membership in a Binary Relation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r03-relation-0011","batch_id":"batch-20261003-advance-20261004-r03-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"09ae533cd34d619784e0fd3ad25851c1ed1751c304b42fd25ccf108e98ea8abf","family_id":"qf-relation-pair-membership","seed":202610043011,"semantic_template":"qf-relation-pair-membership/{\"A\":[-2,-1,0],\"B\":[2,3,4],\"R\":[[-2,2],[-1,3],[0,4]],\"operation\":\"pair_membership\",\"x\":2,\"y\":-2}"},"fields":{"title":"Ordered-Pair Membership in a Binary Relation","stem":"The binary relation has source A={-2, -1, 0}, target B={2, 3, 4}, and R={(-2, 2), (-1, 3), (0, 4)}⊆A×B. Does the ordered pair (2, -2) belong to R?","answer":{"kind":"short_text","value":"No","explanation":"Compare the entire ordered pair with the declared relation, keeping component order. Membership gives No."},"skills":["Ordered-Pair Membership in a Binary Relation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r03-relation-0012","batch_id":"batch-20261003-advance-20261004-r03-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4b6313e0ee73528f27658998311945756ad78ecea9290e58825a678fbaef7d33","family_id":"qf-relation-pair-membership","seed":202610043012,"semantic_template":"qf-relation-pair-membership/{\"A\":[-1,0,1],\"B\":[3,4,5],\"R\":[[0,3],[1,4]],\"operation\":\"pair_membership\",\"x\":1,\"y\":3}"},"fields":{"title":"Ordered-Pair Membership in a Binary Relation","stem":"The binary relation has source A={-1, 0, 1}, target B={3, 4, 5}, and R={(0, 3), (1, 4)}⊆A×B. Does the ordered pair (1, 3) belong to R?","answer":{"kind":"short_text","value":"No","explanation":"Compare the entire ordered pair with the declared relation, keeping component order. Membership gives No."},"skills":["Ordered-Pair Membership in a Binary Relation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r03-relation-0013","batch_id":"batch-20261003-advance-20261004-r03-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b31e3651d8db340390a22d3b14660dfd2b78bb467078823e6fac7d6ab6eec72c","family_id":"qf-relation-pair-membership","seed":202610043013,"semantic_template":"qf-relation-pair-membership/{\"A\":[0,1,2],\"B\":[4,5,6],\"R\":[],\"operation\":\"pair_membership\",\"x\":0,\"y\":4}"},"fields":{"title":"Ordered-Pair Membership in a Binary Relation","stem":"The binary relation has source A={0, 1, 2}, target B={4, 5, 6}, and R=∅⊆A×B. Does the ordered pair (0, 4) belong to R?","answer":{"kind":"short_text","value":"No","explanation":"Compare the entire ordered pair with the declared relation, keeping component order. Membership gives No."},"skills":["Ordered-Pair Membership in a Binary Relation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r03-relation-0014","batch_id":"batch-20261003-advance-20261004-r03-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"442770e853c8e31a38acfe11568b7d6fb45d61c43586c8f73c23652bf2f64626","family_id":"qf-relation-pair-membership","seed":202610043014,"semantic_template":"qf-relation-pair-membership/{\"A\":[1,2,3],\"B\":[5,6,7],\"R\":[[1,5]],\"operation\":\"pair_membership\",\"x\":5,\"y\":1}"},"fields":{"title":"Ordered-Pair Membership in a Binary Relation","stem":"The binary relation has source A={1, 2, 3}, target B={5, 6, 7}, and R={(1, 5)}⊆A×B. Does the ordered pair (5, 1) belong to R?","answer":{"kind":"short_text","value":"No","explanation":"Compare the entire ordered pair with the declared relation, keeping component order. Membership gives No."},"skills":["Ordered-Pair Membership in a Binary Relation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r03-relation-0015","batch_id":"batch-20261003-advance-20261004-r03-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"86f60b93e2b84aeb147b5f37b35fc752abcf43bdc4181cb04525011856d60b50","family_id":"qf-relation-pair-membership","seed":202610043015,"semantic_template":"qf-relation-pair-membership/{\"A\":[2,3,4],\"B\":[6,7,8],\"R\":[[2,6],[2,7]],\"operation\":\"pair_membership\",\"x\":4,\"y\":6}"},"fields":{"title":"Ordered-Pair Membership in a Binary Relation","stem":"The binary relation has source A={2, 3, 4}, target B={6, 7, 8}, and R={(2, 6), (2, 7)}⊆A×B. Does the ordered pair (4, 6) belong to R?","answer":{"kind":"short_text","value":"No","explanation":"Compare the entire ordered pair with the declared relation, keeping component order. Membership gives No."},"skills":["Ordered-Pair Membership in a Binary Relation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r03-relation-0016","batch_id":"batch-20261003-advance-20261004-r03-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2785b1eafc06704fcef3dbedae6e470e05e6f99ad744428a67598bb165f830b6","family_id":"qf-relation-pair-membership","seed":202610043016,"semantic_template":"qf-relation-pair-membership/{\"A\":[3,4,5],\"B\":[7,8,9],\"R\":[[3,8],[4,8]],\"operation\":\"pair_membership\",\"x\":3,\"y\":8}"},"fields":{"title":"Ordered-Pair Membership in a Binary Relation","stem":"The binary relation has source A={3, 4, 5}, target B={7, 8, 9}, and R={(3, 8), (4, 8)}⊆A×B. Does the ordered pair (3, 8) belong to R?","answer":{"kind":"short_text","value":"Yes","explanation":"Compare the entire ordered pair with the declared relation, keeping component order. Membership gives Yes."},"skills":["Ordered-Pair Membership in a Binary Relation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r03-relation-0017","batch_id":"batch-20261003-advance-20261004-r03-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4653346bdd64d449ece0c77d7e1ad30de5df5dbb1ed0219b433e635e79732d27","family_id":"qf-relation-pair-membership","seed":202610043017,"semantic_template":"qf-relation-pair-membership/{\"A\":[4,5,6],\"B\":[8,9,10],\"R\":[[4,8],[5,9],[6,10]],\"operation\":\"pair_membership\",\"x\":8,\"y\":4}"},"fields":{"title":"Ordered-Pair Membership in a Binary Relation","stem":"The binary relation has source A={4, 5, 6}, target B={8, 9, 10}, and R={(4, 8), (5, 9), (6, 10)}⊆A×B. Does the ordered pair (8, 4) belong to R?","answer":{"kind":"short_text","value":"No","explanation":"Compare the entire ordered pair with the declared relation, keeping component order. Membership gives No."},"skills":["Ordered-Pair Membership in a Binary Relation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r03-relation-0018","batch_id":"batch-20261003-advance-20261004-r03-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"431ef61d83bc9415e85aba62ea1a5bfa41d16e59d626ff0115aba3399caeb99d","family_id":"qf-relation-pair-membership","seed":202610043018,"semantic_template":"qf-relation-pair-membership/{\"A\":[5,6,7],\"B\":[9,10,11],\"R\":[[6,9],[7,10]],\"operation\":\"pair_membership\",\"x\":7,\"y\":9}"},"fields":{"title":"Ordered-Pair Membership in a Binary Relation","stem":"The binary relation has source A={5, 6, 7}, target B={9, 10, 11}, and R={(6, 9), (7, 10)}⊆A×B. Does the ordered pair (7, 9) belong to R?","answer":{"kind":"short_text","value":"No","explanation":"Compare the entire ordered pair with the declared relation, keeping component order. Membership gives No."},"skills":["Ordered-Pair Membership in a Binary Relation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r03-relation-0019","batch_id":"batch-20261003-advance-20261004-r03-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f22af2c1aef9eac026642674de4402e76807f795afe667b63258741be42f213e","family_id":"qf-relation-pair-membership","seed":202610043019,"semantic_template":"qf-relation-pair-membership/{\"A\":[6,7,8],\"B\":[10,11,12],\"R\":[],\"operation\":\"pair_membership\",\"x\":6,\"y\":10}"},"fields":{"title":"Ordered-Pair Membership in a Binary Relation","stem":"The binary relation has source A={6, 7, 8}, target B={10, 11, 12}, and R=∅⊆A×B. Does the ordered pair (6, 10) belong to R?","answer":{"kind":"short_text","value":"No","explanation":"Compare the entire ordered pair with the declared relation, keeping component order. Membership gives No."},"skills":["Ordered-Pair Membership in a Binary Relation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r03-relation-0020","batch_id":"batch-20261003-advance-20261004-r03-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"021058b8084a57d7754665c35980ca0832dd33daee877e575a0ce7280e7eb2ec","family_id":"qf-relation-pair-membership","seed":202610043020,"semantic_template":"qf-relation-pair-membership/{\"A\":[7,8,9],\"B\":[11,12,13],\"R\":[[7,11]],\"operation\":\"pair_membership\",\"x\":11,\"y\":7}"},"fields":{"title":"Ordered-Pair Membership in a Binary Relation","stem":"The binary relation has source A={7, 8, 9}, target B={11, 12, 13}, and R={(7, 11)}⊆A×B. Does the ordered pair (11, 7) belong to R?","answer":{"kind":"short_text","value":"No","explanation":"Compare the entire ordered pair with the declared relation, keeping component order. Membership gives No."},"skills":["Ordered-Pair Membership in a Binary Relation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r03-relation-0021","batch_id":"batch-20261003-advance-20261004-r03-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8a567105d83a03f904afcf5398797f7044ca8ecb3d44bd358ed575ef9868597f","family_id":"qf-relation-pair-membership","seed":202610043021,"semantic_template":"qf-relation-pair-membership/{\"A\":[8,9,10],\"B\":[12,13,14],\"R\":[[8,12],[8,13]],\"operation\":\"pair_membership\",\"x\":10,\"y\":12}"},"fields":{"title":"Ordered-Pair Membership in a Binary Relation","stem":"The binary relation has source A={8, 9, 10}, target B={12, 13, 14}, and R={(8, 12), (8, 13)}⊆A×B. Does the ordered pair (10, 12) belong to R?","answer":{"kind":"short_text","value":"No","explanation":"Compare the entire ordered pair with the declared relation, keeping component order. Membership gives No."},"skills":["Ordered-Pair Membership in a Binary Relation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r03-relation-0022","batch_id":"batch-20261003-advance-20261004-r03-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e1993fe565ee4bb542d588651339726d88265f28999ec77143353a132eff80ed","family_id":"qf-relation-pair-membership","seed":202610043022,"semantic_template":"qf-relation-pair-membership/{\"A\":[9,10,11],\"B\":[13,14,15],\"R\":[[9,14],[10,14]],\"operation\":\"pair_membership\",\"x\":9,\"y\":14}"},"fields":{"title":"Ordered-Pair Membership in a Binary Relation","stem":"The binary relation has source A={9, 10, 11}, target B={13, 14, 15}, and R={(9, 14), (10, 14)}⊆A×B. Does the ordered pair (9, 14) belong to R?","answer":{"kind":"short_text","value":"Yes","explanation":"Compare the entire ordered pair with the declared relation, keeping component order. Membership gives Yes."},"skills":["Ordered-Pair Membership in a Binary Relation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r03-relation-0023","batch_id":"batch-20261003-advance-20261004-r03-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"17043f9bbf2faacc4735d2d9d495419ad84e26b73f206ef12b76dd525e96b23d","family_id":"qf-relation-pair-membership","seed":202610043023,"semantic_template":"qf-relation-pair-membership/{\"A\":[10,11,12],\"B\":[14,15,16],\"R\":[[10,14],[11,15],[12,16]],\"operation\":\"pair_membership\",\"x\":14,\"y\":10}"},"fields":{"title":"Ordered-Pair Membership in a Binary Relation","stem":"The binary relation has source A={10, 11, 12}, target B={14, 15, 16}, and R={(10, 14), (11, 15), (12, 16)}⊆A×B. Does the ordered pair (14, 10) belong to R?","answer":{"kind":"short_text","value":"No","explanation":"Compare the entire ordered pair with the declared relation, keeping component order. Membership gives No."},"skills":["Ordered-Pair Membership in a Binary Relation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r03-relation-0024","batch_id":"batch-20261003-advance-20261004-r03-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a271d2ac7555c160cb2d6b7f6d41499d4603905ace73df4446bb2b21967cdcd9","family_id":"qf-relation-pair-membership","seed":202610043024,"semantic_template":"qf-relation-pair-membership/{\"A\":[11,12,13],\"B\":[15,16,17],\"R\":[[12,15],[13,16]],\"operation\":\"pair_membership\",\"x\":13,\"y\":15}"},"fields":{"title":"Ordered-Pair Membership in a Binary Relation","stem":"The binary relation has source A={11, 12, 13}, target B={15, 16, 17}, and R={(12, 15), (13, 16)}⊆A×B. Does the ordered pair (13, 15) belong to R?","answer":{"kind":"short_text","value":"No","explanation":"Compare the entire ordered pair with the declared relation, keeping component order. Membership gives No."},"skills":["Ordered-Pair Membership in a Binary Relation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r03-relation-0025","batch_id":"batch-20261003-advance-20261004-r03-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"530fd87a279e9f4ed9d8cdc9faa7e751ef73d5feb4d8ca48be558705a924d482","family_id":"qf-relation-pair-membership","seed":202610043025,"semantic_template":"qf-relation-pair-membership/{\"A\":[12,13,14],\"B\":[16,17,18],\"R\":[],\"operation\":\"pair_membership\",\"x\":12,\"y\":16}"},"fields":{"title":"Ordered-Pair Membership in a Binary Relation","stem":"The binary relation has source A={12, 13, 14}, target B={16, 17, 18}, and R=∅⊆A×B. Does the ordered pair (12, 16) belong to R?","answer":{"kind":"short_text","value":"No","explanation":"Compare the entire ordered pair with the declared relation, keeping component order. Membership gives No."},"skills":["Ordered-Pair Membership in a Binary Relation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r03-relation-0026","batch_id":"batch-20261003-advance-20261004-r03-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1bd6e8b63298f6461b9482e0382ccca4b742ef55a68755a4fa72b6397359f8aa","family_id":"qf-relation-domain-projection","seed":202610043026,"semantic_template":"qf-relation-domain-projection/{\"A\":[],\"B\":[],\"R\":[],\"operation\":\"domain_projection\"}"},"fields":{"title":"First Projection of a Binary Relation","stem":"The binary relation has source A=∅, target B=∅, and R=∅⊆A×B. List its actual domain Dom(R) in increasing order with each element once.","answer":{"kind":"expression","value":"∅","explanation":"Collect only first components that actually occur in R. Dom(R)=∅; the declared source A need not equal this projection."},"skills":["First Projection of a Binary Relation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r03-relation-0027","batch_id":"batch-20261003-advance-20261004-r03-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"63ef023e6166e7837a1fe917e6be68a4af10f71fd65e99ca96a866e392422100","family_id":"qf-relation-domain-projection","seed":202610043027,"semantic_template":"qf-relation-domain-projection/{\"A\":[0],\"B\":[1],\"R\":[],\"operation\":\"domain_projection\"}"},"fields":{"title":"First Projection of a Binary Relation","stem":"The binary relation has source A={0}, target B={1}, and R=∅⊆A×B. List its actual domain Dom(R) in increasing order with each element once.","answer":{"kind":"expression","value":"∅","explanation":"Collect only first components that actually occur in R. Dom(R)=∅; the declared source A need not equal this projection."},"skills":["First Projection of a Binary Relation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r03-relation-0028","batch_id":"batch-20261003-advance-20261004-r03-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e59a87d7ce64f2edb45f917a7324553862d93dbf546068ff2d15f769bfcdb24c","family_id":"qf-relation-domain-projection","seed":202610043028,"semantic_template":"qf-relation-domain-projection/{\"A\":[-10,-9,-8],\"B\":[-6,-5,-4],\"R\":[[-10,-6],[-10,-5]],\"operation\":\"domain_projection\"}"},"fields":{"title":"First Projection of a Binary Relation","stem":"The binary relation has source A={-10, -9, -8}, target B={-6, -5, -4}, and R={(-10, -6), (-10, -5)}⊆A×B. List its actual domain Dom(R) in increasing order with each element once.","answer":{"kind":"expression","value":"{-10}","explanation":"Collect only first components that actually occur in R. Dom(R)={-10}; the declared source A need not equal this projection."},"skills":["First Projection of a Binary Relation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r03-relation-0029","batch_id":"batch-20261003-advance-20261004-r03-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8deba3ca8b395f7f5837a9554cd89c02fd519749409d1a46cbfc6371c1eefbc6","family_id":"qf-relation-domain-projection","seed":202610043029,"semantic_template":"qf-relation-domain-projection/{\"A\":[-9,-8,-7],\"B\":[-5,-4,-3],\"R\":[[-9,-4],[-8,-4]],\"operation\":\"domain_projection\"}"},"fields":{"title":"First Projection of a Binary Relation","stem":"The binary relation has source A={-9, -8, -7}, target B={-5, -4, -3}, and R={(-9, -4), (-8, -4)}⊆A×B. List its actual domain Dom(R) in increasing order with each element once.","answer":{"kind":"expression","value":"{-9, -8}","explanation":"Collect only first components that actually occur in R. Dom(R)={-9, -8}; the declared source A need not equal this projection."},"skills":["First Projection of a Binary Relation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r03-relation-0030","batch_id":"batch-20261003-advance-20261004-r03-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"142079e1cb2d88f20f8a3eb4b3ad6aa4055df7b7140afaf8e0c33307f0aa59d1","family_id":"qf-relation-domain-projection","seed":202610043030,"semantic_template":"qf-relation-domain-projection/{\"A\":[-8,-7,-6],\"B\":[-4,-3,-2],\"R\":[[-8,-4],[-7,-3],[-6,-2]],\"operation\":\"domain_projection\"}"},"fields":{"title":"First Projection of a Binary Relation","stem":"The binary relation has source A={-8, -7, -6}, target B={-4, -3, -2}, and R={(-8, -4), (-7, -3), (-6, -2)}⊆A×B. List its actual domain Dom(R) in increasing order with each element once.","answer":{"kind":"expression","value":"{-8, -7, -6}","explanation":"Collect only first components that actually occur in R. Dom(R)={-8, -7, -6}; the declared source A need not equal this projection."},"skills":["First Projection of a Binary Relation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r03-relation-0031","batch_id":"batch-20261003-advance-20261004-r03-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"50d22f52d6f227158d5519c474b0b49798303309a4183f6406e641e09e5ac28a","family_id":"qf-relation-domain-projection","seed":202610043031,"semantic_template":"qf-relation-domain-projection/{\"A\":[-7,-6,-5],\"B\":[-3,-2,-1],\"R\":[[-6,-3],[-5,-2]],\"operation\":\"domain_projection\"}"},"fields":{"title":"First Projection of a Binary Relation","stem":"The binary relation has source A={-7, -6, -5}, target B={-3, -2, -1}, and R={(-6, -3), (-5, -2)}⊆A×B. List its actual domain Dom(R) in increasing order with each element once.","answer":{"kind":"expression","value":"{-6, -5}","explanation":"Collect only first components that actually occur in R. Dom(R)={-6, -5}; the declared source A need not equal this projection."},"skills":["First Projection of a Binary Relation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r03-relation-0032","batch_id":"batch-20261003-advance-20261004-r03-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"308c1bc2bb315cf79716f36c40e892e2ce53fe2a45d85f8a2a45402d942b9f2f","family_id":"qf-relation-domain-projection","seed":202610043032,"semantic_template":"qf-relation-domain-projection/{\"A\":[-6,-5,-4],\"B\":[-2,-1,0],\"R\":[],\"operation\":\"domain_projection\"}"},"fields":{"title":"First Projection of a Binary Relation","stem":"The binary relation has source A={-6, -5, -4}, target B={-2, -1, 0}, and R=∅⊆A×B. List its actual domain Dom(R) in increasing order with each element once.","answer":{"kind":"expression","value":"∅","explanation":"Collect only first components that actually occur in R. Dom(R)=∅; the declared source A need not equal this projection."},"skills":["First Projection of a Binary Relation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r03-relation-0033","batch_id":"batch-20261003-advance-20261004-r03-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1cc0d4693e243ff419302c10a434af0671d3c4ced4995fea625874f9177cb880","family_id":"qf-relation-domain-projection","seed":202610043033,"semantic_template":"qf-relation-domain-projection/{\"A\":[-5,-4,-3],\"B\":[-1,0,1],\"R\":[[-5,-1]],\"operation\":\"domain_projection\"}"},"fields":{"title":"First Projection of a Binary Relation","stem":"The binary relation has source A={-5, -4, -3}, target B={-1, 0, 1}, and R={(-5, -1)}⊆A×B. List its actual domain Dom(R) in increasing order with each element once.","answer":{"kind":"expression","value":"{-5}","explanation":"Collect only first components that actually occur in R. Dom(R)={-5}; the declared source A need not equal this projection."},"skills":["First Projection of a Binary Relation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r03-relation-0034","batch_id":"batch-20261003-advance-20261004-r03-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5cfb5926b0021d29d8289fd536aecb5833030eba80081c41dac49a49687e9546","family_id":"qf-relation-domain-projection","seed":202610043034,"semantic_template":"qf-relation-domain-projection/{\"A\":[-4,-3,-2],\"B\":[0,1,2],\"R\":[[-4,0],[-4,1]],\"operation\":\"domain_projection\"}"},"fields":{"title":"First Projection of a Binary Relation","stem":"The binary relation has source A={-4, -3, -2}, target B={0, 1, 2}, and R={(-4, 0), (-4, 1)}⊆A×B. List its actual domain Dom(R) in increasing order with each element once.","answer":{"kind":"expression","value":"{-4}","explanation":"Collect only first components that actually occur in R. Dom(R)={-4}; the declared source A need not equal this projection."},"skills":["First Projection of a Binary Relation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r03-relation-0035","batch_id":"batch-20261003-advance-20261004-r03-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5d8c5efa5711f7d710d5b77bfb8ed7ddc7d45b33fa11e2ddba88058787c905be","family_id":"qf-relation-domain-projection","seed":202610043035,"semantic_template":"qf-relation-domain-projection/{\"A\":[-3,-2,-1],\"B\":[1,2,3],\"R\":[[-3,2],[-2,2]],\"operation\":\"domain_projection\"}"},"fields":{"title":"First Projection of a Binary Relation","stem":"The binary relation has source A={-3, -2, -1}, target B={1, 2, 3}, and R={(-3, 2), (-2, 2)}⊆A×B. List its actual domain Dom(R) in increasing order with each element once.","answer":{"kind":"expression","value":"{-3, -2}","explanation":"Collect only first components that actually occur in R. Dom(R)={-3, -2}; the declared source A need not equal this projection."},"skills":["First Projection of a Binary Relation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r03-relation-0036","batch_id":"batch-20261003-advance-20261004-r03-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cb4887227f5e8095443f13a4cd7e99d383ab5c51320f3e09465ec277c5c6e9fa","family_id":"qf-relation-domain-projection","seed":202610043036,"semantic_template":"qf-relation-domain-projection/{\"A\":[-2,-1,0],\"B\":[2,3,4],\"R\":[[-2,2],[-1,3],[0,4]],\"operation\":\"domain_projection\"}"},"fields":{"title":"First Projection of a Binary Relation","stem":"The binary relation has source A={-2, -1, 0}, target B={2, 3, 4}, and R={(-2, 2), (-1, 3), (0, 4)}⊆A×B. List its actual domain Dom(R) in increasing order with each element once.","answer":{"kind":"expression","value":"{-2, -1, 0}","explanation":"Collect only first components that actually occur in R. Dom(R)={-2, -1, 0}; the declared source A need not equal this projection."},"skills":["First Projection of a Binary Relation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r03-relation-0037","batch_id":"batch-20261003-advance-20261004-r03-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ec3ebb1bb9d9da954ed9b98c19855f8ff522b7186bddebb2b3812e54bccd0dec","family_id":"qf-relation-domain-projection","seed":202610043037,"semantic_template":"qf-relation-domain-projection/{\"A\":[-1,0,1],\"B\":[3,4,5],\"R\":[[0,3],[1,4]],\"operation\":\"domain_projection\"}"},"fields":{"title":"First Projection of a Binary Relation","stem":"The binary relation has source A={-1, 0, 1}, target B={3, 4, 5}, and R={(0, 3), (1, 4)}⊆A×B. List its actual domain Dom(R) in increasing order with each element once.","answer":{"kind":"expression","value":"{0, 1}","explanation":"Collect only first components that actually occur in R. Dom(R)={0, 1}; the declared source A need not equal this projection."},"skills":["First Projection of a Binary Relation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r03-relation-0038","batch_id":"batch-20261003-advance-20261004-r03-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f3b508b719bd54ab7e4454601b42a5d7abf342d181a34e78c5763b67ecbd833a","family_id":"qf-relation-domain-projection","seed":202610043038,"semantic_template":"qf-relation-domain-projection/{\"A\":[0,1,2],\"B\":[4,5,6],\"R\":[],\"operation\":\"domain_projection\"}"},"fields":{"title":"First Projection of a Binary Relation","stem":"The binary relation has source A={0, 1, 2}, target B={4, 5, 6}, and R=∅⊆A×B. List its actual domain Dom(R) in increasing order with each element once.","answer":{"kind":"expression","value":"∅","explanation":"Collect only first components that actually occur in R. Dom(R)=∅; the declared source A need not equal this projection."},"skills":["First Projection of a Binary Relation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r03-relation-0039","batch_id":"batch-20261003-advance-20261004-r03-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"60a8c8bab60d1b77e5a1084200e9ed51c49d37da8477c1584f872ddba959a1d2","family_id":"qf-relation-domain-projection","seed":202610043039,"semantic_template":"qf-relation-domain-projection/{\"A\":[1,2,3],\"B\":[5,6,7],\"R\":[[1,5]],\"operation\":\"domain_projection\"}"},"fields":{"title":"First Projection of a Binary Relation","stem":"The binary relation has source A={1, 2, 3}, target B={5, 6, 7}, and R={(1, 5)}⊆A×B. List its actual domain Dom(R) in increasing order with each element once.","answer":{"kind":"expression","value":"{1}","explanation":"Collect only first components that actually occur in R. Dom(R)={1}; the declared source A need not equal this projection."},"skills":["First Projection of a Binary Relation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r03-relation-0040","batch_id":"batch-20261003-advance-20261004-r03-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"02a2ce090b67f7d124dfb39ab2410301f912cb5640374cb85c4fe6497fad02ca","family_id":"qf-relation-domain-projection","seed":202610043040,"semantic_template":"qf-relation-domain-projection/{\"A\":[2,3,4],\"B\":[6,7,8],\"R\":[[2,6],[2,7]],\"operation\":\"domain_projection\"}"},"fields":{"title":"First Projection of a Binary Relation","stem":"The binary relation has source A={2, 3, 4}, target B={6, 7, 8}, and R={(2, 6), (2, 7)}⊆A×B. List its actual domain Dom(R) in increasing order with each element once.","answer":{"kind":"expression","value":"{2}","explanation":"Collect only first components that actually occur in R. Dom(R)={2}; the declared source A need not equal this projection."},"skills":["First Projection of a Binary Relation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r03-relation-0041","batch_id":"batch-20261003-advance-20261004-r03-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c2c86282fc35631be140b75addfa446d8ca73ecfc4e015f0e9cd7059006acfb9","family_id":"qf-relation-domain-projection","seed":202610043041,"semantic_template":"qf-relation-domain-projection/{\"A\":[3,4,5],\"B\":[7,8,9],\"R\":[[3,8],[4,8]],\"operation\":\"domain_projection\"}"},"fields":{"title":"First Projection of a Binary Relation","stem":"The binary relation has source A={3, 4, 5}, target B={7, 8, 9}, and R={(3, 8), (4, 8)}⊆A×B. List its actual domain Dom(R) in increasing order with each element once.","answer":{"kind":"expression","value":"{3, 4}","explanation":"Collect only first components that actually occur in R. Dom(R)={3, 4}; the declared source A need not equal this projection."},"skills":["First Projection of a Binary Relation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r03-relation-0042","batch_id":"batch-20261003-advance-20261004-r03-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"87c332bc54e9dbed5a7d03246c074c96a89138b604172b534cc1fd878f41a282","family_id":"qf-relation-domain-projection","seed":202610043042,"semantic_template":"qf-relation-domain-projection/{\"A\":[4,5,6],\"B\":[8,9,10],\"R\":[[4,8],[5,9],[6,10]],\"operation\":\"domain_projection\"}"},"fields":{"title":"First Projection of a Binary Relation","stem":"The binary relation has source A={4, 5, 6}, target B={8, 9, 10}, and R={(4, 8), (5, 9), (6, 10)}⊆A×B. List its actual domain Dom(R) in increasing order with each element once.","answer":{"kind":"expression","value":"{4, 5, 6}","explanation":"Collect only first components that actually occur in R. Dom(R)={4, 5, 6}; the declared source A need not equal this projection."},"skills":["First Projection of a Binary Relation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r03-relation-0043","batch_id":"batch-20261003-advance-20261004-r03-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"75624e5f1c1d074f6e4a3e07660bf3d74bf8d4ae76afef815ee74a6e3cf60e48","family_id":"qf-relation-domain-projection","seed":202610043043,"semantic_template":"qf-relation-domain-projection/{\"A\":[5,6,7],\"B\":[9,10,11],\"R\":[[6,9],[7,10]],\"operation\":\"domain_projection\"}"},"fields":{"title":"First Projection of a Binary Relation","stem":"The binary relation has source A={5, 6, 7}, target B={9, 10, 11}, and R={(6, 9), (7, 10)}⊆A×B. List its actual domain Dom(R) in increasing order with each element once.","answer":{"kind":"expression","value":"{6, 7}","explanation":"Collect only first components that actually occur in R. Dom(R)={6, 7}; the declared source A need not equal this projection."},"skills":["First Projection of a Binary Relation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r03-relation-0044","batch_id":"batch-20261003-advance-20261004-r03-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a38069ab1e4ee5c31df3ef0031cd738d468894e1a42efd31c9e05608e4650db3","family_id":"qf-relation-domain-projection","seed":202610043044,"semantic_template":"qf-relation-domain-projection/{\"A\":[6,7,8],\"B\":[10,11,12],\"R\":[],\"operation\":\"domain_projection\"}"},"fields":{"title":"First Projection of a Binary Relation","stem":"The binary relation has source A={6, 7, 8}, target B={10, 11, 12}, and R=∅⊆A×B. List its actual domain Dom(R) in increasing order with each element once.","answer":{"kind":"expression","value":"∅","explanation":"Collect only first components that actually occur in R. Dom(R)=∅; the declared source A need not equal this projection."},"skills":["First Projection of a Binary Relation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r03-relation-0045","batch_id":"batch-20261003-advance-20261004-r03-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e451616805c5e3abed30c6bdbe535c906fbd55b992ddfc18b6a005944bc2c535","family_id":"qf-relation-domain-projection","seed":202610043045,"semantic_template":"qf-relation-domain-projection/{\"A\":[7,8,9],\"B\":[11,12,13],\"R\":[[7,11]],\"operation\":\"domain_projection\"}"},"fields":{"title":"First Projection of a Binary Relation","stem":"The binary relation has source A={7, 8, 9}, target B={11, 12, 13}, and R={(7, 11)}⊆A×B. List its actual domain Dom(R) in increasing order with each element once.","answer":{"kind":"expression","value":"{7}","explanation":"Collect only first components that actually occur in R. Dom(R)={7}; the declared source A need not equal this projection."},"skills":["First Projection of a Binary Relation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r03-relation-0046","batch_id":"batch-20261003-advance-20261004-r03-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"68008475e09578f80c33d1b531a97c834867630afdfe163d1fd54e90108b89cb","family_id":"qf-relation-domain-projection","seed":202610043046,"semantic_template":"qf-relation-domain-projection/{\"A\":[8,9,10],\"B\":[12,13,14],\"R\":[[8,12],[8,13]],\"operation\":\"domain_projection\"}"},"fields":{"title":"First Projection of a Binary Relation","stem":"The binary relation has source A={8, 9, 10}, target B={12, 13, 14}, and R={(8, 12), (8, 13)}⊆A×B. List its actual domain Dom(R) in increasing order with each element once.","answer":{"kind":"expression","value":"{8}","explanation":"Collect only first components that actually occur in R. Dom(R)={8}; the declared source A need not equal this projection."},"skills":["First Projection of a Binary Relation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r03-relation-0047","batch_id":"batch-20261003-advance-20261004-r03-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"593531ed5934b6beefd930da0df0723e6974134fff8a526b9c1de2b05eef3ae3","family_id":"qf-relation-domain-projection","seed":202610043047,"semantic_template":"qf-relation-domain-projection/{\"A\":[9,10,11],\"B\":[13,14,15],\"R\":[[9,14],[10,14]],\"operation\":\"domain_projection\"}"},"fields":{"title":"First Projection of a Binary Relation","stem":"The binary relation has source A={9, 10, 11}, target B={13, 14, 15}, and R={(9, 14), (10, 14)}⊆A×B. List its actual domain Dom(R) in increasing order with each element once.","answer":{"kind":"expression","value":"{9, 10}","explanation":"Collect only first components that actually occur in R. Dom(R)={9, 10}; the declared source A need not equal this projection."},"skills":["First Projection of a Binary Relation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r03-relation-0048","batch_id":"batch-20261003-advance-20261004-r03-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c39e8a8fa0cd65a5726c90d2774805ee87352a6b0ef8bbb14ceef1ea5e6dee3f","family_id":"qf-relation-domain-projection","seed":202610043048,"semantic_template":"qf-relation-domain-projection/{\"A\":[10,11,12],\"B\":[14,15,16],\"R\":[[10,14],[11,15],[12,16]],\"operation\":\"domain_projection\"}"},"fields":{"title":"First Projection of a Binary Relation","stem":"The binary relation has source A={10, 11, 12}, target B={14, 15, 16}, and R={(10, 14), (11, 15), (12, 16)}⊆A×B. List its actual domain Dom(R) in increasing order with each element once.","answer":{"kind":"expression","value":"{10, 11, 12}","explanation":"Collect only first components that actually occur in R. Dom(R)={10, 11, 12}; the declared source A need not equal this projection."},"skills":["First Projection of a Binary Relation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r03-relation-0049","batch_id":"batch-20261003-advance-20261004-r03-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c496426b965873d90b72fcccc6cef6b5de5500e6fb8e294ce89e6c6fcfb15c9b","family_id":"qf-relation-domain-projection","seed":202610043049,"semantic_template":"qf-relation-domain-projection/{\"A\":[11,12,13],\"B\":[15,16,17],\"R\":[[12,15],[13,16]],\"operation\":\"domain_projection\"}"},"fields":{"title":"First Projection of a Binary Relation","stem":"The binary relation has source A={11, 12, 13}, target B={15, 16, 17}, and R={(12, 15), (13, 16)}⊆A×B. List its actual domain Dom(R) in increasing order with each element once.","answer":{"kind":"expression","value":"{12, 13}","explanation":"Collect only first components that actually occur in R. Dom(R)={12, 13}; the declared source A need not equal this projection."},"skills":["First Projection of a Binary Relation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r03-relation-0050","batch_id":"batch-20261003-advance-20261004-r03-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8be0c6bd7561f7528ce45845f264b908970fa6d73861202222f5b87c1a81b626","family_id":"qf-relation-domain-projection","seed":202610043050,"semantic_template":"qf-relation-domain-projection/{\"A\":[12,13,14],\"B\":[16,17,18],\"R\":[],\"operation\":\"domain_projection\"}"},"fields":{"title":"First Projection of a Binary Relation","stem":"The binary relation has source A={12, 13, 14}, target B={16, 17, 18}, and R=∅⊆A×B. List its actual domain Dom(R) in increasing order with each element once.","answer":{"kind":"expression","value":"∅","explanation":"Collect only first components that actually occur in R. Dom(R)=∅; the declared source A need not equal this projection."},"skills":["First Projection of a Binary Relation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r03-relation-0051","batch_id":"batch-20261003-advance-20261004-r03-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7c4faa1d055777408ce2dd5c5539ed877d3d02f8e7fdeeb56734248806c0d33d","family_id":"qf-relation-subset-image","seed":202610043051,"semantic_template":"qf-relation-subset-image/{\"A\":[],\"B\":[],\"C\":[],\"R\":[],\"operation\":\"subset_image\"}"},"fields":{"title":"Image of a Subset under a Binary Relation","stem":"The binary relation has source A=∅, target B=∅, and R=∅⊆A×B. For the explicit C=∅⊆A, list R[C] in increasing order with no duplicates.","answer":{"kind":"expression","value":"∅","explanation":"Select relation pairs whose first component lies in C and collect their second components. R[C]=∅; a relation need not have a unique output for each input."},"skills":["Image of a Subset under a Binary Relation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r03-relation-0052","batch_id":"batch-20261003-advance-20261004-r03-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9fb59ac05eb95f0be56d5c638e9937829db2aecfb8b425caedf9d5791d422049","family_id":"qf-relation-subset-image","seed":202610043052,"semantic_template":"qf-relation-subset-image/{\"A\":[0],\"B\":[1],\"C\":[0],\"R\":[],\"operation\":\"subset_image\"}"},"fields":{"title":"Image of a Subset under a Binary Relation","stem":"The binary relation has source A={0}, target B={1}, and R=∅⊆A×B. For the explicit C={0}⊆A, list R[C] in increasing order with no duplicates.","answer":{"kind":"expression","value":"∅","explanation":"Select relation pairs whose first component lies in C and collect their second components. R[C]=∅; a relation need not have a unique output for each input."},"skills":["Image of a Subset under a Binary Relation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r03-relation-0053","batch_id":"batch-20261003-advance-20261004-r03-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f6385ab414052da792a616fbfaf0d65b12c03a3b976a7c0640c4d283f753ed3a","family_id":"qf-relation-subset-image","seed":202610043053,"semantic_template":"qf-relation-subset-image/{\"A\":[-10,-9,-8],\"B\":[-6,-5,-4],\"C\":[-10],\"R\":[[-10,-6],[-10,-5]],\"operation\":\"subset_image\"}"},"fields":{"title":"Image of a Subset under a Binary Relation","stem":"The binary relation has source A={-10, -9, -8}, target B={-6, -5, -4}, and R={(-10, -6), (-10, -5)}⊆A×B. For the explicit C={-10}⊆A, list R[C] in increasing order with no duplicates.","answer":{"kind":"expression","value":"{-6, -5}","explanation":"Select relation pairs whose first component lies in C and collect their second components. R[C]={-6, -5}; a relation need not have a unique output for each input."},"skills":["Image of a Subset under a Binary Relation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r03-relation-0054","batch_id":"batch-20261003-advance-20261004-r03-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d58947fb65271703f8d593a95bd0c0694434d2aaddbbfaf51bbebb5b451bf01f","family_id":"qf-relation-subset-image","seed":202610043054,"semantic_template":"qf-relation-subset-image/{\"A\":[-9,-8,-7],\"B\":[-5,-4,-3],\"C\":[-8,-7],\"R\":[[-9,-4],[-8,-4]],\"operation\":\"subset_image\"}"},"fields":{"title":"Image of a Subset under a Binary Relation","stem":"The binary relation has source A={-9, -8, -7}, target B={-5, -4, -3}, and R={(-9, -4), (-8, -4)}⊆A×B. For the explicit C={-8, -7}⊆A, list R[C] in increasing order with no duplicates.","answer":{"kind":"expression","value":"{-4}","explanation":"Select relation pairs whose first component lies in C and collect their second components. R[C]={-4}; a relation need not have a unique output for each input."},"skills":["Image of a Subset under a Binary Relation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r03-relation-0055","batch_id":"batch-20261003-advance-20261004-r03-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"24b92662f48dbef5f25ab09df9fb2bcfd844aad61bfdfd667ab295a0d7ae122a","family_id":"qf-relation-subset-image","seed":202610043055,"semantic_template":"qf-relation-subset-image/{\"A\":[-8,-7,-6],\"B\":[-4,-3,-2],\"C\":[],\"R\":[[-8,-4],[-7,-3],[-6,-2]],\"operation\":\"subset_image\"}"},"fields":{"title":"Image of a Subset under a Binary Relation","stem":"The binary relation has source A={-8, -7, -6}, target B={-4, -3, -2}, and R={(-8, -4), (-7, -3), (-6, -2)}⊆A×B. For the explicit C=∅⊆A, list R[C] in increasing order with no duplicates.","answer":{"kind":"expression","value":"∅","explanation":"Select relation pairs whose first component lies in C and collect their second components. R[C]=∅; a relation need not have a unique output for each input."},"skills":["Image of a Subset under a Binary Relation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r03-relation-0056","batch_id":"batch-20261003-advance-20261004-r03-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"59003f7968c70f5064a5f55e9df12f5b8f4744b1279fe0a514fadacaf003e63e","family_id":"qf-relation-subset-image","seed":202610043056,"semantic_template":"qf-relation-subset-image/{\"A\":[-7,-6,-5],\"B\":[-3,-2,-1],\"C\":[-7,-6,-5],\"R\":[[-6,-3],[-5,-2]],\"operation\":\"subset_image\"}"},"fields":{"title":"Image of a Subset under a Binary Relation","stem":"The binary relation has source A={-7, -6, -5}, target B={-3, -2, -1}, and R={(-6, -3), (-5, -2)}⊆A×B. For the explicit C={-7, -6, -5}⊆A, list R[C] in increasing order with no duplicates.","answer":{"kind":"expression","value":"{-3, -2}","explanation":"Select relation pairs whose first component lies in C and collect their second components. R[C]={-3, -2}; a relation need not have a unique output for each input."},"skills":["Image of a Subset under a Binary Relation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r03-relation-0057","batch_id":"batch-20261003-advance-20261004-r03-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9bc40f8b6586006feca151370a0554b11f2b9a479cd1a56beee24fd8f5eab4f6","family_id":"qf-relation-subset-image","seed":202610043057,"semantic_template":"qf-relation-subset-image/{\"A\":[-6,-5,-4],\"B\":[-2,-1,0],\"C\":[-6],\"R\":[],\"operation\":\"subset_image\"}"},"fields":{"title":"Image of a Subset under a Binary Relation","stem":"The binary relation has source A={-6, -5, -4}, target B={-2, -1, 0}, and R=∅⊆A×B. For the explicit C={-6}⊆A, list R[C] in increasing order with no duplicates.","answer":{"kind":"expression","value":"∅","explanation":"Select relation pairs whose first component lies in C and collect their second components. R[C]=∅; a relation need not have a unique output for each input."},"skills":["Image of a Subset under a Binary Relation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r03-relation-0058","batch_id":"batch-20261003-advance-20261004-r03-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3be97eb315a9662b22c98bafc7dc4d933c933356d94af26983eac335d95b2171","family_id":"qf-relation-subset-image","seed":202610043058,"semantic_template":"qf-relation-subset-image/{\"A\":[-5,-4,-3],\"B\":[-1,0,1],\"C\":[-4,-3],\"R\":[[-5,-1]],\"operation\":\"subset_image\"}"},"fields":{"title":"Image of a Subset under a Binary Relation","stem":"The binary relation has source A={-5, -4, -3}, target B={-1, 0, 1}, and R={(-5, -1)}⊆A×B. For the explicit C={-4, -3}⊆A, list R[C] in increasing order with no duplicates.","answer":{"kind":"expression","value":"∅","explanation":"Select relation pairs whose first component lies in C and collect their second components. R[C]=∅; a relation need not have a unique output for each input."},"skills":["Image of a Subset under a Binary Relation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r03-relation-0059","batch_id":"batch-20261003-advance-20261004-r03-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c891d112807e998fb974b1e379bb356dbfc39ea378658ce5f87f219fd8c540b4","family_id":"qf-relation-subset-image","seed":202610043059,"semantic_template":"qf-relation-subset-image/{\"A\":[-4,-3,-2],\"B\":[0,1,2],\"C\":[],\"R\":[[-4,0],[-4,1]],\"operation\":\"subset_image\"}"},"fields":{"title":"Image of a Subset under a Binary Relation","stem":"The binary relation has source A={-4, -3, -2}, target B={0, 1, 2}, and R={(-4, 0), (-4, 1)}⊆A×B. For the explicit C=∅⊆A, list R[C] in increasing order with no duplicates.","answer":{"kind":"expression","value":"∅","explanation":"Select relation pairs whose first component lies in C and collect their second components. R[C]=∅; a relation need not have a unique output for each input."},"skills":["Image of a Subset under a Binary Relation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r03-relation-0060","batch_id":"batch-20261003-advance-20261004-r03-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"03437cf4d39504db98cd0f3d41aa3413d466530276da86e4fb60bb61f0a48a70","family_id":"qf-relation-subset-image","seed":202610043060,"semantic_template":"qf-relation-subset-image/{\"A\":[-3,-2,-1],\"B\":[1,2,3],\"C\":[-3,-2,-1],\"R\":[[-3,2],[-2,2]],\"operation\":\"subset_image\"}"},"fields":{"title":"Image of a Subset under a Binary Relation","stem":"The binary relation has source A={-3, -2, -1}, target B={1, 2, 3}, and R={(-3, 2), (-2, 2)}⊆A×B. For the explicit C={-3, -2, -1}⊆A, list R[C] in increasing order with no duplicates.","answer":{"kind":"expression","value":"{2}","explanation":"Select relation pairs whose first component lies in C and collect their second components. R[C]={2}; a relation need not have a unique output for each input."},"skills":["Image of a Subset under a Binary Relation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r03-relation-0061","batch_id":"batch-20261003-advance-20261004-r03-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fefa444c9efff8cf48c39a00509ab9f8622e4dd01fcf15d0d5eb80d0e4f37b2e","family_id":"qf-relation-subset-image","seed":202610043061,"semantic_template":"qf-relation-subset-image/{\"A\":[-2,-1,0],\"B\":[2,3,4],\"C\":[-2],\"R\":[[-2,2],[-1,3],[0,4]],\"operation\":\"subset_image\"}"},"fields":{"title":"Image of a Subset under a Binary Relation","stem":"The binary relation has source A={-2, -1, 0}, target B={2, 3, 4}, and R={(-2, 2), (-1, 3), (0, 4)}⊆A×B. For the explicit C={-2}⊆A, list R[C] in increasing order with no duplicates.","answer":{"kind":"expression","value":"{2}","explanation":"Select relation pairs whose first component lies in C and collect their second components. R[C]={2}; a relation need not have a unique output for each input."},"skills":["Image of a Subset under a Binary Relation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r03-relation-0062","batch_id":"batch-20261003-advance-20261004-r03-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8d2cb5068634fdee20cfcc8711022f598aaf9146b5221e529a8945e3bd7fd15a","family_id":"qf-relation-subset-image","seed":202610043062,"semantic_template":"qf-relation-subset-image/{\"A\":[-1,0,1],\"B\":[3,4,5],\"C\":[0,1],\"R\":[[0,3],[1,4]],\"operation\":\"subset_image\"}"},"fields":{"title":"Image of a Subset under a Binary Relation","stem":"The binary relation has source A={-1, 0, 1}, target B={3, 4, 5}, and R={(0, 3), (1, 4)}⊆A×B. For the explicit C={0, 1}⊆A, list R[C] in increasing order with no duplicates.","answer":{"kind":"expression","value":"{3, 4}","explanation":"Select relation pairs whose first component lies in C and collect their second components. R[C]={3, 4}; a relation need not have a unique output for each input."},"skills":["Image of a Subset under a Binary Relation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r03-relation-0063","batch_id":"batch-20261003-advance-20261004-r03-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7d374639b96991af5bc34ede7840c82206733ce0ec3ddbd73140e48a92c854f4","family_id":"qf-relation-subset-image","seed":202610043063,"semantic_template":"qf-relation-subset-image/{\"A\":[0,1,2],\"B\":[4,5,6],\"C\":[],\"R\":[],\"operation\":\"subset_image\"}"},"fields":{"title":"Image of a Subset under a Binary Relation","stem":"The binary relation has source A={0, 1, 2}, target B={4, 5, 6}, and R=∅⊆A×B. For the explicit C=∅⊆A, list R[C] in increasing order with no duplicates.","answer":{"kind":"expression","value":"∅","explanation":"Select relation pairs whose first component lies in C and collect their second components. R[C]=∅; a relation need not have a unique output for each input."},"skills":["Image of a Subset under a Binary Relation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r03-relation-0064","batch_id":"batch-20261003-advance-20261004-r03-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"dae84b64aff2b5cbe6b92d0987e04d92bf4c01c06736f8cdc45b19feef8a9d22","family_id":"qf-relation-subset-image","seed":202610043064,"semantic_template":"qf-relation-subset-image/{\"A\":[1,2,3],\"B\":[5,6,7],\"C\":[1,2,3],\"R\":[[1,5]],\"operation\":\"subset_image\"}"},"fields":{"title":"Image of a Subset under a Binary Relation","stem":"The binary relation has source A={1, 2, 3}, target B={5, 6, 7}, and R={(1, 5)}⊆A×B. For the explicit C={1, 2, 3}⊆A, list R[C] in increasing order with no duplicates.","answer":{"kind":"expression","value":"{5}","explanation":"Select relation pairs whose first component lies in C and collect their second components. R[C]={5}; a relation need not have a unique output for each input."},"skills":["Image of a Subset under a Binary Relation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r03-relation-0065","batch_id":"batch-20261003-advance-20261004-r03-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c84de39d036486d1105da89c337eba429b9d0766224e2370a32aed4d35ef74fb","family_id":"qf-relation-subset-image","seed":202610043065,"semantic_template":"qf-relation-subset-image/{\"A\":[2,3,4],\"B\":[6,7,8],\"C\":[2],\"R\":[[2,6],[2,7]],\"operation\":\"subset_image\"}"},"fields":{"title":"Image of a Subset under a Binary Relation","stem":"The binary relation has source A={2, 3, 4}, target B={6, 7, 8}, and R={(2, 6), (2, 7)}⊆A×B. For the explicit C={2}⊆A, list R[C] in increasing order with no duplicates.","answer":{"kind":"expression","value":"{6, 7}","explanation":"Select relation pairs whose first component lies in C and collect their second components. R[C]={6, 7}; a relation need not have a unique output for each input."},"skills":["Image of a Subset under a Binary Relation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r03-relation-0066","batch_id":"batch-20261003-advance-20261004-r03-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4a6fda7916b2a696753381cdca8e038aedef4fa82a0f53f39c93d8669e50a08d","family_id":"qf-relation-subset-image","seed":202610043066,"semantic_template":"qf-relation-subset-image/{\"A\":[3,4,5],\"B\":[7,8,9],\"C\":[4,5],\"R\":[[3,8],[4,8]],\"operation\":\"subset_image\"}"},"fields":{"title":"Image of a Subset under a Binary Relation","stem":"The binary relation has source A={3, 4, 5}, target B={7, 8, 9}, and R={(3, 8), (4, 8)}⊆A×B. For the explicit C={4, 5}⊆A, list R[C] in increasing order with no duplicates.","answer":{"kind":"expression","value":"{8}","explanation":"Select relation pairs whose first component lies in C and collect their second components. R[C]={8}; a relation need not have a unique output for each input."},"skills":["Image of a Subset under a Binary Relation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r03-relation-0067","batch_id":"batch-20261003-advance-20261004-r03-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ec60eece90f9bafaa3b7d64a8191af577eded5562d228099495e3a312e88c06b","family_id":"qf-relation-subset-image","seed":202610043067,"semantic_template":"qf-relation-subset-image/{\"A\":[4,5,6],\"B\":[8,9,10],\"C\":[],\"R\":[[4,8],[5,9],[6,10]],\"operation\":\"subset_image\"}"},"fields":{"title":"Image of a Subset under a Binary Relation","stem":"The binary relation has source A={4, 5, 6}, target B={8, 9, 10}, and R={(4, 8), (5, 9), (6, 10)}⊆A×B. For the explicit C=∅⊆A, list R[C] in increasing order with no duplicates.","answer":{"kind":"expression","value":"∅","explanation":"Select relation pairs whose first component lies in C and collect their second components. R[C]=∅; a relation need not have a unique output for each input."},"skills":["Image of a Subset under a Binary Relation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r03-relation-0068","batch_id":"batch-20261003-advance-20261004-r03-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ed17b6d843faacea4e614bdebe8040dbe52813a0883dad3e3e3e226f11266d80","family_id":"qf-relation-subset-image","seed":202610043068,"semantic_template":"qf-relation-subset-image/{\"A\":[5,6,7],\"B\":[9,10,11],\"C\":[5,6,7],\"R\":[[6,9],[7,10]],\"operation\":\"subset_image\"}"},"fields":{"title":"Image of a Subset under a Binary Relation","stem":"The binary relation has source A={5, 6, 7}, target B={9, 10, 11}, and R={(6, 9), (7, 10)}⊆A×B. For the explicit C={5, 6, 7}⊆A, list R[C] in increasing order with no duplicates.","answer":{"kind":"expression","value":"{9, 10}","explanation":"Select relation pairs whose first component lies in C and collect their second components. R[C]={9, 10}; a relation need not have a unique output for each input."},"skills":["Image of a Subset under a Binary Relation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r03-relation-0069","batch_id":"batch-20261003-advance-20261004-r03-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b41635ed268e12ef1f464779e8a095c73825cc2fcddc4f9db2e95ad913735901","family_id":"qf-relation-subset-image","seed":202610043069,"semantic_template":"qf-relation-subset-image/{\"A\":[6,7,8],\"B\":[10,11,12],\"C\":[6],\"R\":[],\"operation\":\"subset_image\"}"},"fields":{"title":"Image of a Subset under a Binary Relation","stem":"The binary relation has source A={6, 7, 8}, target B={10, 11, 12}, and R=∅⊆A×B. For the explicit C={6}⊆A, list R[C] in increasing order with no duplicates.","answer":{"kind":"expression","value":"∅","explanation":"Select relation pairs whose first component lies in C and collect their second components. R[C]=∅; a relation need not have a unique output for each input."},"skills":["Image of a Subset under a Binary Relation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r03-relation-0070","batch_id":"batch-20261003-advance-20261004-r03-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"af6ad9c315205ca85c7a1fd6f1ec714b145bcc663b34e0816a787c56a04a957b","family_id":"qf-relation-subset-image","seed":202610043070,"semantic_template":"qf-relation-subset-image/{\"A\":[7,8,9],\"B\":[11,12,13],\"C\":[8,9],\"R\":[[7,11]],\"operation\":\"subset_image\"}"},"fields":{"title":"Image of a Subset under a Binary Relation","stem":"The binary relation has source A={7, 8, 9}, target B={11, 12, 13}, and R={(7, 11)}⊆A×B. For the explicit C={8, 9}⊆A, list R[C] in increasing order with no duplicates.","answer":{"kind":"expression","value":"∅","explanation":"Select relation pairs whose first component lies in C and collect their second components. R[C]=∅; a relation need not have a unique output for each input."},"skills":["Image of a Subset under a Binary Relation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r03-relation-0071","batch_id":"batch-20261003-advance-20261004-r03-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3e02f98cf55c12c08066f381db9f7dd04a3182340be685d8b3d58e67a85cedc7","family_id":"qf-relation-subset-image","seed":202610043071,"semantic_template":"qf-relation-subset-image/{\"A\":[8,9,10],\"B\":[12,13,14],\"C\":[],\"R\":[[8,12],[8,13]],\"operation\":\"subset_image\"}"},"fields":{"title":"Image of a Subset under a Binary Relation","stem":"The binary relation has source A={8, 9, 10}, target B={12, 13, 14}, and R={(8, 12), (8, 13)}⊆A×B. For the explicit C=∅⊆A, list R[C] in increasing order with no duplicates.","answer":{"kind":"expression","value":"∅","explanation":"Select relation pairs whose first component lies in C and collect their second components. R[C]=∅; a relation need not have a unique output for each input."},"skills":["Image of a Subset under a Binary Relation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r03-relation-0072","batch_id":"batch-20261003-advance-20261004-r03-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b9acf02d1bf87f1af2805a90dd7d005fb4af7a4d85bb4393c0f5d06bb64d77c4","family_id":"qf-relation-subset-image","seed":202610043072,"semantic_template":"qf-relation-subset-image/{\"A\":[9,10,11],\"B\":[13,14,15],\"C\":[9,10,11],\"R\":[[9,14],[10,14]],\"operation\":\"subset_image\"}"},"fields":{"title":"Image of a Subset under a Binary Relation","stem":"The binary relation has source A={9, 10, 11}, target B={13, 14, 15}, and R={(9, 14), (10, 14)}⊆A×B. For the explicit C={9, 10, 11}⊆A, list R[C] in increasing order with no duplicates.","answer":{"kind":"expression","value":"{14}","explanation":"Select relation pairs whose first component lies in C and collect their second components. R[C]={14}; a relation need not have a unique output for each input."},"skills":["Image of a Subset under a Binary Relation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r03-relation-0073","batch_id":"batch-20261003-advance-20261004-r03-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c68a0d95ece3cac4aabb131b0997a2f62be2e19501d572f59df24146d490eb3f","family_id":"qf-relation-subset-image","seed":202610043073,"semantic_template":"qf-relation-subset-image/{\"A\":[10,11,12],\"B\":[14,15,16],\"C\":[10],\"R\":[[10,14],[11,15],[12,16]],\"operation\":\"subset_image\"}"},"fields":{"title":"Image of a Subset under a Binary Relation","stem":"The binary relation has source A={10, 11, 12}, target B={14, 15, 16}, and R={(10, 14), (11, 15), (12, 16)}⊆A×B. For the explicit C={10}⊆A, list R[C] in increasing order with no duplicates.","answer":{"kind":"expression","value":"{14}","explanation":"Select relation pairs whose first component lies in C and collect their second components. R[C]={14}; a relation need not have a unique output for each input."},"skills":["Image of a Subset under a Binary Relation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r03-relation-0074","batch_id":"batch-20261003-advance-20261004-r03-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9a31481db263f21b52906e7d8458cfefd0f0ae53f5a1917ee2d1ee2d54d77b26","family_id":"qf-relation-subset-image","seed":202610043074,"semantic_template":"qf-relation-subset-image/{\"A\":[11,12,13],\"B\":[15,16,17],\"C\":[12,13],\"R\":[[12,15],[13,16]],\"operation\":\"subset_image\"}"},"fields":{"title":"Image of a Subset under a Binary Relation","stem":"The binary relation has source A={11, 12, 13}, target B={15, 16, 17}, and R={(12, 15), (13, 16)}⊆A×B. For the explicit C={12, 13}⊆A, list R[C] in increasing order with no duplicates.","answer":{"kind":"expression","value":"{15, 16}","explanation":"Select relation pairs whose first component lies in C and collect their second components. R[C]={15, 16}; a relation need not have a unique output for each input."},"skills":["Image of a Subset under a Binary Relation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r03-relation-0075","batch_id":"batch-20261003-advance-20261004-r03-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ef007f59782a6c8fb91f129d34514ed4c5d2caba7ba4087de7f71a741fc78bde","family_id":"qf-relation-subset-image","seed":202610043075,"semantic_template":"qf-relation-subset-image/{\"A\":[12,13,14],\"B\":[16,17,18],\"C\":[],\"R\":[],\"operation\":\"subset_image\"}"},"fields":{"title":"Image of a Subset under a Binary Relation","stem":"The binary relation has source A={12, 13, 14}, target B={16, 17, 18}, and R=∅⊆A×B. For the explicit C=∅⊆A, list R[C] in increasing order with no duplicates.","answer":{"kind":"expression","value":"∅","explanation":"Select relation pairs whose first component lies in C and collect their second components. R[C]=∅; a relation need not have a unique output for each input."},"skills":["Image of a Subset under a Binary Relation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r03-relation-0076","batch_id":"batch-20261003-advance-20261004-r03-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"99c312d448bd3d55bf11c50896a9f27d96e4fdd34c97ce5c506f97f189224c01","family_id":"qf-relation-converse-set","seed":202610043076,"semantic_template":"qf-relation-converse-set/{\"A\":[],\"B\":[],\"R\":[],\"operation\":\"converse\"}"},"fields":{"title":"Converse of a Binary Relation","stem":"The binary relation has source A=∅, target B=∅, and R=∅⊆A×B. After exchanging source and target, list the converse R⁻¹ as ordered pairs sorted by first and then second component, with duplicates removed.","answer":{"kind":"expression","value":"∅","explanation":"Exchange every (x,y) for (y,x), giving R⁻¹=∅. A converse relation is not automatically an inverse function."},"skills":["Converse of a Binary Relation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r03-relation-0077","batch_id":"batch-20261003-advance-20261004-r03-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"19a4a4c718d781f000e9fb15568c58f36701c5963e9c7c82d6c4c7d1116a3049","family_id":"qf-relation-converse-set","seed":202610043077,"semantic_template":"qf-relation-converse-set/{\"A\":[0],\"B\":[1],\"R\":[],\"operation\":\"converse\"}"},"fields":{"title":"Converse of a Binary Relation","stem":"The binary relation has source A={0}, target B={1}, and R=∅⊆A×B. After exchanging source and target, list the converse R⁻¹ as ordered pairs sorted by first and then second component, with duplicates removed.","answer":{"kind":"expression","value":"∅","explanation":"Exchange every (x,y) for (y,x), giving R⁻¹=∅. A converse relation is not automatically an inverse function."},"skills":["Converse of a Binary Relation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r03-relation-0078","batch_id":"batch-20261003-advance-20261004-r03-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4e8f98230cb73720af81a0652f46080dce700beb42bb10a82bde1e2e44b0fecf","family_id":"qf-relation-converse-set","seed":202610043078,"semantic_template":"qf-relation-converse-set/{\"A\":[-10,-9,-8],\"B\":[-6,-5,-4],\"R\":[[-10,-6],[-10,-5]],\"operation\":\"converse\"}"},"fields":{"title":"Converse of a Binary Relation","stem":"The binary relation has source A={-10, -9, -8}, target B={-6, -5, -4}, and R={(-10, -6), (-10, -5)}⊆A×B. After exchanging source and target, list the converse R⁻¹ as ordered pairs sorted by first and then second component, with duplicates removed.","answer":{"kind":"expression","value":"{(-6, -10), (-5, -10)}","explanation":"Exchange every (x,y) for (y,x), giving R⁻¹={(-6, -10), (-5, -10)}. A converse relation is not automatically an inverse function."},"skills":["Converse of a Binary Relation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r03-relation-0079","batch_id":"batch-20261003-advance-20261004-r03-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7820f592214397b925248f579df1c220b58b08cb72d578fa5a241695b64af1ae","family_id":"qf-relation-converse-set","seed":202610043079,"semantic_template":"qf-relation-converse-set/{\"A\":[-9,-8,-7],\"B\":[-5,-4,-3],\"R\":[[-9,-4],[-8,-4]],\"operation\":\"converse\"}"},"fields":{"title":"Converse of a Binary Relation","stem":"The binary relation has source A={-9, -8, -7}, target B={-5, -4, -3}, and R={(-9, -4), (-8, -4)}⊆A×B. After exchanging source and target, list the converse R⁻¹ as ordered pairs sorted by first and then second component, with duplicates removed.","answer":{"kind":"expression","value":"{(-4, -9), (-4, -8)}","explanation":"Exchange every (x,y) for (y,x), giving R⁻¹={(-4, -9), (-4, -8)}. A converse relation is not automatically an inverse function."},"skills":["Converse of a Binary Relation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r03-relation-0080","batch_id":"batch-20261003-advance-20261004-r03-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8b1b8537e2dd8230d47d76c8c7eaa166c099ed4787e673dcc5cf6c14a4adb5a8","family_id":"qf-relation-converse-set","seed":202610043080,"semantic_template":"qf-relation-converse-set/{\"A\":[-8,-7,-6],\"B\":[-4,-3,-2],\"R\":[[-8,-4],[-7,-3],[-6,-2]],\"operation\":\"converse\"}"},"fields":{"title":"Converse of a Binary Relation","stem":"The binary relation has source A={-8, -7, -6}, target B={-4, -3, -2}, and R={(-8, -4), (-7, -3), (-6, -2)}⊆A×B. After exchanging source and target, list the converse R⁻¹ as ordered pairs sorted by first and then second component, with duplicates removed.","answer":{"kind":"expression","value":"{(-4, -8), (-3, -7), (-2, -6)}","explanation":"Exchange every (x,y) for (y,x), giving R⁻¹={(-4, -8), (-3, -7), (-2, -6)}. A converse relation is not automatically an inverse function."},"skills":["Converse of a Binary Relation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r03-relation-0081","batch_id":"batch-20261003-advance-20261004-r03-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0da689eaa19a4b03eaf5cb3aaf7f984b005cb37528e2f55461b3c70f2ab35f8a","family_id":"qf-relation-converse-set","seed":202610043081,"semantic_template":"qf-relation-converse-set/{\"A\":[-7,-6,-5],\"B\":[-3,-2,-1],\"R\":[[-6,-3],[-5,-2]],\"operation\":\"converse\"}"},"fields":{"title":"Converse of a Binary Relation","stem":"The binary relation has source A={-7, -6, -5}, target B={-3, -2, -1}, and R={(-6, -3), (-5, -2)}⊆A×B. After exchanging source and target, list the converse R⁻¹ as ordered pairs sorted by first and then second component, with duplicates removed.","answer":{"kind":"expression","value":"{(-3, -6), (-2, -5)}","explanation":"Exchange every (x,y) for (y,x), giving R⁻¹={(-3, -6), (-2, -5)}. A converse relation is not automatically an inverse function."},"skills":["Converse of a Binary Relation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r03-relation-0082","batch_id":"batch-20261003-advance-20261004-r03-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"24eb36edb2e847a84d75765ee716ee17e0d13fdd9d31375b08973cc8bda9b283","family_id":"qf-relation-converse-set","seed":202610043082,"semantic_template":"qf-relation-converse-set/{\"A\":[-6,-5,-4],\"B\":[-2,-1,0],\"R\":[],\"operation\":\"converse\"}"},"fields":{"title":"Converse of a Binary Relation","stem":"The binary relation has source A={-6, -5, -4}, target B={-2, -1, 0}, and R=∅⊆A×B. After exchanging source and target, list the converse R⁻¹ as ordered pairs sorted by first and then second component, with duplicates removed.","answer":{"kind":"expression","value":"∅","explanation":"Exchange every (x,y) for (y,x), giving R⁻¹=∅. A converse relation is not automatically an inverse function."},"skills":["Converse of a Binary Relation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r03-relation-0083","batch_id":"batch-20261003-advance-20261004-r03-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"18ff38297cf7f8401657813e684c6b08aad4986dd049f67752f3aae3638b2b95","family_id":"qf-relation-converse-set","seed":202610043083,"semantic_template":"qf-relation-converse-set/{\"A\":[-5,-4,-3],\"B\":[-1,0,1],\"R\":[[-5,-1]],\"operation\":\"converse\"}"},"fields":{"title":"Converse of a Binary Relation","stem":"The binary relation has source A={-5, -4, -3}, target B={-1, 0, 1}, and R={(-5, -1)}⊆A×B. After exchanging source and target, list the converse R⁻¹ as ordered pairs sorted by first and then second component, with duplicates removed.","answer":{"kind":"expression","value":"{(-1, -5)}","explanation":"Exchange every (x,y) for (y,x), giving R⁻¹={(-1, -5)}. A converse relation is not automatically an inverse function."},"skills":["Converse of a Binary Relation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r03-relation-0084","batch_id":"batch-20261003-advance-20261004-r03-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c68701065e3826ddc43c9dd2640a330c84adee199c650aaaa2baf10af2f6679c","family_id":"qf-relation-converse-set","seed":202610043084,"semantic_template":"qf-relation-converse-set/{\"A\":[-4,-3,-2],\"B\":[0,1,2],\"R\":[[-4,0],[-4,1]],\"operation\":\"converse\"}"},"fields":{"title":"Converse of a Binary Relation","stem":"The binary relation has source A={-4, -3, -2}, target B={0, 1, 2}, and R={(-4, 0), (-4, 1)}⊆A×B. After exchanging source and target, list the converse R⁻¹ as ordered pairs sorted by first and then second component, with duplicates removed.","answer":{"kind":"expression","value":"{(0, -4), (1, -4)}","explanation":"Exchange every (x,y) for (y,x), giving R⁻¹={(0, -4), (1, -4)}. A converse relation is not automatically an inverse function."},"skills":["Converse of a Binary Relation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r03-relation-0085","batch_id":"batch-20261003-advance-20261004-r03-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"297e75b01336e01e48c64c1b44acd0493d14de89f83e4175495ace2af84d1a96","family_id":"qf-relation-converse-set","seed":202610043085,"semantic_template":"qf-relation-converse-set/{\"A\":[-3,-2,-1],\"B\":[1,2,3],\"R\":[[-3,2],[-2,2]],\"operation\":\"converse\"}"},"fields":{"title":"Converse of a Binary Relation","stem":"The binary relation has source A={-3, -2, -1}, target B={1, 2, 3}, and R={(-3, 2), (-2, 2)}⊆A×B. After exchanging source and target, list the converse R⁻¹ as ordered pairs sorted by first and then second component, with duplicates removed.","answer":{"kind":"expression","value":"{(2, -3), (2, -2)}","explanation":"Exchange every (x,y) for (y,x), giving R⁻¹={(2, -3), (2, -2)}. A converse relation is not automatically an inverse function."},"skills":["Converse of a Binary Relation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r03-relation-0086","batch_id":"batch-20261003-advance-20261004-r03-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1ac5a2d47a2af27af25add3bdb136cb768bd07181f74ffe362d62e99fb0df8cb","family_id":"qf-relation-converse-set","seed":202610043086,"semantic_template":"qf-relation-converse-set/{\"A\":[-2,-1,0],\"B\":[2,3,4],\"R\":[[-2,2],[-1,3],[0,4]],\"operation\":\"converse\"}"},"fields":{"title":"Converse of a Binary Relation","stem":"The binary relation has source A={-2, -1, 0}, target B={2, 3, 4}, and R={(-2, 2), (-1, 3), (0, 4)}⊆A×B. After exchanging source and target, list the converse R⁻¹ as ordered pairs sorted by first and then second component, with duplicates removed.","answer":{"kind":"expression","value":"{(2, -2), (3, -1), (4, 0)}","explanation":"Exchange every (x,y) for (y,x), giving R⁻¹={(2, -2), (3, -1), (4, 0)}. A converse relation is not automatically an inverse function."},"skills":["Converse of a Binary Relation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r03-relation-0087","batch_id":"batch-20261003-advance-20261004-r03-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b3a36b0a9535705496c7d2cf510f3c255101e2c23556adc4c20199c011cb0edc","family_id":"qf-relation-converse-set","seed":202610043087,"semantic_template":"qf-relation-converse-set/{\"A\":[-1,0,1],\"B\":[3,4,5],\"R\":[[0,3],[1,4]],\"operation\":\"converse\"}"},"fields":{"title":"Converse of a Binary Relation","stem":"The binary relation has source A={-1, 0, 1}, target B={3, 4, 5}, and R={(0, 3), (1, 4)}⊆A×B. After exchanging source and target, list the converse R⁻¹ as ordered pairs sorted by first and then second component, with duplicates removed.","answer":{"kind":"expression","value":"{(3, 0), (4, 1)}","explanation":"Exchange every (x,y) for (y,x), giving R⁻¹={(3, 0), (4, 1)}. A converse relation is not automatically an inverse function."},"skills":["Converse of a Binary Relation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r03-relation-0088","batch_id":"batch-20261003-advance-20261004-r03-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1833b4ac1ce2e8dc19f6ab79c6d9852021117bcb9ae2e9869fd870083b1621e2","family_id":"qf-relation-converse-set","seed":202610043088,"semantic_template":"qf-relation-converse-set/{\"A\":[0,1,2],\"B\":[4,5,6],\"R\":[],\"operation\":\"converse\"}"},"fields":{"title":"Converse of a Binary Relation","stem":"The binary relation has source A={0, 1, 2}, target B={4, 5, 6}, and R=∅⊆A×B. After exchanging source and target, list the converse R⁻¹ as ordered pairs sorted by first and then second component, with duplicates removed.","answer":{"kind":"expression","value":"∅","explanation":"Exchange every (x,y) for (y,x), giving R⁻¹=∅. A converse relation is not automatically an inverse function."},"skills":["Converse of a Binary Relation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r03-relation-0089","batch_id":"batch-20261003-advance-20261004-r03-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b40c17d8784e4c8552828c2e84a3097c7f6d63a0d3aef2c548f8616c9f7f66cb","family_id":"qf-relation-converse-set","seed":202610043089,"semantic_template":"qf-relation-converse-set/{\"A\":[1,2,3],\"B\":[5,6,7],\"R\":[[1,5]],\"operation\":\"converse\"}"},"fields":{"title":"Converse of a Binary Relation","stem":"The binary relation has source A={1, 2, 3}, target B={5, 6, 7}, and R={(1, 5)}⊆A×B. After exchanging source and target, list the converse R⁻¹ as ordered pairs sorted by first and then second component, with duplicates removed.","answer":{"kind":"expression","value":"{(5, 1)}","explanation":"Exchange every (x,y) for (y,x), giving R⁻¹={(5, 1)}. A converse relation is not automatically an inverse function."},"skills":["Converse of a Binary Relation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r03-relation-0090","batch_id":"batch-20261003-advance-20261004-r03-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1c19a0628ca3b8e1107282c8668a2297128e9d07622fa438917d28aa3dd50e8e","family_id":"qf-relation-converse-set","seed":202610043090,"semantic_template":"qf-relation-converse-set/{\"A\":[2,3,4],\"B\":[6,7,8],\"R\":[[2,6],[2,7]],\"operation\":\"converse\"}"},"fields":{"title":"Converse of a Binary Relation","stem":"The binary relation has source A={2, 3, 4}, target B={6, 7, 8}, and R={(2, 6), (2, 7)}⊆A×B. After exchanging source and target, list the converse R⁻¹ as ordered pairs sorted by first and then second component, with duplicates removed.","answer":{"kind":"expression","value":"{(6, 2), (7, 2)}","explanation":"Exchange every (x,y) for (y,x), giving R⁻¹={(6, 2), (7, 2)}. A converse relation is not automatically an inverse function."},"skills":["Converse of a Binary Relation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r03-relation-0091","batch_id":"batch-20261003-advance-20261004-r03-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b7daaa365551bb3c26bb4142439aae32debd6d28598632f073c033d938be0269","family_id":"qf-relation-converse-set","seed":202610043091,"semantic_template":"qf-relation-converse-set/{\"A\":[3,4,5],\"B\":[7,8,9],\"R\":[[3,8],[4,8]],\"operation\":\"converse\"}"},"fields":{"title":"Converse of a Binary Relation","stem":"The binary relation has source A={3, 4, 5}, target B={7, 8, 9}, and R={(3, 8), (4, 8)}⊆A×B. After exchanging source and target, list the converse R⁻¹ as ordered pairs sorted by first and then second component, with duplicates removed.","answer":{"kind":"expression","value":"{(8, 3), (8, 4)}","explanation":"Exchange every (x,y) for (y,x), giving R⁻¹={(8, 3), (8, 4)}. A converse relation is not automatically an inverse function."},"skills":["Converse of a Binary Relation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r03-relation-0092","batch_id":"batch-20261003-advance-20261004-r03-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6eb31fca665c48aead60fff0310f680f35929ef9193d69782bb6433668083649","family_id":"qf-relation-converse-set","seed":202610043092,"semantic_template":"qf-relation-converse-set/{\"A\":[4,5,6],\"B\":[8,9,10],\"R\":[[4,8],[5,9],[6,10]],\"operation\":\"converse\"}"},"fields":{"title":"Converse of a Binary Relation","stem":"The binary relation has source A={4, 5, 6}, target B={8, 9, 10}, and R={(4, 8), (5, 9), (6, 10)}⊆A×B. After exchanging source and target, list the converse R⁻¹ as ordered pairs sorted by first and then second component, with duplicates removed.","answer":{"kind":"expression","value":"{(8, 4), (9, 5), (10, 6)}","explanation":"Exchange every (x,y) for (y,x), giving R⁻¹={(8, 4), (9, 5), (10, 6)}. A converse relation is not automatically an inverse function."},"skills":["Converse of a Binary Relation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r03-relation-0093","batch_id":"batch-20261003-advance-20261004-r03-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a8cd8cb98c43878a52b942e8d33cc6c808b9458ef844eb44f333ba5d4df8d787","family_id":"qf-relation-converse-set","seed":202610043093,"semantic_template":"qf-relation-converse-set/{\"A\":[5,6,7],\"B\":[9,10,11],\"R\":[[6,9],[7,10]],\"operation\":\"converse\"}"},"fields":{"title":"Converse of a Binary Relation","stem":"The binary relation has source A={5, 6, 7}, target B={9, 10, 11}, and R={(6, 9), (7, 10)}⊆A×B. After exchanging source and target, list the converse R⁻¹ as ordered pairs sorted by first and then second component, with duplicates removed.","answer":{"kind":"expression","value":"{(9, 6), (10, 7)}","explanation":"Exchange every (x,y) for (y,x), giving R⁻¹={(9, 6), (10, 7)}. A converse relation is not automatically an inverse function."},"skills":["Converse of a Binary Relation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r03-relation-0094","batch_id":"batch-20261003-advance-20261004-r03-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4b8f63ab6ec2c47469cbc05cc0bdad2ba613c90d31c046cfcdbe81c4515ca494","family_id":"qf-relation-converse-set","seed":202610043094,"semantic_template":"qf-relation-converse-set/{\"A\":[6,7,8],\"B\":[10,11,12],\"R\":[],\"operation\":\"converse\"}"},"fields":{"title":"Converse of a Binary Relation","stem":"The binary relation has source A={6, 7, 8}, target B={10, 11, 12}, and R=∅⊆A×B. After exchanging source and target, list the converse R⁻¹ as ordered pairs sorted by first and then second component, with duplicates removed.","answer":{"kind":"expression","value":"∅","explanation":"Exchange every (x,y) for (y,x), giving R⁻¹=∅. A converse relation is not automatically an inverse function."},"skills":["Converse of a Binary Relation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r03-relation-0095","batch_id":"batch-20261003-advance-20261004-r03-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b4eb786879d0f1d296691f4d4ac09b2d8eac98c741733e164a69a1445a147023","family_id":"qf-relation-converse-set","seed":202610043095,"semantic_template":"qf-relation-converse-set/{\"A\":[7,8,9],\"B\":[11,12,13],\"R\":[[7,11]],\"operation\":\"converse\"}"},"fields":{"title":"Converse of a Binary Relation","stem":"The binary relation has source A={7, 8, 9}, target B={11, 12, 13}, and R={(7, 11)}⊆A×B. After exchanging source and target, list the converse R⁻¹ as ordered pairs sorted by first and then second component, with duplicates removed.","answer":{"kind":"expression","value":"{(11, 7)}","explanation":"Exchange every (x,y) for (y,x), giving R⁻¹={(11, 7)}. A converse relation is not automatically an inverse function."},"skills":["Converse of a Binary Relation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r03-relation-0096","batch_id":"batch-20261003-advance-20261004-r03-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6dd69ed12ef92e7133ad192c28c05fdc823f228683b70757e4f373138a707a55","family_id":"qf-relation-converse-set","seed":202610043096,"semantic_template":"qf-relation-converse-set/{\"A\":[8,9,10],\"B\":[12,13,14],\"R\":[[8,12],[8,13]],\"operation\":\"converse\"}"},"fields":{"title":"Converse of a Binary Relation","stem":"The binary relation has source A={8, 9, 10}, target B={12, 13, 14}, and R={(8, 12), (8, 13)}⊆A×B. After exchanging source and target, list the converse R⁻¹ as ordered pairs sorted by first and then second component, with duplicates removed.","answer":{"kind":"expression","value":"{(12, 8), (13, 8)}","explanation":"Exchange every (x,y) for (y,x), giving R⁻¹={(12, 8), (13, 8)}. A converse relation is not automatically an inverse function."},"skills":["Converse of a Binary Relation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r03-relation-0097","batch_id":"batch-20261003-advance-20261004-r03-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bb4db431218e68e45ccf9754ccfd08e74ca52db8a0868d223895f01ced4ff34b","family_id":"qf-relation-converse-set","seed":202610043097,"semantic_template":"qf-relation-converse-set/{\"A\":[9,10,11],\"B\":[13,14,15],\"R\":[[9,14],[10,14]],\"operation\":\"converse\"}"},"fields":{"title":"Converse of a Binary Relation","stem":"The binary relation has source A={9, 10, 11}, target B={13, 14, 15}, and R={(9, 14), (10, 14)}⊆A×B. After exchanging source and target, list the converse R⁻¹ as ordered pairs sorted by first and then second component, with duplicates removed.","answer":{"kind":"expression","value":"{(14, 9), (14, 10)}","explanation":"Exchange every (x,y) for (y,x), giving R⁻¹={(14, 9), (14, 10)}. A converse relation is not automatically an inverse function."},"skills":["Converse of a Binary Relation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r03-relation-0098","batch_id":"batch-20261003-advance-20261004-r03-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"40111ab048149a3cb246d6749584f36a27fa24127c0446d91d66441dc1b6f552","family_id":"qf-relation-converse-set","seed":202610043098,"semantic_template":"qf-relation-converse-set/{\"A\":[10,11,12],\"B\":[14,15,16],\"R\":[[10,14],[11,15],[12,16]],\"operation\":\"converse\"}"},"fields":{"title":"Converse of a Binary Relation","stem":"The binary relation has source A={10, 11, 12}, target B={14, 15, 16}, and R={(10, 14), (11, 15), (12, 16)}⊆A×B. After exchanging source and target, list the converse R⁻¹ as ordered pairs sorted by first and then second component, with duplicates removed.","answer":{"kind":"expression","value":"{(14, 10), (15, 11), (16, 12)}","explanation":"Exchange every (x,y) for (y,x), giving R⁻¹={(14, 10), (15, 11), (16, 12)}. A converse relation is not automatically an inverse function."},"skills":["Converse of a Binary Relation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r03-relation-0099","batch_id":"batch-20261003-advance-20261004-r03-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"906598a42530f552afa487c289b52e0c435bc3cd771496ba6ac75ad5f0e13ba1","family_id":"qf-relation-converse-set","seed":202610043099,"semantic_template":"qf-relation-converse-set/{\"A\":[11,12,13],\"B\":[15,16,17],\"R\":[[12,15],[13,16]],\"operation\":\"converse\"}"},"fields":{"title":"Converse of a Binary Relation","stem":"The binary relation has source A={11, 12, 13}, target B={15, 16, 17}, and R={(12, 15), (13, 16)}⊆A×B. After exchanging source and target, list the converse R⁻¹ as ordered pairs sorted by first and then second component, with duplicates removed.","answer":{"kind":"expression","value":"{(15, 12), (16, 13)}","explanation":"Exchange every (x,y) for (y,x), giving R⁻¹={(15, 12), (16, 13)}. A converse relation is not automatically an inverse function."},"skills":["Converse of a Binary Relation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261003-advance-20261004-r03-relation-0100","batch_id":"batch-20261003-advance-20261004-r03-relation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"91233ed0fd94bffe025829d7fd54f70bbe8cbe17da9bf641bad450965979269b","family_id":"qf-relation-converse-set","seed":202610043100,"semantic_template":"qf-relation-converse-set/{\"A\":[12,13,14],\"B\":[16,17,18],\"R\":[],\"operation\":\"converse\"}"},"fields":{"title":"Converse of a Binary Relation","stem":"The binary relation has source A={12, 13, 14}, target B={16, 17, 18}, and R=∅⊆A×B. After exchanging source and target, list the converse R⁻¹ as ordered pairs sorted by first and then second component, with duplicates removed.","answer":{"kind":"expression","value":"∅","explanation":"Exchange every (x,y) for (y,x), giving R⁻¹=∅. A converse relation is not automatically an inverse function."},"skills":["Converse of a Binary Relation"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}