{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0001","batch_id":"batch-20261014-advance-20261108-r05-inclusion-exclusion-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e09fa1d3ad0c31920f0796f915800ee0b92fa6ea920f90c8fd3dcf26c2ca4179","family_id":"qf-mod39-incexc-two-set-formula","seed":202611085001,"semantic_template":"qf-mod39-incexc-two-set-formula/{\"sets\":[[0,1,2],[0,2,3]]}"},"fields":{"title":"Two-Set Inclusion-Exclusion Keeps Both Sizes and Their Intersection","stem":"Use the complete indexed finite integer-set family sets=[[0, 1, 2], [0, 2, 3]] in supplied index order. Each set has distinct elements; equal sets at different indices remain separate indexed entries. Every nonempty index subset contributes its intersection cardinality, with positive odd-order and negative even-order sign. Keep zero-size intersections and source increasing-bitmask term order. Give (set_sizes, intersection_size, union_size, alternating_sum, equal).","answer":{"kind":"expression","value":"([3, 3], 2, 4, 4, Yes)","explanation":"Require exactly two indexed sets. Add their sizes, subtract the full intersection size and compare with the actual union cardinality. The exact result, retaining the requested order, is ([3, 3], 2, 4, 4, Yes)."},"skills":["Two-Set Inclusion-Exclusion Keeps Both Sizes and Their Intersection"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0002","batch_id":"batch-20261014-advance-20261108-r05-inclusion-exclusion-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"021ff7d97ecd17dca79144de1b34f0613b012b13a0610f0b1b384bfcc5c4d771","family_id":"qf-mod39-incexc-two-set-formula","seed":202611085002,"semantic_template":"qf-mod39-incexc-two-set-formula/{\"sets\":[[1,2,3],[1,3,4]]}"},"fields":{"title":"Two-Set Inclusion-Exclusion Keeps Both Sizes and Their Intersection","stem":"Use the complete indexed finite integer-set family sets=[[1, 2, 3], [1, 3, 4]] in supplied index order. Each set has distinct elements; equal sets at different indices remain separate indexed entries. Every nonempty index subset contributes its intersection cardinality, with positive odd-order and negative even-order sign. Keep zero-size intersections and source increasing-bitmask term order. Give (set_sizes, intersection_size, union_size, alternating_sum, equal).","answer":{"kind":"expression","value":"([3, 3], 2, 4, 4, Yes)","explanation":"Require exactly two indexed sets. Add their sizes, subtract the full intersection size and compare with the actual union cardinality. The exact result, retaining the requested order, is ([3, 3], 2, 4, 4, Yes)."},"skills":["Two-Set Inclusion-Exclusion Keeps Both Sizes and Their Intersection"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0003","batch_id":"batch-20261014-advance-20261108-r05-inclusion-exclusion-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a502fefe0162d9c548dff7ec88734e0a3f76f6e6c864dd35cea39fb7d6e2dea4","family_id":"qf-mod39-incexc-two-set-formula","seed":202611085003,"semantic_template":"qf-mod39-incexc-two-set-formula/{\"sets\":[[2,3,4],[2,4,5]]}"},"fields":{"title":"Two-Set Inclusion-Exclusion Keeps Both Sizes and Their Intersection","stem":"Use the complete indexed finite integer-set family sets=[[2, 3, 4], [2, 4, 5]] in supplied index order. Each set has distinct elements; equal sets at different indices remain separate indexed entries. Every nonempty index subset contributes its intersection cardinality, with positive odd-order and negative even-order sign. Keep zero-size intersections and source increasing-bitmask term order. Give (set_sizes, intersection_size, union_size, alternating_sum, equal).","answer":{"kind":"expression","value":"([3, 3], 2, 4, 4, Yes)","explanation":"Require exactly two indexed sets. Add their sizes, subtract the full intersection size and compare with the actual union cardinality. The exact result, retaining the requested order, is ([3, 3], 2, 4, 4, Yes)."},"skills":["Two-Set Inclusion-Exclusion Keeps Both Sizes and Their Intersection"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0004","batch_id":"batch-20261014-advance-20261108-r05-inclusion-exclusion-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3beda58b864d4047131720de1f3159f1f8c52a3208bfd692af7687cfc3343525","family_id":"qf-mod39-incexc-two-set-formula","seed":202611085004,"semantic_template":"qf-mod39-incexc-two-set-formula/{\"sets\":[[3,4,5],[3,5,6]]}"},"fields":{"title":"Two-Set Inclusion-Exclusion Keeps Both Sizes and Their Intersection","stem":"Use the complete indexed finite integer-set family sets=[[3, 4, 5], [3, 5, 6]] in supplied index order. Each set has distinct elements; equal sets at different indices remain separate indexed entries. Every nonempty index subset contributes its intersection cardinality, with positive odd-order and negative even-order sign. Keep zero-size intersections and source increasing-bitmask term order. Give (set_sizes, intersection_size, union_size, alternating_sum, equal).","answer":{"kind":"expression","value":"([3, 3], 2, 4, 4, Yes)","explanation":"Require exactly two indexed sets. Add their sizes, subtract the full intersection size and compare with the actual union cardinality. The exact result, retaining the requested order, is ([3, 3], 2, 4, 4, Yes)."},"skills":["Two-Set Inclusion-Exclusion Keeps Both Sizes and Their Intersection"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0005","batch_id":"batch-20261014-advance-20261108-r05-inclusion-exclusion-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f4f0b53b143c4947544014e749a8c6ea423e25f14092afe6ef5084fddd95781b","family_id":"qf-mod39-incexc-two-set-formula","seed":202611085005,"semantic_template":"qf-mod39-incexc-two-set-formula/{\"sets\":[[4,5,6],[4,6,7]]}"},"fields":{"title":"Two-Set Inclusion-Exclusion Keeps Both Sizes and Their Intersection","stem":"Use the complete indexed finite integer-set family sets=[[4, 5, 6], [4, 6, 7]] in supplied index order. Each set has distinct elements; equal sets at different indices remain separate indexed entries. Every nonempty index subset contributes its intersection cardinality, with positive odd-order and negative even-order sign. Keep zero-size intersections and source increasing-bitmask term order. Give (set_sizes, intersection_size, union_size, alternating_sum, equal).","answer":{"kind":"expression","value":"([3, 3], 2, 4, 4, Yes)","explanation":"Require exactly two indexed sets. Add their sizes, subtract the full intersection size and compare with the actual union cardinality. The exact result, retaining the requested order, is ([3, 3], 2, 4, 4, Yes)."},"skills":["Two-Set Inclusion-Exclusion Keeps Both Sizes and Their Intersection"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0006","batch_id":"batch-20261014-advance-20261108-r05-inclusion-exclusion-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c00059906c862f8c6ad6cb50e0159ba547f3b9ff43dd1e4f57d41a7ef543ec9a","family_id":"qf-mod39-incexc-two-set-formula","seed":202611085006,"semantic_template":"qf-mod39-incexc-two-set-formula/{\"sets\":[[5,6,7],[5,7,8]]}"},"fields":{"title":"Two-Set Inclusion-Exclusion Keeps Both Sizes and Their Intersection","stem":"Use the complete indexed finite integer-set family sets=[[5, 6, 7], [5, 7, 8]] in supplied index order. Each set has distinct elements; equal sets at different indices remain separate indexed entries. Every nonempty index subset contributes its intersection cardinality, with positive odd-order and negative even-order sign. Keep zero-size intersections and source increasing-bitmask term order. Give (set_sizes, intersection_size, union_size, alternating_sum, equal).","answer":{"kind":"expression","value":"([3, 3], 2, 4, 4, Yes)","explanation":"Require exactly two indexed sets. Add their sizes, subtract the full intersection size and compare with the actual union cardinality. The exact result, retaining the requested order, is ([3, 3], 2, 4, 4, Yes)."},"skills":["Two-Set Inclusion-Exclusion Keeps Both Sizes and Their Intersection"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0007","batch_id":"batch-20261014-advance-20261108-r05-inclusion-exclusion-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1b6f83e43f054f3f06c3410d0a6241ddf7ce1b5bd1df1bb694085c74452e0e72","family_id":"qf-mod39-incexc-two-set-formula","seed":202611085007,"semantic_template":"qf-mod39-incexc-two-set-formula/{\"sets\":[[6,7,8],[6,8,9]]}"},"fields":{"title":"Two-Set Inclusion-Exclusion Keeps Both Sizes and Their Intersection","stem":"Use the complete indexed finite integer-set family sets=[[6, 7, 8], [6, 8, 9]] in supplied index order. Each set has distinct elements; equal sets at different indices remain separate indexed entries. Every nonempty index subset contributes its intersection cardinality, with positive odd-order and negative even-order sign. Keep zero-size intersections and source increasing-bitmask term order. Give (set_sizes, intersection_size, union_size, alternating_sum, equal).","answer":{"kind":"expression","value":"([3, 3], 2, 4, 4, Yes)","explanation":"Require exactly two indexed sets. Add their sizes, subtract the full intersection size and compare with the actual union cardinality. The exact result, retaining the requested order, is ([3, 3], 2, 4, 4, Yes)."},"skills":["Two-Set Inclusion-Exclusion Keeps Both Sizes and Their Intersection"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0008","batch_id":"batch-20261014-advance-20261108-r05-inclusion-exclusion-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a7eb284031fefeb54a87e48a28a95cf61b85212c0be7c1e2b9c001a06d5f2e70","family_id":"qf-mod39-incexc-two-set-formula","seed":202611085008,"semantic_template":"qf-mod39-incexc-two-set-formula/{\"sets\":[[7,8,9],[7,9,10]]}"},"fields":{"title":"Two-Set Inclusion-Exclusion Keeps Both Sizes and Their Intersection","stem":"Use the complete indexed finite integer-set family sets=[[7, 8, 9], [7, 9, 10]] in supplied index order. Each set has distinct elements; equal sets at different indices remain separate indexed entries. Every nonempty index subset contributes its intersection cardinality, with positive odd-order and negative even-order sign. Keep zero-size intersections and source increasing-bitmask term order. Give (set_sizes, intersection_size, union_size, alternating_sum, equal).","answer":{"kind":"expression","value":"([3, 3], 2, 4, 4, Yes)","explanation":"Require exactly two indexed sets. Add their sizes, subtract the full intersection size and compare with the actual union cardinality. The exact result, retaining the requested order, is ([3, 3], 2, 4, 4, Yes)."},"skills":["Two-Set Inclusion-Exclusion Keeps Both Sizes and Their Intersection"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0009","batch_id":"batch-20261014-advance-20261108-r05-inclusion-exclusion-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"abc3a8c7f865ba65043b8e50b0d6483b207223baa4cfa715d75a2699cde796ab","family_id":"qf-mod39-incexc-two-set-formula","seed":202611085009,"semantic_template":"qf-mod39-incexc-two-set-formula/{\"sets\":[[8,10,11],[8,9,10]]}"},"fields":{"title":"Two-Set Inclusion-Exclusion Keeps Both Sizes and Their Intersection","stem":"Use the complete indexed finite integer-set family sets=[[8, 9, 10], [8, 10, 11]] in supplied index order. Each set has distinct elements; equal sets at different indices remain separate indexed entries. Every nonempty index subset contributes its intersection cardinality, with positive odd-order and negative even-order sign. Keep zero-size intersections and source increasing-bitmask term order. Give (set_sizes, intersection_size, union_size, alternating_sum, equal).","answer":{"kind":"expression","value":"([3, 3], 2, 4, 4, Yes)","explanation":"Require exactly two indexed sets. Add their sizes, subtract the full intersection size and compare with the actual union cardinality. The exact result, retaining the requested order, is ([3, 3], 2, 4, 4, Yes)."},"skills":["Two-Set Inclusion-Exclusion Keeps Both Sizes and Their Intersection"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0010","batch_id":"batch-20261014-advance-20261108-r05-inclusion-exclusion-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9c2d43f3fc1e5dfac4396d7faf24dd0555b66018af76b4240185b8df8c5af89f","family_id":"qf-mod39-incexc-two-set-formula","seed":202611085010,"semantic_template":"qf-mod39-incexc-two-set-formula/{\"sets\":[[9,10,11],[9,11,12]]}"},"fields":{"title":"Two-Set Inclusion-Exclusion Keeps Both Sizes and Their Intersection","stem":"Use the complete indexed finite integer-set family sets=[[9, 10, 11], [9, 11, 12]] in supplied index order. Each set has distinct elements; equal sets at different indices remain separate indexed entries. Every nonempty index subset contributes its intersection cardinality, with positive odd-order and negative even-order sign. Keep zero-size intersections and source increasing-bitmask term order. Give (set_sizes, intersection_size, union_size, alternating_sum, equal).","answer":{"kind":"expression","value":"([3, 3], 2, 4, 4, Yes)","explanation":"Require exactly two indexed sets. Add their sizes, subtract the full intersection size and compare with the actual union cardinality. The exact result, retaining the requested order, is ([3, 3], 2, 4, 4, Yes)."},"skills":["Two-Set Inclusion-Exclusion Keeps Both Sizes and Their Intersection"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0011","batch_id":"batch-20261014-advance-20261108-r05-inclusion-exclusion-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e0d75f3ec0d89938b93ff6c1620d5101848f6644bb320263fffdb3c1041d6ca2","family_id":"qf-mod39-incexc-two-set-formula","seed":202611085011,"semantic_template":"qf-mod39-incexc-two-set-formula/{\"sets\":[[10,11,12],[10,12,13]]}"},"fields":{"title":"Two-Set Inclusion-Exclusion Keeps Both Sizes and Their Intersection","stem":"Use the complete indexed finite integer-set family sets=[[10, 11, 12], [10, 12, 13]] in supplied index order. Each set has distinct elements; equal sets at different indices remain separate indexed entries. Every nonempty index subset contributes its intersection cardinality, with positive odd-order and negative even-order sign. Keep zero-size intersections and source increasing-bitmask term order. Give (set_sizes, intersection_size, union_size, alternating_sum, equal).","answer":{"kind":"expression","value":"([3, 3], 2, 4, 4, Yes)","explanation":"Require exactly two indexed sets. Add their sizes, subtract the full intersection size and compare with the actual union cardinality. The exact result, retaining the requested order, is ([3, 3], 2, 4, 4, Yes)."},"skills":["Two-Set Inclusion-Exclusion Keeps Both Sizes and Their Intersection"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0012","batch_id":"batch-20261014-advance-20261108-r05-inclusion-exclusion-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"74ce24ad75e84af733f4efd41b862dc34e67c86fbbc431aa73f67c70d96c7e82","family_id":"qf-mod39-incexc-two-set-formula","seed":202611085012,"semantic_template":"qf-mod39-incexc-two-set-formula/{\"sets\":[[11,12,13],[11,13,14]]}"},"fields":{"title":"Two-Set Inclusion-Exclusion Keeps Both Sizes and Their Intersection","stem":"Use the complete indexed finite integer-set family sets=[[11, 12, 13], [11, 13, 14]] in supplied index order. Each set has distinct elements; equal sets at different indices remain separate indexed entries. Every nonempty index subset contributes its intersection cardinality, with positive odd-order and negative even-order sign. Keep zero-size intersections and source increasing-bitmask term order. Give (set_sizes, intersection_size, union_size, alternating_sum, equal).","answer":{"kind":"expression","value":"([3, 3], 2, 4, 4, Yes)","explanation":"Require exactly two indexed sets. Add their sizes, subtract the full intersection size and compare with the actual union cardinality. The exact result, retaining the requested order, is ([3, 3], 2, 4, 4, Yes)."},"skills":["Two-Set Inclusion-Exclusion Keeps Both Sizes and Their Intersection"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0013","batch_id":"batch-20261014-advance-20261108-r05-inclusion-exclusion-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c2d85ab9e5260a3ebb8ab02f5c203f9637b525f97152ed7f4601ebfea37f96b7","family_id":"qf-mod39-incexc-two-set-formula","seed":202611085013,"semantic_template":"qf-mod39-incexc-two-set-formula/{\"sets\":[[12,13,14],[12,14,15]]}"},"fields":{"title":"Two-Set Inclusion-Exclusion Keeps Both Sizes and Their Intersection","stem":"Use the complete indexed finite integer-set family sets=[[12, 13, 14], [12, 14, 15]] in supplied index order. Each set has distinct elements; equal sets at different indices remain separate indexed entries. Every nonempty index subset contributes its intersection cardinality, with positive odd-order and negative even-order sign. Keep zero-size intersections and source increasing-bitmask term order. Give (set_sizes, intersection_size, union_size, alternating_sum, equal).","answer":{"kind":"expression","value":"([3, 3], 2, 4, 4, Yes)","explanation":"Require exactly two indexed sets. Add their sizes, subtract the full intersection size and compare with the actual union cardinality. The exact result, retaining the requested order, is ([3, 3], 2, 4, 4, Yes)."},"skills":["Two-Set Inclusion-Exclusion Keeps Both Sizes and Their Intersection"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0014","batch_id":"batch-20261014-advance-20261108-r05-inclusion-exclusion-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b8032e6d15858a2a75a24d5e9c976830bed966ed18128777465b8a5b36e541fe","family_id":"qf-mod39-incexc-two-set-formula","seed":202611085014,"semantic_template":"qf-mod39-incexc-two-set-formula/{\"sets\":[[13,14,15],[13,15,16]]}"},"fields":{"title":"Two-Set Inclusion-Exclusion Keeps Both Sizes and Their Intersection","stem":"Use the complete indexed finite integer-set family sets=[[13, 14, 15], [13, 15, 16]] in supplied index order. Each set has distinct elements; equal sets at different indices remain separate indexed entries. Every nonempty index subset contributes its intersection cardinality, with positive odd-order and negative even-order sign. Keep zero-size intersections and source increasing-bitmask term order. Give (set_sizes, intersection_size, union_size, alternating_sum, equal).","answer":{"kind":"expression","value":"([3, 3], 2, 4, 4, Yes)","explanation":"Require exactly two indexed sets. Add their sizes, subtract the full intersection size and compare with the actual union cardinality. The exact result, retaining the requested order, is ([3, 3], 2, 4, 4, Yes)."},"skills":["Two-Set Inclusion-Exclusion Keeps Both Sizes and Their Intersection"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0015","batch_id":"batch-20261014-advance-20261108-r05-inclusion-exclusion-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9b6da4164b712b8e5592ebc270b5f19e8edce79fd28a30e885ff3f48c4607216","family_id":"qf-mod39-incexc-two-set-formula","seed":202611085015,"semantic_template":"qf-mod39-incexc-two-set-formula/{\"sets\":[[14,15,16],[14,16,17]]}"},"fields":{"title":"Two-Set Inclusion-Exclusion Keeps Both Sizes and Their Intersection","stem":"Use the complete indexed finite integer-set family sets=[[14, 15, 16], [14, 16, 17]] in supplied index order. Each set has distinct elements; equal sets at different indices remain separate indexed entries. Every nonempty index subset contributes its intersection cardinality, with positive odd-order and negative even-order sign. Keep zero-size intersections and source increasing-bitmask term order. Give (set_sizes, intersection_size, union_size, alternating_sum, equal).","answer":{"kind":"expression","value":"([3, 3], 2, 4, 4, Yes)","explanation":"Require exactly two indexed sets. Add their sizes, subtract the full intersection size and compare with the actual union cardinality. The exact result, retaining the requested order, is ([3, 3], 2, 4, 4, Yes)."},"skills":["Two-Set Inclusion-Exclusion Keeps Both Sizes and Their Intersection"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0016","batch_id":"batch-20261014-advance-20261108-r05-inclusion-exclusion-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"41f009d51fc25176d0761da9bc61552ac31b3911d98b47fac5cf33e82a23dfbf","family_id":"qf-mod39-incexc-two-set-formula","seed":202611085016,"semantic_template":"qf-mod39-incexc-two-set-formula/{\"sets\":[[15,16,17],[15,17,18]]}"},"fields":{"title":"Two-Set Inclusion-Exclusion Keeps Both Sizes and Their Intersection","stem":"Use the complete indexed finite integer-set family sets=[[15, 16, 17], [15, 17, 18]] in supplied index order. Each set has distinct elements; equal sets at different indices remain separate indexed entries. Every nonempty index subset contributes its intersection cardinality, with positive odd-order and negative even-order sign. Keep zero-size intersections and source increasing-bitmask term order. Give (set_sizes, intersection_size, union_size, alternating_sum, equal).","answer":{"kind":"expression","value":"([3, 3], 2, 4, 4, Yes)","explanation":"Require exactly two indexed sets. Add their sizes, subtract the full intersection size and compare with the actual union cardinality. The exact result, retaining the requested order, is ([3, 3], 2, 4, 4, Yes)."},"skills":["Two-Set Inclusion-Exclusion Keeps Both Sizes and Their Intersection"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0017","batch_id":"batch-20261014-advance-20261108-r05-inclusion-exclusion-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"54ca2490f9d7b69a6aea171d9153a2aadee282606804ccb29216d6838efe149e","family_id":"qf-mod39-incexc-two-set-formula","seed":202611085017,"semantic_template":"qf-mod39-incexc-two-set-formula/{\"sets\":[[16,17,18],[16,18,19]]}"},"fields":{"title":"Two-Set Inclusion-Exclusion Keeps Both Sizes and Their Intersection","stem":"Use the complete indexed finite integer-set family sets=[[16, 17, 18], [16, 18, 19]] in supplied index order. Each set has distinct elements; equal sets at different indices remain separate indexed entries. Every nonempty index subset contributes its intersection cardinality, with positive odd-order and negative even-order sign. Keep zero-size intersections and source increasing-bitmask term order. Give (set_sizes, intersection_size, union_size, alternating_sum, equal).","answer":{"kind":"expression","value":"([3, 3], 2, 4, 4, Yes)","explanation":"Require exactly two indexed sets. Add their sizes, subtract the full intersection size and compare with the actual union cardinality. The exact result, retaining the requested order, is ([3, 3], 2, 4, 4, Yes)."},"skills":["Two-Set Inclusion-Exclusion Keeps Both Sizes and Their Intersection"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0018","batch_id":"batch-20261014-advance-20261108-r05-inclusion-exclusion-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"946105dba2677ff4d4cdccb9bab2aa8eca75af9f5195b76a5c733a60eb9010da","family_id":"qf-mod39-incexc-two-set-formula","seed":202611085018,"semantic_template":"qf-mod39-incexc-two-set-formula/{\"sets\":[[17,18,19],[17,19,20]]}"},"fields":{"title":"Two-Set Inclusion-Exclusion Keeps Both Sizes and Their Intersection","stem":"Use the complete indexed finite integer-set family sets=[[17, 18, 19], [17, 19, 20]] in supplied index order. Each set has distinct elements; equal sets at different indices remain separate indexed entries. Every nonempty index subset contributes its intersection cardinality, with positive odd-order and negative even-order sign. Keep zero-size intersections and source increasing-bitmask term order. Give (set_sizes, intersection_size, union_size, alternating_sum, equal).","answer":{"kind":"expression","value":"([3, 3], 2, 4, 4, Yes)","explanation":"Require exactly two indexed sets. Add their sizes, subtract the full intersection size and compare with the actual union cardinality. The exact result, retaining the requested order, is ([3, 3], 2, 4, 4, Yes)."},"skills":["Two-Set Inclusion-Exclusion Keeps Both Sizes and Their Intersection"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0019","batch_id":"batch-20261014-advance-20261108-r05-inclusion-exclusion-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3fe68042256ac265a2d1b21f066a9c311dd7f241b7b7ebbc51bed69204816e8d","family_id":"qf-mod39-incexc-two-set-formula","seed":202611085019,"semantic_template":"qf-mod39-incexc-two-set-formula/{\"sets\":[[18,19,20],[18,20,21]]}"},"fields":{"title":"Two-Set Inclusion-Exclusion Keeps Both Sizes and Their Intersection","stem":"Use the complete indexed finite integer-set family sets=[[18, 19, 20], [18, 20, 21]] in supplied index order. Each set has distinct elements; equal sets at different indices remain separate indexed entries. Every nonempty index subset contributes its intersection cardinality, with positive odd-order and negative even-order sign. Keep zero-size intersections and source increasing-bitmask term order. Give (set_sizes, intersection_size, union_size, alternating_sum, equal).","answer":{"kind":"expression","value":"([3, 3], 2, 4, 4, Yes)","explanation":"Require exactly two indexed sets. Add their sizes, subtract the full intersection size and compare with the actual union cardinality. The exact result, retaining the requested order, is ([3, 3], 2, 4, 4, Yes)."},"skills":["Two-Set Inclusion-Exclusion Keeps Both Sizes and Their Intersection"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0020","batch_id":"batch-20261014-advance-20261108-r05-inclusion-exclusion-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"83e89344636b91128f448581324c697592f8b00e56a8696db402830fe5afedd3","family_id":"qf-mod39-incexc-two-set-formula","seed":202611085020,"semantic_template":"qf-mod39-incexc-two-set-formula/{\"sets\":[[19,20,21],[19,21,22]]}"},"fields":{"title":"Two-Set Inclusion-Exclusion Keeps Both Sizes and Their Intersection","stem":"Use the complete indexed finite integer-set family sets=[[19, 20, 21], [19, 21, 22]] in supplied index order. Each set has distinct elements; equal sets at different indices remain separate indexed entries. Every nonempty index subset contributes its intersection cardinality, with positive odd-order and negative even-order sign. Keep zero-size intersections and source increasing-bitmask term order. Give (set_sizes, intersection_size, union_size, alternating_sum, equal).","answer":{"kind":"expression","value":"([3, 3], 2, 4, 4, Yes)","explanation":"Require exactly two indexed sets. Add their sizes, subtract the full intersection size and compare with the actual union cardinality. The exact result, retaining the requested order, is ([3, 3], 2, 4, 4, Yes)."},"skills":["Two-Set Inclusion-Exclusion Keeps Both Sizes and Their Intersection"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0021","batch_id":"batch-20261014-advance-20261108-r05-inclusion-exclusion-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6a25d41a4f096b5c8e78154ac58d7fc676649562239a6f332be86275db9145d1","family_id":"qf-mod39-incexc-two-set-formula","seed":202611085021,"semantic_template":"qf-mod39-incexc-two-set-formula/{\"sets\":[[20,21,22],[20,22,23]]}"},"fields":{"title":"Two-Set Inclusion-Exclusion Keeps Both Sizes and Their Intersection","stem":"Use the complete indexed finite integer-set family sets=[[20, 21, 22], [20, 22, 23]] in supplied index order. Each set has distinct elements; equal sets at different indices remain separate indexed entries. Every nonempty index subset contributes its intersection cardinality, with positive odd-order and negative even-order sign. Keep zero-size intersections and source increasing-bitmask term order. Give (set_sizes, intersection_size, union_size, alternating_sum, equal).","answer":{"kind":"expression","value":"([3, 3], 2, 4, 4, Yes)","explanation":"Require exactly two indexed sets. Add their sizes, subtract the full intersection size and compare with the actual union cardinality. The exact result, retaining the requested order, is ([3, 3], 2, 4, 4, Yes)."},"skills":["Two-Set Inclusion-Exclusion Keeps Both Sizes and Their Intersection"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0022","batch_id":"batch-20261014-advance-20261108-r05-inclusion-exclusion-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"16e53dad7b2ff723c422d57529887c802054917a46c356b9f70ed811b51e65cf","family_id":"qf-mod39-incexc-two-set-formula","seed":202611085022,"semantic_template":"qf-mod39-incexc-two-set-formula/{\"sets\":[[21,22,23],[21,23,24]]}"},"fields":{"title":"Two-Set Inclusion-Exclusion Keeps Both Sizes and Their Intersection","stem":"Use the complete indexed finite integer-set family sets=[[21, 22, 23], [21, 23, 24]] in supplied index order. Each set has distinct elements; equal sets at different indices remain separate indexed entries. Every nonempty index subset contributes its intersection cardinality, with positive odd-order and negative even-order sign. Keep zero-size intersections and source increasing-bitmask term order. Give (set_sizes, intersection_size, union_size, alternating_sum, equal).","answer":{"kind":"expression","value":"([3, 3], 2, 4, 4, Yes)","explanation":"Require exactly two indexed sets. Add their sizes, subtract the full intersection size and compare with the actual union cardinality. The exact result, retaining the requested order, is ([3, 3], 2, 4, 4, Yes)."},"skills":["Two-Set Inclusion-Exclusion Keeps Both Sizes and Their Intersection"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0023","batch_id":"batch-20261014-advance-20261108-r05-inclusion-exclusion-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"138e3378009d46110f676ba302e34dd780ca68f388c748bf95c90b3516381f46","family_id":"qf-mod39-incexc-two-set-formula","seed":202611085023,"semantic_template":"qf-mod39-incexc-two-set-formula/{\"sets\":[[22,23,24],[22,24,25]]}"},"fields":{"title":"Two-Set Inclusion-Exclusion Keeps Both Sizes and Their Intersection","stem":"Use the complete indexed finite integer-set family sets=[[22, 23, 24], [22, 24, 25]] in supplied index order. Each set has distinct elements; equal sets at different indices remain separate indexed entries. Every nonempty index subset contributes its intersection cardinality, with positive odd-order and negative even-order sign. Keep zero-size intersections and source increasing-bitmask term order. Give (set_sizes, intersection_size, union_size, alternating_sum, equal).","answer":{"kind":"expression","value":"([3, 3], 2, 4, 4, Yes)","explanation":"Require exactly two indexed sets. Add their sizes, subtract the full intersection size and compare with the actual union cardinality. The exact result, retaining the requested order, is ([3, 3], 2, 4, 4, Yes)."},"skills":["Two-Set Inclusion-Exclusion Keeps Both Sizes and Their Intersection"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0024","batch_id":"batch-20261014-advance-20261108-r05-inclusion-exclusion-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"13b47ea3b8688adf65b939e22d48198d46596865b561a5ed09ffb3926acd9e17","family_id":"qf-mod39-incexc-two-set-formula","seed":202611085024,"semantic_template":"qf-mod39-incexc-two-set-formula/{\"sets\":[[23,24,25],[23,25,26]]}"},"fields":{"title":"Two-Set Inclusion-Exclusion Keeps Both Sizes and Their Intersection","stem":"Use the complete indexed finite integer-set family sets=[[23, 24, 25], [23, 25, 26]] in supplied index order. Each set has distinct elements; equal sets at different indices remain separate indexed entries. Every nonempty index subset contributes its intersection cardinality, with positive odd-order and negative even-order sign. Keep zero-size intersections and source increasing-bitmask term order. Give (set_sizes, intersection_size, union_size, alternating_sum, equal).","answer":{"kind":"expression","value":"([3, 3], 2, 4, 4, Yes)","explanation":"Require exactly two indexed sets. Add their sizes, subtract the full intersection size and compare with the actual union cardinality. The exact result, retaining the requested order, is ([3, 3], 2, 4, 4, Yes)."},"skills":["Two-Set Inclusion-Exclusion Keeps Both Sizes and Their Intersection"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0025","batch_id":"batch-20261014-advance-20261108-r05-inclusion-exclusion-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ced1e5704fbc65df98fc7cf316e54ff71552269a00764e4a9f843b57846f4fcf","family_id":"qf-mod39-incexc-two-set-formula","seed":202611085025,"semantic_template":"qf-mod39-incexc-two-set-formula/{\"sets\":[[24,25,26],[24,26,27]]}"},"fields":{"title":"Two-Set Inclusion-Exclusion Keeps Both Sizes and Their Intersection","stem":"Use the complete indexed finite integer-set family sets=[[24, 25, 26], [24, 26, 27]] in supplied index order. Each set has distinct elements; equal sets at different indices remain separate indexed entries. Every nonempty index subset contributes its intersection cardinality, with positive odd-order and negative even-order sign. Keep zero-size intersections and source increasing-bitmask term order. Give (set_sizes, intersection_size, union_size, alternating_sum, equal).","answer":{"kind":"expression","value":"([3, 3], 2, 4, 4, Yes)","explanation":"Require exactly two indexed sets. Add their sizes, subtract the full intersection size and compare with the actual union cardinality. The exact result, retaining the requested order, is ([3, 3], 2, 4, 4, Yes)."},"skills":["Two-Set Inclusion-Exclusion Keeps Both Sizes and Their Intersection"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0026","batch_id":"batch-20261014-advance-20261108-r05-inclusion-exclusion-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"98cdfed21f44cde1f6260ed90667c1519ccf2009599a04b5f7579c4610266a9c","family_id":"qf-mod39-incexc-three-set-formula","seed":202611085026,"semantic_template":"qf-mod39-incexc-three-set-formula/{\"sets\":[[0,1,2],[0,2,3],[0,2,4]]}"},"fields":{"title":"Three-Set Inclusion-Exclusion Needs the Triple Intersection","stem":"Use the complete indexed finite integer-set family sets=[[0, 1, 2], [0, 2, 3], [0, 2, 4]] in supplied index order. Each set has distinct elements; equal sets at different indices remain separate indexed entries. Every nonempty index subset contributes its intersection cardinality, with positive odd-order and negative even-order sign. Keep zero-size intersections and source increasing-bitmask term order. Give (terms, union_size, alternating_sum, equal, certificate).","answer":{"kind":"expression","value":"([[[0], 1, 3], [[1], 1, 3], [[0, 1], -1, 2], [[2], 1, 3], [[0, 2], -1, 2], [[1, 2], -1, 2], [[0, 1, 2], 1, 2]], 5, 5, Yes, odd_intersections_positive_even_intersections_negative)","explanation":"Require exactly three sets. Retain every [index_subset,sign,intersection_size] term in source bitmask order, including the positive triple-intersection term, and compare the alternating sum with the actual union size. The exact result, retaining the requested order, is ([[[0], 1, 3], [[1], 1, 3], [[0, 1], -1, 2], [[2], 1, 3], [[0, 2], -1, 2], [[1, 2], -1, 2], [[0, 1, 2], 1, 2]], 5, 5, Yes, odd_intersections_positive_even_intersections_negative)."},"skills":["Three-Set Inclusion-Exclusion Needs the Triple Intersection"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0027","batch_id":"batch-20261014-advance-20261108-r05-inclusion-exclusion-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a2ad9ff48bcb49fb3d12a58336f59a5ac322d3d9737d425bedf631364b460b97","family_id":"qf-mod39-incexc-three-set-formula","seed":202611085027,"semantic_template":"qf-mod39-incexc-three-set-formula/{\"sets\":[[1,2,3],[1,3,4],[1,3,5]]}"},"fields":{"title":"Three-Set Inclusion-Exclusion Needs the Triple Intersection","stem":"Use the complete indexed finite integer-set family sets=[[1, 2, 3], [1, 3, 4], [1, 3, 5]] in supplied index order. Each set has distinct elements; equal sets at different indices remain separate indexed entries. Every nonempty index subset contributes its intersection cardinality, with positive odd-order and negative even-order sign. Keep zero-size intersections and source increasing-bitmask term order. Give (terms, union_size, alternating_sum, equal, certificate).","answer":{"kind":"expression","value":"([[[0], 1, 3], [[1], 1, 3], [[0, 1], -1, 2], [[2], 1, 3], [[0, 2], -1, 2], [[1, 2], -1, 2], [[0, 1, 2], 1, 2]], 5, 5, Yes, odd_intersections_positive_even_intersections_negative)","explanation":"Require exactly three sets. Retain every [index_subset,sign,intersection_size] term in source bitmask order, including the positive triple-intersection term, and compare the alternating sum with the actual union size. The exact result, retaining the requested order, is ([[[0], 1, 3], [[1], 1, 3], [[0, 1], -1, 2], [[2], 1, 3], [[0, 2], -1, 2], [[1, 2], -1, 2], [[0, 1, 2], 1, 2]], 5, 5, Yes, odd_intersections_positive_even_intersections_negative)."},"skills":["Three-Set Inclusion-Exclusion Needs the Triple Intersection"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0028","batch_id":"batch-20261014-advance-20261108-r05-inclusion-exclusion-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fa29d0530d1a148a741ae701b0dd252d783604ed5396a1abb26e1be0e0093acc","family_id":"qf-mod39-incexc-three-set-formula","seed":202611085028,"semantic_template":"qf-mod39-incexc-three-set-formula/{\"sets\":[[2,3,4],[2,4,5],[2,4,6]]}"},"fields":{"title":"Three-Set Inclusion-Exclusion Needs the Triple Intersection","stem":"Use the complete indexed finite integer-set family sets=[[2, 3, 4], [2, 4, 5], [2, 4, 6]] in supplied index order. Each set has distinct elements; equal sets at different indices remain separate indexed entries. Every nonempty index subset contributes its intersection cardinality, with positive odd-order and negative even-order sign. Keep zero-size intersections and source increasing-bitmask term order. Give (terms, union_size, alternating_sum, equal, certificate).","answer":{"kind":"expression","value":"([[[0], 1, 3], [[1], 1, 3], [[0, 1], -1, 2], [[2], 1, 3], [[0, 2], -1, 2], [[1, 2], -1, 2], [[0, 1, 2], 1, 2]], 5, 5, Yes, odd_intersections_positive_even_intersections_negative)","explanation":"Require exactly three sets. Retain every [index_subset,sign,intersection_size] term in source bitmask order, including the positive triple-intersection term, and compare the alternating sum with the actual union size. The exact result, retaining the requested order, is ([[[0], 1, 3], [[1], 1, 3], [[0, 1], -1, 2], [[2], 1, 3], [[0, 2], -1, 2], [[1, 2], -1, 2], [[0, 1, 2], 1, 2]], 5, 5, Yes, odd_intersections_positive_even_intersections_negative)."},"skills":["Three-Set Inclusion-Exclusion Needs the Triple Intersection"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0029","batch_id":"batch-20261014-advance-20261108-r05-inclusion-exclusion-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b0da6b15965410cb3f75519dd934c3d47972da15e8a70e4103624a1cd013ddc5","family_id":"qf-mod39-incexc-three-set-formula","seed":202611085029,"semantic_template":"qf-mod39-incexc-three-set-formula/{\"sets\":[[3,4,5],[3,5,6],[3,5,7]]}"},"fields":{"title":"Three-Set Inclusion-Exclusion Needs the Triple Intersection","stem":"Use the complete indexed finite integer-set family sets=[[3, 4, 5], [3, 5, 6], [3, 5, 7]] in supplied index order. Each set has distinct elements; equal sets at different indices remain separate indexed entries. Every nonempty index subset contributes its intersection cardinality, with positive odd-order and negative even-order sign. Keep zero-size intersections and source increasing-bitmask term order. Give (terms, union_size, alternating_sum, equal, certificate).","answer":{"kind":"expression","value":"([[[0], 1, 3], [[1], 1, 3], [[0, 1], -1, 2], [[2], 1, 3], [[0, 2], -1, 2], [[1, 2], -1, 2], [[0, 1, 2], 1, 2]], 5, 5, Yes, odd_intersections_positive_even_intersections_negative)","explanation":"Require exactly three sets. Retain every [index_subset,sign,intersection_size] term in source bitmask order, including the positive triple-intersection term, and compare the alternating sum with the actual union size. The exact result, retaining the requested order, is ([[[0], 1, 3], [[1], 1, 3], [[0, 1], -1, 2], [[2], 1, 3], [[0, 2], -1, 2], [[1, 2], -1, 2], [[0, 1, 2], 1, 2]], 5, 5, Yes, odd_intersections_positive_even_intersections_negative)."},"skills":["Three-Set Inclusion-Exclusion Needs the Triple Intersection"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0030","batch_id":"batch-20261014-advance-20261108-r05-inclusion-exclusion-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"90c73462a3b9b59aac323bbd287449e83e24ded53fc0ec8a2d27b59920235f79","family_id":"qf-mod39-incexc-three-set-formula","seed":202611085030,"semantic_template":"qf-mod39-incexc-three-set-formula/{\"sets\":[[4,5,6],[4,6,7],[4,6,8]]}"},"fields":{"title":"Three-Set Inclusion-Exclusion Needs the Triple Intersection","stem":"Use the complete indexed finite integer-set family sets=[[4, 5, 6], [4, 6, 7], [4, 6, 8]] in supplied index order. Each set has distinct elements; equal sets at different indices remain separate indexed entries. Every nonempty index subset contributes its intersection cardinality, with positive odd-order and negative even-order sign. Keep zero-size intersections and source increasing-bitmask term order. Give (terms, union_size, alternating_sum, equal, certificate).","answer":{"kind":"expression","value":"([[[0], 1, 3], [[1], 1, 3], [[0, 1], -1, 2], [[2], 1, 3], [[0, 2], -1, 2], [[1, 2], -1, 2], [[0, 1, 2], 1, 2]], 5, 5, Yes, odd_intersections_positive_even_intersections_negative)","explanation":"Require exactly three sets. Retain every [index_subset,sign,intersection_size] term in source bitmask order, including the positive triple-intersection term, and compare the alternating sum with the actual union size. The exact result, retaining the requested order, is ([[[0], 1, 3], [[1], 1, 3], [[0, 1], -1, 2], [[2], 1, 3], [[0, 2], -1, 2], [[1, 2], -1, 2], [[0, 1, 2], 1, 2]], 5, 5, Yes, odd_intersections_positive_even_intersections_negative)."},"skills":["Three-Set Inclusion-Exclusion Needs the Triple Intersection"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0031","batch_id":"batch-20261014-advance-20261108-r05-inclusion-exclusion-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3364ef6d152c853301339ac8cb90ea9567dc159f3a06fa835e405426e7e13e98","family_id":"qf-mod39-incexc-three-set-formula","seed":202611085031,"semantic_template":"qf-mod39-incexc-three-set-formula/{\"sets\":[[5,6,7],[5,7,8],[5,7,9]]}"},"fields":{"title":"Three-Set Inclusion-Exclusion Needs the Triple Intersection","stem":"Use the complete indexed finite integer-set family sets=[[5, 6, 7], [5, 7, 8], [5, 7, 9]] in supplied index order. Each set has distinct elements; equal sets at different indices remain separate indexed entries. Every nonempty index subset contributes its intersection cardinality, with positive odd-order and negative even-order sign. Keep zero-size intersections and source increasing-bitmask term order. Give (terms, union_size, alternating_sum, equal, certificate).","answer":{"kind":"expression","value":"([[[0], 1, 3], [[1], 1, 3], [[0, 1], -1, 2], [[2], 1, 3], [[0, 2], -1, 2], [[1, 2], -1, 2], [[0, 1, 2], 1, 2]], 5, 5, Yes, odd_intersections_positive_even_intersections_negative)","explanation":"Require exactly three sets. Retain every [index_subset,sign,intersection_size] term in source bitmask order, including the positive triple-intersection term, and compare the alternating sum with the actual union size. The exact result, retaining the requested order, is ([[[0], 1, 3], [[1], 1, 3], [[0, 1], -1, 2], [[2], 1, 3], [[0, 2], -1, 2], [[1, 2], -1, 2], [[0, 1, 2], 1, 2]], 5, 5, Yes, odd_intersections_positive_even_intersections_negative)."},"skills":["Three-Set Inclusion-Exclusion Needs the Triple Intersection"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0032","batch_id":"batch-20261014-advance-20261108-r05-inclusion-exclusion-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8a594a86c2084f5d6422453a6995bc35a8f8e1d9cee23320248da619fe374ce5","family_id":"qf-mod39-incexc-three-set-formula","seed":202611085032,"semantic_template":"qf-mod39-incexc-three-set-formula/{\"sets\":[[6,7,8],[6,8,10],[6,8,9]]}"},"fields":{"title":"Three-Set Inclusion-Exclusion Needs the Triple Intersection","stem":"Use the complete indexed finite integer-set family sets=[[6, 7, 8], [6, 8, 9], [6, 8, 10]] in supplied index order. Each set has distinct elements; equal sets at different indices remain separate indexed entries. Every nonempty index subset contributes its intersection cardinality, with positive odd-order and negative even-order sign. Keep zero-size intersections and source increasing-bitmask term order. Give (terms, union_size, alternating_sum, equal, certificate).","answer":{"kind":"expression","value":"([[[0], 1, 3], [[1], 1, 3], [[0, 1], -1, 2], [[2], 1, 3], [[0, 2], -1, 2], [[1, 2], -1, 2], [[0, 1, 2], 1, 2]], 5, 5, Yes, odd_intersections_positive_even_intersections_negative)","explanation":"Require exactly three sets. Retain every [index_subset,sign,intersection_size] term in source bitmask order, including the positive triple-intersection term, and compare the alternating sum with the actual union size. The exact result, retaining the requested order, is ([[[0], 1, 3], [[1], 1, 3], [[0, 1], -1, 2], [[2], 1, 3], [[0, 2], -1, 2], [[1, 2], -1, 2], [[0, 1, 2], 1, 2]], 5, 5, Yes, odd_intersections_positive_even_intersections_negative)."},"skills":["Three-Set Inclusion-Exclusion Needs the Triple Intersection"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0033","batch_id":"batch-20261014-advance-20261108-r05-inclusion-exclusion-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5486f798f0298b122d6767a822523ee30c009e215352a04be12ff3b1e15ca457","family_id":"qf-mod39-incexc-three-set-formula","seed":202611085033,"semantic_template":"qf-mod39-incexc-three-set-formula/{\"sets\":[[7,8,9],[7,9,10],[7,9,11]]}"},"fields":{"title":"Three-Set Inclusion-Exclusion Needs the Triple Intersection","stem":"Use the complete indexed finite integer-set family sets=[[7, 8, 9], [7, 9, 10], [7, 9, 11]] in supplied index order. Each set has distinct elements; equal sets at different indices remain separate indexed entries. Every nonempty index subset contributes its intersection cardinality, with positive odd-order and negative even-order sign. Keep zero-size intersections and source increasing-bitmask term order. Give (terms, union_size, alternating_sum, equal, certificate).","answer":{"kind":"expression","value":"([[[0], 1, 3], [[1], 1, 3], [[0, 1], -1, 2], [[2], 1, 3], [[0, 2], -1, 2], [[1, 2], -1, 2], [[0, 1, 2], 1, 2]], 5, 5, Yes, odd_intersections_positive_even_intersections_negative)","explanation":"Require exactly three sets. Retain every [index_subset,sign,intersection_size] term in source bitmask order, including the positive triple-intersection term, and compare the alternating sum with the actual union size. The exact result, retaining the requested order, is ([[[0], 1, 3], [[1], 1, 3], [[0, 1], -1, 2], [[2], 1, 3], [[0, 2], -1, 2], [[1, 2], -1, 2], [[0, 1, 2], 1, 2]], 5, 5, Yes, odd_intersections_positive_even_intersections_negative)."},"skills":["Three-Set Inclusion-Exclusion Needs the Triple Intersection"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0034","batch_id":"batch-20261014-advance-20261108-r05-inclusion-exclusion-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1dd5650fe7d4c926c0535fcf504ef05f230a5c912143d96f14315e5afe84c2dd","family_id":"qf-mod39-incexc-three-set-formula","seed":202611085034,"semantic_template":"qf-mod39-incexc-three-set-formula/{\"sets\":[[8,10,11],[8,10,12],[8,9,10]]}"},"fields":{"title":"Three-Set Inclusion-Exclusion Needs the Triple Intersection","stem":"Use the complete indexed finite integer-set family sets=[[8, 9, 10], [8, 10, 11], [8, 10, 12]] in supplied index order. Each set has distinct elements; equal sets at different indices remain separate indexed entries. Every nonempty index subset contributes its intersection cardinality, with positive odd-order and negative even-order sign. Keep zero-size intersections and source increasing-bitmask term order. Give (terms, union_size, alternating_sum, equal, certificate).","answer":{"kind":"expression","value":"([[[0], 1, 3], [[1], 1, 3], [[0, 1], -1, 2], [[2], 1, 3], [[0, 2], -1, 2], [[1, 2], -1, 2], [[0, 1, 2], 1, 2]], 5, 5, Yes, odd_intersections_positive_even_intersections_negative)","explanation":"Require exactly three sets. Retain every [index_subset,sign,intersection_size] term in source bitmask order, including the positive triple-intersection term, and compare the alternating sum with the actual union size. The exact result, retaining the requested order, is ([[[0], 1, 3], [[1], 1, 3], [[0, 1], -1, 2], [[2], 1, 3], [[0, 2], -1, 2], [[1, 2], -1, 2], [[0, 1, 2], 1, 2]], 5, 5, Yes, odd_intersections_positive_even_intersections_negative)."},"skills":["Three-Set Inclusion-Exclusion Needs the Triple Intersection"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0035","batch_id":"batch-20261014-advance-20261108-r05-inclusion-exclusion-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9a2f50174aea1a3e5106ecd4f80d027f4cca4ceded3f303602494b42b0d8bc93","family_id":"qf-mod39-incexc-three-set-formula","seed":202611085035,"semantic_template":"qf-mod39-incexc-three-set-formula/{\"sets\":[[9,10,11],[9,11,12],[9,11,13]]}"},"fields":{"title":"Three-Set Inclusion-Exclusion Needs the Triple Intersection","stem":"Use the complete indexed finite integer-set family sets=[[9, 10, 11], [9, 11, 12], [9, 11, 13]] in supplied index order. Each set has distinct elements; equal sets at different indices remain separate indexed entries. Every nonempty index subset contributes its intersection cardinality, with positive odd-order and negative even-order sign. Keep zero-size intersections and source increasing-bitmask term order. Give (terms, union_size, alternating_sum, equal, certificate).","answer":{"kind":"expression","value":"([[[0], 1, 3], [[1], 1, 3], [[0, 1], -1, 2], [[2], 1, 3], [[0, 2], -1, 2], [[1, 2], -1, 2], [[0, 1, 2], 1, 2]], 5, 5, Yes, odd_intersections_positive_even_intersections_negative)","explanation":"Require exactly three sets. Retain every [index_subset,sign,intersection_size] term in source bitmask order, including the positive triple-intersection term, and compare the alternating sum with the actual union size. The exact result, retaining the requested order, is ([[[0], 1, 3], [[1], 1, 3], [[0, 1], -1, 2], [[2], 1, 3], [[0, 2], -1, 2], [[1, 2], -1, 2], [[0, 1, 2], 1, 2]], 5, 5, Yes, odd_intersections_positive_even_intersections_negative)."},"skills":["Three-Set Inclusion-Exclusion Needs the Triple Intersection"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0036","batch_id":"batch-20261014-advance-20261108-r05-inclusion-exclusion-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a1b5c985248794927bb5a93e7e2e765e2e8e2b4f48a6f98f779178b7ea934e01","family_id":"qf-mod39-incexc-three-set-formula","seed":202611085036,"semantic_template":"qf-mod39-incexc-three-set-formula/{\"sets\":[[10,11,12],[10,12,13],[10,12,14]]}"},"fields":{"title":"Three-Set Inclusion-Exclusion Needs the Triple Intersection","stem":"Use the complete indexed finite integer-set family sets=[[10, 11, 12], [10, 12, 13], [10, 12, 14]] in supplied index order. Each set has distinct elements; equal sets at different indices remain separate indexed entries. Every nonempty index subset contributes its intersection cardinality, with positive odd-order and negative even-order sign. Keep zero-size intersections and source increasing-bitmask term order. Give (terms, union_size, alternating_sum, equal, certificate).","answer":{"kind":"expression","value":"([[[0], 1, 3], [[1], 1, 3], [[0, 1], -1, 2], [[2], 1, 3], [[0, 2], -1, 2], [[1, 2], -1, 2], [[0, 1, 2], 1, 2]], 5, 5, Yes, odd_intersections_positive_even_intersections_negative)","explanation":"Require exactly three sets. Retain every [index_subset,sign,intersection_size] term in source bitmask order, including the positive triple-intersection term, and compare the alternating sum with the actual union size. The exact result, retaining the requested order, is ([[[0], 1, 3], [[1], 1, 3], [[0, 1], -1, 2], [[2], 1, 3], [[0, 2], -1, 2], [[1, 2], -1, 2], [[0, 1, 2], 1, 2]], 5, 5, Yes, odd_intersections_positive_even_intersections_negative)."},"skills":["Three-Set Inclusion-Exclusion Needs the Triple Intersection"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0037","batch_id":"batch-20261014-advance-20261108-r05-inclusion-exclusion-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"129e04dc76e477cdb06fab09e07c1c011ac99152e0cb064c59859a817dff92c5","family_id":"qf-mod39-incexc-three-set-formula","seed":202611085037,"semantic_template":"qf-mod39-incexc-three-set-formula/{\"sets\":[[11,12,13],[11,13,14],[11,13,15]]}"},"fields":{"title":"Three-Set Inclusion-Exclusion Needs the Triple Intersection","stem":"Use the complete indexed finite integer-set family sets=[[11, 12, 13], [11, 13, 14], [11, 13, 15]] in supplied index order. Each set has distinct elements; equal sets at different indices remain separate indexed entries. Every nonempty index subset contributes its intersection cardinality, with positive odd-order and negative even-order sign. Keep zero-size intersections and source increasing-bitmask term order. Give (terms, union_size, alternating_sum, equal, certificate).","answer":{"kind":"expression","value":"([[[0], 1, 3], [[1], 1, 3], [[0, 1], -1, 2], [[2], 1, 3], [[0, 2], -1, 2], [[1, 2], -1, 2], [[0, 1, 2], 1, 2]], 5, 5, Yes, odd_intersections_positive_even_intersections_negative)","explanation":"Require exactly three sets. Retain every [index_subset,sign,intersection_size] term in source bitmask order, including the positive triple-intersection term, and compare the alternating sum with the actual union size. The exact result, retaining the requested order, is ([[[0], 1, 3], [[1], 1, 3], [[0, 1], -1, 2], [[2], 1, 3], [[0, 2], -1, 2], [[1, 2], -1, 2], [[0, 1, 2], 1, 2]], 5, 5, Yes, odd_intersections_positive_even_intersections_negative)."},"skills":["Three-Set Inclusion-Exclusion Needs the Triple Intersection"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0038","batch_id":"batch-20261014-advance-20261108-r05-inclusion-exclusion-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"71386396ce6eedc77beaab877a0dc22a804fbd6162c3decd94cfde9d62eefa16","family_id":"qf-mod39-incexc-three-set-formula","seed":202611085038,"semantic_template":"qf-mod39-incexc-three-set-formula/{\"sets\":[[12,13,14],[12,14,15],[12,14,16]]}"},"fields":{"title":"Three-Set Inclusion-Exclusion Needs the Triple Intersection","stem":"Use the complete indexed finite integer-set family sets=[[12, 13, 14], [12, 14, 15], [12, 14, 16]] in supplied index order. Each set has distinct elements; equal sets at different indices remain separate indexed entries. Every nonempty index subset contributes its intersection cardinality, with positive odd-order and negative even-order sign. Keep zero-size intersections and source increasing-bitmask term order. Give (terms, union_size, alternating_sum, equal, certificate).","answer":{"kind":"expression","value":"([[[0], 1, 3], [[1], 1, 3], [[0, 1], -1, 2], [[2], 1, 3], [[0, 2], -1, 2], [[1, 2], -1, 2], [[0, 1, 2], 1, 2]], 5, 5, Yes, odd_intersections_positive_even_intersections_negative)","explanation":"Require exactly three sets. Retain every [index_subset,sign,intersection_size] term in source bitmask order, including the positive triple-intersection term, and compare the alternating sum with the actual union size. The exact result, retaining the requested order, is ([[[0], 1, 3], [[1], 1, 3], [[0, 1], -1, 2], [[2], 1, 3], [[0, 2], -1, 2], [[1, 2], -1, 2], [[0, 1, 2], 1, 2]], 5, 5, Yes, odd_intersections_positive_even_intersections_negative)."},"skills":["Three-Set Inclusion-Exclusion Needs the Triple Intersection"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0039","batch_id":"batch-20261014-advance-20261108-r05-inclusion-exclusion-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"747ef6be0d69baf015a9e85c6730e318ff7b9a822d17df4023839e0b83d6fe78","family_id":"qf-mod39-incexc-three-set-formula","seed":202611085039,"semantic_template":"qf-mod39-incexc-three-set-formula/{\"sets\":[[13,14,15],[13,15,16],[13,15,17]]}"},"fields":{"title":"Three-Set Inclusion-Exclusion Needs the Triple Intersection","stem":"Use the complete indexed finite integer-set family sets=[[13, 14, 15], [13, 15, 16], [13, 15, 17]] in supplied index order. Each set has distinct elements; equal sets at different indices remain separate indexed entries. Every nonempty index subset contributes its intersection cardinality, with positive odd-order and negative even-order sign. Keep zero-size intersections and source increasing-bitmask term order. Give (terms, union_size, alternating_sum, equal, certificate).","answer":{"kind":"expression","value":"([[[0], 1, 3], [[1], 1, 3], [[0, 1], -1, 2], [[2], 1, 3], [[0, 2], -1, 2], [[1, 2], -1, 2], [[0, 1, 2], 1, 2]], 5, 5, Yes, odd_intersections_positive_even_intersections_negative)","explanation":"Require exactly three sets. Retain every [index_subset,sign,intersection_size] term in source bitmask order, including the positive triple-intersection term, and compare the alternating sum with the actual union size. The exact result, retaining the requested order, is ([[[0], 1, 3], [[1], 1, 3], [[0, 1], -1, 2], [[2], 1, 3], [[0, 2], -1, 2], [[1, 2], -1, 2], [[0, 1, 2], 1, 2]], 5, 5, Yes, odd_intersections_positive_even_intersections_negative)."},"skills":["Three-Set Inclusion-Exclusion Needs the Triple Intersection"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0040","batch_id":"batch-20261014-advance-20261108-r05-inclusion-exclusion-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9f69210b39d2244fa0b185d7f9a9054e67885ca24176d4a6948e8d4a219b730a","family_id":"qf-mod39-incexc-three-set-formula","seed":202611085040,"semantic_template":"qf-mod39-incexc-three-set-formula/{\"sets\":[[14,15,16],[14,16,17],[14,16,18]]}"},"fields":{"title":"Three-Set Inclusion-Exclusion Needs the Triple Intersection","stem":"Use the complete indexed finite integer-set family sets=[[14, 15, 16], [14, 16, 17], [14, 16, 18]] in supplied index order. Each set has distinct elements; equal sets at different indices remain separate indexed entries. Every nonempty index subset contributes its intersection cardinality, with positive odd-order and negative even-order sign. Keep zero-size intersections and source increasing-bitmask term order. Give (terms, union_size, alternating_sum, equal, certificate).","answer":{"kind":"expression","value":"([[[0], 1, 3], [[1], 1, 3], [[0, 1], -1, 2], [[2], 1, 3], [[0, 2], -1, 2], [[1, 2], -1, 2], [[0, 1, 2], 1, 2]], 5, 5, Yes, odd_intersections_positive_even_intersections_negative)","explanation":"Require exactly three sets. Retain every [index_subset,sign,intersection_size] term in source bitmask order, including the positive triple-intersection term, and compare the alternating sum with the actual union size. The exact result, retaining the requested order, is ([[[0], 1, 3], [[1], 1, 3], [[0, 1], -1, 2], [[2], 1, 3], [[0, 2], -1, 2], [[1, 2], -1, 2], [[0, 1, 2], 1, 2]], 5, 5, Yes, odd_intersections_positive_even_intersections_negative)."},"skills":["Three-Set Inclusion-Exclusion Needs the Triple Intersection"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0041","batch_id":"batch-20261014-advance-20261108-r05-inclusion-exclusion-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"57f830546069bb741f57120cccadcf3e4d087068b1aaac68624b25eda66a63a6","family_id":"qf-mod39-incexc-three-set-formula","seed":202611085041,"semantic_template":"qf-mod39-incexc-three-set-formula/{\"sets\":[[15,16,17],[15,17,18],[15,17,19]]}"},"fields":{"title":"Three-Set Inclusion-Exclusion Needs the Triple Intersection","stem":"Use the complete indexed finite integer-set family sets=[[15, 16, 17], [15, 17, 18], [15, 17, 19]] in supplied index order. Each set has distinct elements; equal sets at different indices remain separate indexed entries. Every nonempty index subset contributes its intersection cardinality, with positive odd-order and negative even-order sign. Keep zero-size intersections and source increasing-bitmask term order. Give (terms, union_size, alternating_sum, equal, certificate).","answer":{"kind":"expression","value":"([[[0], 1, 3], [[1], 1, 3], [[0, 1], -1, 2], [[2], 1, 3], [[0, 2], -1, 2], [[1, 2], -1, 2], [[0, 1, 2], 1, 2]], 5, 5, Yes, odd_intersections_positive_even_intersections_negative)","explanation":"Require exactly three sets. Retain every [index_subset,sign,intersection_size] term in source bitmask order, including the positive triple-intersection term, and compare the alternating sum with the actual union size. The exact result, retaining the requested order, is ([[[0], 1, 3], [[1], 1, 3], [[0, 1], -1, 2], [[2], 1, 3], [[0, 2], -1, 2], [[1, 2], -1, 2], [[0, 1, 2], 1, 2]], 5, 5, Yes, odd_intersections_positive_even_intersections_negative)."},"skills":["Three-Set Inclusion-Exclusion Needs the Triple Intersection"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0042","batch_id":"batch-20261014-advance-20261108-r05-inclusion-exclusion-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4c2895401cd76601f71fb63aa307e9a37f921a9fa1676172fea18f9d19966836","family_id":"qf-mod39-incexc-three-set-formula","seed":202611085042,"semantic_template":"qf-mod39-incexc-three-set-formula/{\"sets\":[[16,17,18],[16,18,19],[16,18,20]]}"},"fields":{"title":"Three-Set Inclusion-Exclusion Needs the Triple Intersection","stem":"Use the complete indexed finite integer-set family sets=[[16, 17, 18], [16, 18, 19], [16, 18, 20]] in supplied index order. Each set has distinct elements; equal sets at different indices remain separate indexed entries. Every nonempty index subset contributes its intersection cardinality, with positive odd-order and negative even-order sign. Keep zero-size intersections and source increasing-bitmask term order. Give (terms, union_size, alternating_sum, equal, certificate).","answer":{"kind":"expression","value":"([[[0], 1, 3], [[1], 1, 3], [[0, 1], -1, 2], [[2], 1, 3], [[0, 2], -1, 2], [[1, 2], -1, 2], [[0, 1, 2], 1, 2]], 5, 5, Yes, odd_intersections_positive_even_intersections_negative)","explanation":"Require exactly three sets. Retain every [index_subset,sign,intersection_size] term in source bitmask order, including the positive triple-intersection term, and compare the alternating sum with the actual union size. The exact result, retaining the requested order, is ([[[0], 1, 3], [[1], 1, 3], [[0, 1], -1, 2], [[2], 1, 3], [[0, 2], -1, 2], [[1, 2], -1, 2], [[0, 1, 2], 1, 2]], 5, 5, Yes, odd_intersections_positive_even_intersections_negative)."},"skills":["Three-Set Inclusion-Exclusion Needs the Triple Intersection"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0043","batch_id":"batch-20261014-advance-20261108-r05-inclusion-exclusion-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9d26840471a6c1a233a54bf0f3c9de28aba70db17074350d86882830a6f54e30","family_id":"qf-mod39-incexc-three-set-formula","seed":202611085043,"semantic_template":"qf-mod39-incexc-three-set-formula/{\"sets\":[[17,18,19],[17,19,20],[17,19,21]]}"},"fields":{"title":"Three-Set Inclusion-Exclusion Needs the Triple Intersection","stem":"Use the complete indexed finite integer-set family sets=[[17, 18, 19], [17, 19, 20], [17, 19, 21]] in supplied index order. Each set has distinct elements; equal sets at different indices remain separate indexed entries. Every nonempty index subset contributes its intersection cardinality, with positive odd-order and negative even-order sign. Keep zero-size intersections and source increasing-bitmask term order. Give (terms, union_size, alternating_sum, equal, certificate).","answer":{"kind":"expression","value":"([[[0], 1, 3], [[1], 1, 3], [[0, 1], -1, 2], [[2], 1, 3], [[0, 2], -1, 2], [[1, 2], -1, 2], [[0, 1, 2], 1, 2]], 5, 5, Yes, odd_intersections_positive_even_intersections_negative)","explanation":"Require exactly three sets. Retain every [index_subset,sign,intersection_size] term in source bitmask order, including the positive triple-intersection term, and compare the alternating sum with the actual union size. The exact result, retaining the requested order, is ([[[0], 1, 3], [[1], 1, 3], [[0, 1], -1, 2], [[2], 1, 3], [[0, 2], -1, 2], [[1, 2], -1, 2], [[0, 1, 2], 1, 2]], 5, 5, Yes, odd_intersections_positive_even_intersections_negative)."},"skills":["Three-Set Inclusion-Exclusion Needs the Triple Intersection"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0044","batch_id":"batch-20261014-advance-20261108-r05-inclusion-exclusion-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"11eeaebb7f90dbecacb3cc3618fd828a1ee778258ce994e7a1d43697f748f2c7","family_id":"qf-mod39-incexc-three-set-formula","seed":202611085044,"semantic_template":"qf-mod39-incexc-three-set-formula/{\"sets\":[[18,19,20],[18,20,21],[18,20,22]]}"},"fields":{"title":"Three-Set Inclusion-Exclusion Needs the Triple Intersection","stem":"Use the complete indexed finite integer-set family sets=[[18, 19, 20], [18, 20, 21], [18, 20, 22]] in supplied index order. Each set has distinct elements; equal sets at different indices remain separate indexed entries. Every nonempty index subset contributes its intersection cardinality, with positive odd-order and negative even-order sign. Keep zero-size intersections and source increasing-bitmask term order. Give (terms, union_size, alternating_sum, equal, certificate).","answer":{"kind":"expression","value":"([[[0], 1, 3], [[1], 1, 3], [[0, 1], -1, 2], [[2], 1, 3], [[0, 2], -1, 2], [[1, 2], -1, 2], [[0, 1, 2], 1, 2]], 5, 5, Yes, odd_intersections_positive_even_intersections_negative)","explanation":"Require exactly three sets. Retain every [index_subset,sign,intersection_size] term in source bitmask order, including the positive triple-intersection term, and compare the alternating sum with the actual union size. The exact result, retaining the requested order, is ([[[0], 1, 3], [[1], 1, 3], [[0, 1], -1, 2], [[2], 1, 3], [[0, 2], -1, 2], [[1, 2], -1, 2], [[0, 1, 2], 1, 2]], 5, 5, Yes, odd_intersections_positive_even_intersections_negative)."},"skills":["Three-Set Inclusion-Exclusion Needs the Triple Intersection"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0045","batch_id":"batch-20261014-advance-20261108-r05-inclusion-exclusion-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"45f4929efb95d02960a89c380215c3fd5468c931724e2f41912ee33c497711ab","family_id":"qf-mod39-incexc-three-set-formula","seed":202611085045,"semantic_template":"qf-mod39-incexc-three-set-formula/{\"sets\":[[19,20,21],[19,21,22],[19,21,23]]}"},"fields":{"title":"Three-Set Inclusion-Exclusion Needs the Triple Intersection","stem":"Use the complete indexed finite integer-set family sets=[[19, 20, 21], [19, 21, 22], [19, 21, 23]] in supplied index order. Each set has distinct elements; equal sets at different indices remain separate indexed entries. Every nonempty index subset contributes its intersection cardinality, with positive odd-order and negative even-order sign. Keep zero-size intersections and source increasing-bitmask term order. Give (terms, union_size, alternating_sum, equal, certificate).","answer":{"kind":"expression","value":"([[[0], 1, 3], [[1], 1, 3], [[0, 1], -1, 2], [[2], 1, 3], [[0, 2], -1, 2], [[1, 2], -1, 2], [[0, 1, 2], 1, 2]], 5, 5, Yes, odd_intersections_positive_even_intersections_negative)","explanation":"Require exactly three sets. Retain every [index_subset,sign,intersection_size] term in source bitmask order, including the positive triple-intersection term, and compare the alternating sum with the actual union size. The exact result, retaining the requested order, is ([[[0], 1, 3], [[1], 1, 3], [[0, 1], -1, 2], [[2], 1, 3], [[0, 2], -1, 2], [[1, 2], -1, 2], [[0, 1, 2], 1, 2]], 5, 5, Yes, odd_intersections_positive_even_intersections_negative)."},"skills":["Three-Set Inclusion-Exclusion Needs the Triple Intersection"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0046","batch_id":"batch-20261014-advance-20261108-r05-inclusion-exclusion-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e107fb21d3c10ff557d99f80ffe6e43be25f5e64e11e1bb3ce1da99f7d0049ef","family_id":"qf-mod39-incexc-three-set-formula","seed":202611085046,"semantic_template":"qf-mod39-incexc-three-set-formula/{\"sets\":[[20,21,22],[20,22,23],[20,22,24]]}"},"fields":{"title":"Three-Set Inclusion-Exclusion Needs the Triple Intersection","stem":"Use the complete indexed finite integer-set family sets=[[20, 21, 22], [20, 22, 23], [20, 22, 24]] in supplied index order. Each set has distinct elements; equal sets at different indices remain separate indexed entries. Every nonempty index subset contributes its intersection cardinality, with positive odd-order and negative even-order sign. Keep zero-size intersections and source increasing-bitmask term order. Give (terms, union_size, alternating_sum, equal, certificate).","answer":{"kind":"expression","value":"([[[0], 1, 3], [[1], 1, 3], [[0, 1], -1, 2], [[2], 1, 3], [[0, 2], -1, 2], [[1, 2], -1, 2], [[0, 1, 2], 1, 2]], 5, 5, Yes, odd_intersections_positive_even_intersections_negative)","explanation":"Require exactly three sets. Retain every [index_subset,sign,intersection_size] term in source bitmask order, including the positive triple-intersection term, and compare the alternating sum with the actual union size. The exact result, retaining the requested order, is ([[[0], 1, 3], [[1], 1, 3], [[0, 1], -1, 2], [[2], 1, 3], [[0, 2], -1, 2], [[1, 2], -1, 2], [[0, 1, 2], 1, 2]], 5, 5, Yes, odd_intersections_positive_even_intersections_negative)."},"skills":["Three-Set Inclusion-Exclusion Needs the Triple Intersection"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0047","batch_id":"batch-20261014-advance-20261108-r05-inclusion-exclusion-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6c76bd476d544f912fecbb9f506aff8430b22d13eed1aa8a6e65aa4db1782f55","family_id":"qf-mod39-incexc-three-set-formula","seed":202611085047,"semantic_template":"qf-mod39-incexc-three-set-formula/{\"sets\":[[21,22,23],[21,23,24],[21,23,25]]}"},"fields":{"title":"Three-Set Inclusion-Exclusion Needs the Triple Intersection","stem":"Use the complete indexed finite integer-set family sets=[[21, 22, 23], [21, 23, 24], [21, 23, 25]] in supplied index order. Each set has distinct elements; equal sets at different indices remain separate indexed entries. Every nonempty index subset contributes its intersection cardinality, with positive odd-order and negative even-order sign. Keep zero-size intersections and source increasing-bitmask term order. Give (terms, union_size, alternating_sum, equal, certificate).","answer":{"kind":"expression","value":"([[[0], 1, 3], [[1], 1, 3], [[0, 1], -1, 2], [[2], 1, 3], [[0, 2], -1, 2], [[1, 2], -1, 2], [[0, 1, 2], 1, 2]], 5, 5, Yes, odd_intersections_positive_even_intersections_negative)","explanation":"Require exactly three sets. Retain every [index_subset,sign,intersection_size] term in source bitmask order, including the positive triple-intersection term, and compare the alternating sum with the actual union size. The exact result, retaining the requested order, is ([[[0], 1, 3], [[1], 1, 3], [[0, 1], -1, 2], [[2], 1, 3], [[0, 2], -1, 2], [[1, 2], -1, 2], [[0, 1, 2], 1, 2]], 5, 5, Yes, odd_intersections_positive_even_intersections_negative)."},"skills":["Three-Set Inclusion-Exclusion Needs the Triple Intersection"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0048","batch_id":"batch-20261014-advance-20261108-r05-inclusion-exclusion-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"530cd26db3f1fb325a046fc5f96ee65fdfa675436e3af0f2be08f0449c9d4e3f","family_id":"qf-mod39-incexc-three-set-formula","seed":202611085048,"semantic_template":"qf-mod39-incexc-three-set-formula/{\"sets\":[[22,23,24],[22,24,25],[22,24,26]]}"},"fields":{"title":"Three-Set Inclusion-Exclusion Needs the Triple Intersection","stem":"Use the complete indexed finite integer-set family sets=[[22, 23, 24], [22, 24, 25], [22, 24, 26]] in supplied index order. Each set has distinct elements; equal sets at different indices remain separate indexed entries. Every nonempty index subset contributes its intersection cardinality, with positive odd-order and negative even-order sign. Keep zero-size intersections and source increasing-bitmask term order. Give (terms, union_size, alternating_sum, equal, certificate).","answer":{"kind":"expression","value":"([[[0], 1, 3], [[1], 1, 3], [[0, 1], -1, 2], [[2], 1, 3], [[0, 2], -1, 2], [[1, 2], -1, 2], [[0, 1, 2], 1, 2]], 5, 5, Yes, odd_intersections_positive_even_intersections_negative)","explanation":"Require exactly three sets. Retain every [index_subset,sign,intersection_size] term in source bitmask order, including the positive triple-intersection term, and compare the alternating sum with the actual union size. The exact result, retaining the requested order, is ([[[0], 1, 3], [[1], 1, 3], [[0, 1], -1, 2], [[2], 1, 3], [[0, 2], -1, 2], [[1, 2], -1, 2], [[0, 1, 2], 1, 2]], 5, 5, Yes, odd_intersections_positive_even_intersections_negative)."},"skills":["Three-Set Inclusion-Exclusion Needs the Triple Intersection"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0049","batch_id":"batch-20261014-advance-20261108-r05-inclusion-exclusion-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"18807ae3018475befeb5d622b01d85813c2c1720b273f2365cf55bb71f8593ab","family_id":"qf-mod39-incexc-three-set-formula","seed":202611085049,"semantic_template":"qf-mod39-incexc-three-set-formula/{\"sets\":[[23,24,25],[23,25,26],[23,25,27]]}"},"fields":{"title":"Three-Set Inclusion-Exclusion Needs the Triple Intersection","stem":"Use the complete indexed finite integer-set family sets=[[23, 24, 25], [23, 25, 26], [23, 25, 27]] in supplied index order. Each set has distinct elements; equal sets at different indices remain separate indexed entries. Every nonempty index subset contributes its intersection cardinality, with positive odd-order and negative even-order sign. Keep zero-size intersections and source increasing-bitmask term order. Give (terms, union_size, alternating_sum, equal, certificate).","answer":{"kind":"expression","value":"([[[0], 1, 3], [[1], 1, 3], [[0, 1], -1, 2], [[2], 1, 3], [[0, 2], -1, 2], [[1, 2], -1, 2], [[0, 1, 2], 1, 2]], 5, 5, Yes, odd_intersections_positive_even_intersections_negative)","explanation":"Require exactly three sets. Retain every [index_subset,sign,intersection_size] term in source bitmask order, including the positive triple-intersection term, and compare the alternating sum with the actual union size. The exact result, retaining the requested order, is ([[[0], 1, 3], [[1], 1, 3], [[0, 1], -1, 2], [[2], 1, 3], [[0, 2], -1, 2], [[1, 2], -1, 2], [[0, 1, 2], 1, 2]], 5, 5, Yes, odd_intersections_positive_even_intersections_negative)."},"skills":["Three-Set Inclusion-Exclusion Needs the Triple Intersection"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0050","batch_id":"batch-20261014-advance-20261108-r05-inclusion-exclusion-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d67816615665656cdc5881706a208d7b59868f673b8179b0e10e3b073abe37b0","family_id":"qf-mod39-incexc-three-set-formula","seed":202611085050,"semantic_template":"qf-mod39-incexc-three-set-formula/{\"sets\":[[24,25,26],[24,26,27],[24,26,28]]}"},"fields":{"title":"Three-Set Inclusion-Exclusion Needs the Triple Intersection","stem":"Use the complete indexed finite integer-set family sets=[[24, 25, 26], [24, 26, 27], [24, 26, 28]] in supplied index order. Each set has distinct elements; equal sets at different indices remain separate indexed entries. Every nonempty index subset contributes its intersection cardinality, with positive odd-order and negative even-order sign. Keep zero-size intersections and source increasing-bitmask term order. Give (terms, union_size, alternating_sum, equal, certificate).","answer":{"kind":"expression","value":"([[[0], 1, 3], [[1], 1, 3], [[0, 1], -1, 2], [[2], 1, 3], [[0, 2], -1, 2], [[1, 2], -1, 2], [[0, 1, 2], 1, 2]], 5, 5, Yes, odd_intersections_positive_even_intersections_negative)","explanation":"Require exactly three sets. Retain every [index_subset,sign,intersection_size] term in source bitmask order, including the positive triple-intersection term, and compare the alternating sum with the actual union size. The exact result, retaining the requested order, is ([[[0], 1, 3], [[1], 1, 3], [[0, 1], -1, 2], [[2], 1, 3], [[0, 2], -1, 2], [[1, 2], -1, 2], [[0, 1, 2], 1, 2]], 5, 5, Yes, odd_intersections_positive_even_intersections_negative)."},"skills":["Three-Set Inclusion-Exclusion Needs the Triple Intersection"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0051","batch_id":"batch-20261014-advance-20261108-r05-inclusion-exclusion-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0319e5c054e1931356045fdccefc4de80442064f2e2cd7afbf8131d9fea08d61","family_id":"qf-mod39-incexc-general-alternating-sum","seed":202611085051,"semantic_template":"qf-mod39-incexc-general-alternating-sum/{\"sets\":[[0,1,2]]}"},"fields":{"title":"All Nonempty Index Subsets Are Needed for the Exact Alternating Sum","stem":"Use the complete indexed finite integer-set family sets=[[0, 1, 2]] in supplied index order. Each set has distinct elements; equal sets at different indices remain separate indexed entries. Every nonempty index subset contributes its intersection cardinality, with positive odd-order and negative even-order sign. Keep zero-size intersections and source increasing-bitmask term order. Give (family_size, terms, union_size, alternating_sum, equal, certificate).","answer":{"kind":"expression","value":"(1, [[[0], 1, 3]], 3, 3, Yes, all_nonempty_index_subsets_included)","explanation":"Include all2^k−1 nonempty index subsets for a family of size k. Equal indexed sets are not deduplicated. The empty indexed family has empty union, no terms and alternating sum0. The exact result, retaining the requested order, is (1, [[[0], 1, 3]], 3, 3, Yes, all_nonempty_index_subsets_included)."},"skills":["All Nonempty Index Subsets Are Needed for the Exact Alternating Sum"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0052","batch_id":"batch-20261014-advance-20261108-r05-inclusion-exclusion-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ebab00bfebb4ab7f62468df60b778e7256268dca2a5c57a5b50c7f2dc0786945","family_id":"qf-mod39-incexc-general-alternating-sum","seed":202611085052,"semantic_template":"qf-mod39-incexc-general-alternating-sum/{\"sets\":[[1,2,3],[1,3,4]]}"},"fields":{"title":"All Nonempty Index Subsets Are Needed for the Exact Alternating Sum","stem":"Use the complete indexed finite integer-set family sets=[[1, 2, 3], [1, 3, 4]] in supplied index order. Each set has distinct elements; equal sets at different indices remain separate indexed entries. Every nonempty index subset contributes its intersection cardinality, with positive odd-order and negative even-order sign. Keep zero-size intersections and source increasing-bitmask term order. Give (family_size, terms, union_size, alternating_sum, equal, certificate).","answer":{"kind":"expression","value":"(2, [[[0], 1, 3], [[1], 1, 3], [[0, 1], -1, 2]], 4, 4, Yes, all_nonempty_index_subsets_included)","explanation":"Include all2^k−1 nonempty index subsets for a family of size k. Equal indexed sets are not deduplicated. The empty indexed family has empty union, no terms and alternating sum0. The exact result, retaining the requested order, is (2, [[[0], 1, 3], [[1], 1, 3], [[0, 1], -1, 2]], 4, 4, Yes, all_nonempty_index_subsets_included)."},"skills":["All Nonempty Index Subsets Are Needed for the Exact Alternating Sum"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0053","batch_id":"batch-20261014-advance-20261108-r05-inclusion-exclusion-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d126503270e110cc23fc357910814b28bacb9d8471a5954f6c9ed62412614f22","family_id":"qf-mod39-incexc-general-alternating-sum","seed":202611085053,"semantic_template":"qf-mod39-incexc-general-alternating-sum/{\"sets\":[[2,3,4],[2,4,5],[2,4,6]]}"},"fields":{"title":"All Nonempty Index Subsets Are Needed for the Exact Alternating Sum","stem":"Use the complete indexed finite integer-set family sets=[[2, 3, 4], [2, 4, 5], [2, 4, 6]] in supplied index order. Each set has distinct elements; equal sets at different indices remain separate indexed entries. Every nonempty index subset contributes its intersection cardinality, with positive odd-order and negative even-order sign. Keep zero-size intersections and source increasing-bitmask term order. Give (family_size, terms, union_size, alternating_sum, equal, certificate).","answer":{"kind":"expression","value":"(3, [[[0], 1, 3], [[1], 1, 3], [[0, 1], -1, 2], [[2], 1, 3], [[0, 2], -1, 2], [[1, 2], -1, 2], [[0, 1, 2], 1, 2]], 5, 5, Yes, all_nonempty_index_subsets_included)","explanation":"Include all2^k−1 nonempty index subsets for a family of size k. Equal indexed sets are not deduplicated. The empty indexed family has empty union, no terms and alternating sum0. The exact result, retaining the requested order, is (3, [[[0], 1, 3], [[1], 1, 3], [[0, 1], -1, 2], [[2], 1, 3], [[0, 2], -1, 2], [[1, 2], -1, 2], [[0, 1, 2], 1, 2]], 5, 5, Yes, all_nonempty_index_subsets_included)."},"skills":["All Nonempty Index Subsets Are Needed for the Exact Alternating Sum"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0054","batch_id":"batch-20261014-advance-20261108-r05-inclusion-exclusion-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e276ef2e21aba2fc73847b44b5ec17f79d77a01bf745c32fdede3e1e23477bd6","family_id":"qf-mod39-incexc-general-alternating-sum","seed":202611085054,"semantic_template":"qf-mod39-incexc-general-alternating-sum/{\"sets\":[[3,4,5],[3,5,6],[3,5,7],[3,8]]}"},"fields":{"title":"All Nonempty Index Subsets Are Needed for the Exact Alternating Sum","stem":"Use the complete indexed finite integer-set family sets=[[3, 4, 5], [3, 5, 6], [3, 5, 7], [3, 8]] in supplied index order. Each set has distinct elements; equal sets at different indices remain separate indexed entries. Every nonempty index subset contributes its intersection cardinality, with positive odd-order and negative even-order sign. Keep zero-size intersections and source increasing-bitmask term order. Give (family_size, terms, union_size, alternating_sum, equal, certificate).","answer":{"kind":"expression","value":"(4, [[[0], 1, 3], [[1], 1, 3], [[0, 1], -1, 2], [[2], 1, 3], [[0, 2], -1, 2], [[1, 2], -1, 2], [[0, 1, 2], 1, 2], [[3], 1, 2], [[0, 3], -1, 1], [[1, 3], -1, 1], [[0, 1, 3], 1, 1], [[2, 3], -1, 1], [[0, 2, 3], 1, 1], [[1, 2, 3], 1, 1], [[0, 1, 2, 3], -1, 1]], 6, 6, Yes, all_nonempty_index_subsets_included)","explanation":"Include all2^k−1 nonempty index subsets for a family of size k. Equal indexed sets are not deduplicated. The empty indexed family has empty union, no terms and alternating sum0. The exact result, retaining the requested order, is (4, [[[0], 1, 3], [[1], 1, 3], [[0, 1], -1, 2], [[2], 1, 3], [[0, 2], -1, 2], [[1, 2], -1, 2], [[0, 1, 2], 1, 2], [[3], 1, 2], [[0, 3], -1, 1], [[1, 3], -1, 1], [[0, 1, 3], 1, 1], [[2, 3], -1, 1], [[0, 2, 3], 1, 1], [[1, 2, 3], 1, 1], [[0, 1, 2, 3], -1, 1]], 6, 6, Yes, all_nonempty_index_subsets_included)."},"skills":["All Nonempty Index Subsets Are Needed for the Exact Alternating Sum"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0055","batch_id":"batch-20261014-advance-20261108-r05-inclusion-exclusion-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"557fd7a4007ac504a020c604044fa4fc5feaaab65f99a5fe6554af3ad77e6bb7","family_id":"qf-mod39-incexc-general-alternating-sum","seed":202611085055,"semantic_template":"qf-mod39-incexc-general-alternating-sum/{\"sets\":[[4,5,6]]}"},"fields":{"title":"All Nonempty Index Subsets Are Needed for the Exact Alternating Sum","stem":"Use the complete indexed finite integer-set family sets=[[4, 5, 6]] in supplied index order. Each set has distinct elements; equal sets at different indices remain separate indexed entries. Every nonempty index subset contributes its intersection cardinality, with positive odd-order and negative even-order sign. Keep zero-size intersections and source increasing-bitmask term order. Give (family_size, terms, union_size, alternating_sum, equal, certificate).","answer":{"kind":"expression","value":"(1, [[[0], 1, 3]], 3, 3, Yes, all_nonempty_index_subsets_included)","explanation":"Include all2^k−1 nonempty index subsets for a family of size k. Equal indexed sets are not deduplicated. The empty indexed family has empty union, no terms and alternating sum0. The exact result, retaining the requested order, is (1, [[[0], 1, 3]], 3, 3, Yes, all_nonempty_index_subsets_included)."},"skills":["All Nonempty Index Subsets Are Needed for the Exact Alternating Sum"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0056","batch_id":"batch-20261014-advance-20261108-r05-inclusion-exclusion-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"476e0658e5824e2d8d270ff099ce9a73ef60fd65d2b2aa400791f1e7cd694d39","family_id":"qf-mod39-incexc-general-alternating-sum","seed":202611085056,"semantic_template":"qf-mod39-incexc-general-alternating-sum/{\"sets\":[[5,6,7],[5,7,8]]}"},"fields":{"title":"All Nonempty Index Subsets Are Needed for the Exact Alternating Sum","stem":"Use the complete indexed finite integer-set family sets=[[5, 6, 7], [5, 7, 8]] in supplied index order. Each set has distinct elements; equal sets at different indices remain separate indexed entries. Every nonempty index subset contributes its intersection cardinality, with positive odd-order and negative even-order sign. Keep zero-size intersections and source increasing-bitmask term order. Give (family_size, terms, union_size, alternating_sum, equal, certificate).","answer":{"kind":"expression","value":"(2, [[[0], 1, 3], [[1], 1, 3], [[0, 1], -1, 2]], 4, 4, Yes, all_nonempty_index_subsets_included)","explanation":"Include all2^k−1 nonempty index subsets for a family of size k. Equal indexed sets are not deduplicated. The empty indexed family has empty union, no terms and alternating sum0. The exact result, retaining the requested order, is (2, [[[0], 1, 3], [[1], 1, 3], [[0, 1], -1, 2]], 4, 4, Yes, all_nonempty_index_subsets_included)."},"skills":["All Nonempty Index Subsets Are Needed for the Exact Alternating Sum"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0057","batch_id":"batch-20261014-advance-20261108-r05-inclusion-exclusion-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"90099db4395f54d063b1bc25b8868f46173f46697822195f7ac5805c5fdf22bc","family_id":"qf-mod39-incexc-general-alternating-sum","seed":202611085057,"semantic_template":"qf-mod39-incexc-general-alternating-sum/{\"sets\":[[6,7,8],[6,8,10],[6,8,9]]}"},"fields":{"title":"All Nonempty Index Subsets Are Needed for the Exact Alternating Sum","stem":"Use the complete indexed finite integer-set family sets=[[6, 7, 8], [6, 8, 9], [6, 8, 10]] in supplied index order. Each set has distinct elements; equal sets at different indices remain separate indexed entries. Every nonempty index subset contributes its intersection cardinality, with positive odd-order and negative even-order sign. Keep zero-size intersections and source increasing-bitmask term order. Give (family_size, terms, union_size, alternating_sum, equal, certificate).","answer":{"kind":"expression","value":"(3, [[[0], 1, 3], [[1], 1, 3], [[0, 1], -1, 2], [[2], 1, 3], [[0, 2], -1, 2], [[1, 2], -1, 2], [[0, 1, 2], 1, 2]], 5, 5, Yes, all_nonempty_index_subsets_included)","explanation":"Include all2^k−1 nonempty index subsets for a family of size k. Equal indexed sets are not deduplicated. The empty indexed family has empty union, no terms and alternating sum0. The exact result, retaining the requested order, is (3, [[[0], 1, 3], [[1], 1, 3], [[0, 1], -1, 2], [[2], 1, 3], [[0, 2], -1, 2], [[1, 2], -1, 2], [[0, 1, 2], 1, 2]], 5, 5, Yes, all_nonempty_index_subsets_included)."},"skills":["All Nonempty Index Subsets Are Needed for the Exact Alternating Sum"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0058","batch_id":"batch-20261014-advance-20261108-r05-inclusion-exclusion-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"631743458c2fec50581d6ab7ae1844e891877ce84aff7efd4f456584938cf109","family_id":"qf-mod39-incexc-general-alternating-sum","seed":202611085058,"semantic_template":"qf-mod39-incexc-general-alternating-sum/{\"sets\":[[7,12],[7,8,9],[7,9,10],[7,9,11]]}"},"fields":{"title":"All Nonempty Index Subsets Are Needed for the Exact Alternating Sum","stem":"Use the complete indexed finite integer-set family sets=[[7, 8, 9], [7, 9, 10], [7, 9, 11], [7, 12]] in supplied index order. Each set has distinct elements; equal sets at different indices remain separate indexed entries. Every nonempty index subset contributes its intersection cardinality, with positive odd-order and negative even-order sign. Keep zero-size intersections and source increasing-bitmask term order. Give (family_size, terms, union_size, alternating_sum, equal, certificate).","answer":{"kind":"expression","value":"(4, [[[0], 1, 3], [[1], 1, 3], [[0, 1], -1, 2], [[2], 1, 3], [[0, 2], -1, 2], [[1, 2], -1, 2], [[0, 1, 2], 1, 2], [[3], 1, 2], [[0, 3], -1, 1], [[1, 3], -1, 1], [[0, 1, 3], 1, 1], [[2, 3], -1, 1], [[0, 2, 3], 1, 1], [[1, 2, 3], 1, 1], [[0, 1, 2, 3], -1, 1]], 6, 6, Yes, all_nonempty_index_subsets_included)","explanation":"Include all2^k−1 nonempty index subsets for a family of size k. Equal indexed sets are not deduplicated. The empty indexed family has empty union, no terms and alternating sum0. The exact result, retaining the requested order, is (4, [[[0], 1, 3], [[1], 1, 3], [[0, 1], -1, 2], [[2], 1, 3], [[0, 2], -1, 2], [[1, 2], -1, 2], [[0, 1, 2], 1, 2], [[3], 1, 2], [[0, 3], -1, 1], [[1, 3], -1, 1], [[0, 1, 3], 1, 1], [[2, 3], -1, 1], [[0, 2, 3], 1, 1], [[1, 2, 3], 1, 1], [[0, 1, 2, 3], -1, 1]], 6, 6, Yes, all_nonempty_index_subsets_included)."},"skills":["All Nonempty Index Subsets Are Needed for the Exact Alternating Sum"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0059","batch_id":"batch-20261014-advance-20261108-r05-inclusion-exclusion-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7a7fee97d6a137494caae395a958f7f052ad1a1755d1c2bb06e5f3052d0a34fa","family_id":"qf-mod39-incexc-general-alternating-sum","seed":202611085059,"semantic_template":"qf-mod39-incexc-general-alternating-sum/{\"sets\":[[8,9,10]]}"},"fields":{"title":"All Nonempty Index Subsets Are Needed for the Exact Alternating Sum","stem":"Use the complete indexed finite integer-set family sets=[[8, 9, 10]] in supplied index order. Each set has distinct elements; equal sets at different indices remain separate indexed entries. Every nonempty index subset contributes its intersection cardinality, with positive odd-order and negative even-order sign. Keep zero-size intersections and source increasing-bitmask term order. Give (family_size, terms, union_size, alternating_sum, equal, certificate).","answer":{"kind":"expression","value":"(1, [[[0], 1, 3]], 3, 3, Yes, all_nonempty_index_subsets_included)","explanation":"Include all2^k−1 nonempty index subsets for a family of size k. Equal indexed sets are not deduplicated. The empty indexed family has empty union, no terms and alternating sum0. The exact result, retaining the requested order, is (1, [[[0], 1, 3]], 3, 3, Yes, all_nonempty_index_subsets_included)."},"skills":["All Nonempty Index Subsets Are Needed for the Exact Alternating Sum"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0060","batch_id":"batch-20261014-advance-20261108-r05-inclusion-exclusion-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0fab96977d8c54310fa3d5b85dab3e84b7c9d2737bb9d50ac8d1221e30790822","family_id":"qf-mod39-incexc-general-alternating-sum","seed":202611085060,"semantic_template":"qf-mod39-incexc-general-alternating-sum/{\"sets\":[[9,10,11],[9,11,12]]}"},"fields":{"title":"All Nonempty Index Subsets Are Needed for the Exact Alternating Sum","stem":"Use the complete indexed finite integer-set family sets=[[9, 10, 11], [9, 11, 12]] in supplied index order. Each set has distinct elements; equal sets at different indices remain separate indexed entries. Every nonempty index subset contributes its intersection cardinality, with positive odd-order and negative even-order sign. Keep zero-size intersections and source increasing-bitmask term order. Give (family_size, terms, union_size, alternating_sum, equal, certificate).","answer":{"kind":"expression","value":"(2, [[[0], 1, 3], [[1], 1, 3], [[0, 1], -1, 2]], 4, 4, Yes, all_nonempty_index_subsets_included)","explanation":"Include all2^k−1 nonempty index subsets for a family of size k. Equal indexed sets are not deduplicated. The empty indexed family has empty union, no terms and alternating sum0. The exact result, retaining the requested order, is (2, [[[0], 1, 3], [[1], 1, 3], [[0, 1], -1, 2]], 4, 4, Yes, all_nonempty_index_subsets_included)."},"skills":["All Nonempty Index Subsets Are Needed for the Exact Alternating Sum"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0061","batch_id":"batch-20261014-advance-20261108-r05-inclusion-exclusion-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7aa1e0b1ef781da88a0dd6a98087f6d747ba42e07ce1a3c5315f6cdfa12b18e8","family_id":"qf-mod39-incexc-general-alternating-sum","seed":202611085061,"semantic_template":"qf-mod39-incexc-general-alternating-sum/{\"sets\":[[10,11,12],[10,12,13],[10,12,14]]}"},"fields":{"title":"All Nonempty Index Subsets Are Needed for the Exact Alternating Sum","stem":"Use the complete indexed finite integer-set family sets=[[10, 11, 12], [10, 12, 13], [10, 12, 14]] in supplied index order. Each set has distinct elements; equal sets at different indices remain separate indexed entries. Every nonempty index subset contributes its intersection cardinality, with positive odd-order and negative even-order sign. Keep zero-size intersections and source increasing-bitmask term order. Give (family_size, terms, union_size, alternating_sum, equal, certificate).","answer":{"kind":"expression","value":"(3, [[[0], 1, 3], [[1], 1, 3], [[0, 1], -1, 2], [[2], 1, 3], [[0, 2], -1, 2], [[1, 2], -1, 2], [[0, 1, 2], 1, 2]], 5, 5, Yes, all_nonempty_index_subsets_included)","explanation":"Include all2^k−1 nonempty index subsets for a family of size k. Equal indexed sets are not deduplicated. The empty indexed family has empty union, no terms and alternating sum0. The exact result, retaining the requested order, is (3, [[[0], 1, 3], [[1], 1, 3], [[0, 1], -1, 2], [[2], 1, 3], [[0, 2], -1, 2], [[1, 2], -1, 2], [[0, 1, 2], 1, 2]], 5, 5, Yes, all_nonempty_index_subsets_included)."},"skills":["All Nonempty Index Subsets Are Needed for the Exact Alternating Sum"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0062","batch_id":"batch-20261014-advance-20261108-r05-inclusion-exclusion-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c17cc8fe17f48c13d1796f7b127ab4474608f4fbe5f7adfebeef21a92efa553c","family_id":"qf-mod39-incexc-general-alternating-sum","seed":202611085062,"semantic_template":"qf-mod39-incexc-general-alternating-sum/{\"sets\":[[11,12,13],[11,13,14],[11,13,15],[11,16]]}"},"fields":{"title":"All Nonempty Index Subsets Are Needed for the Exact Alternating Sum","stem":"Use the complete indexed finite integer-set family sets=[[11, 12, 13], [11, 13, 14], [11, 13, 15], [11, 16]] in supplied index order. Each set has distinct elements; equal sets at different indices remain separate indexed entries. Every nonempty index subset contributes its intersection cardinality, with positive odd-order and negative even-order sign. Keep zero-size intersections and source increasing-bitmask term order. Give (family_size, terms, union_size, alternating_sum, equal, certificate).","answer":{"kind":"expression","value":"(4, [[[0], 1, 3], [[1], 1, 3], [[0, 1], -1, 2], [[2], 1, 3], [[0, 2], -1, 2], [[1, 2], -1, 2], [[0, 1, 2], 1, 2], [[3], 1, 2], [[0, 3], -1, 1], [[1, 3], -1, 1], [[0, 1, 3], 1, 1], [[2, 3], -1, 1], [[0, 2, 3], 1, 1], [[1, 2, 3], 1, 1], [[0, 1, 2, 3], -1, 1]], 6, 6, Yes, all_nonempty_index_subsets_included)","explanation":"Include all2^k−1 nonempty index subsets for a family of size k. Equal indexed sets are not deduplicated. The empty indexed family has empty union, no terms and alternating sum0. The exact result, retaining the requested order, is (4, [[[0], 1, 3], [[1], 1, 3], [[0, 1], -1, 2], [[2], 1, 3], [[0, 2], -1, 2], [[1, 2], -1, 2], [[0, 1, 2], 1, 2], [[3], 1, 2], [[0, 3], -1, 1], [[1, 3], -1, 1], [[0, 1, 3], 1, 1], [[2, 3], -1, 1], [[0, 2, 3], 1, 1], [[1, 2, 3], 1, 1], [[0, 1, 2, 3], -1, 1]], 6, 6, Yes, all_nonempty_index_subsets_included)."},"skills":["All Nonempty Index Subsets Are Needed for the Exact Alternating Sum"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0063","batch_id":"batch-20261014-advance-20261108-r05-inclusion-exclusion-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"20b54e470308314c56f10b0c7f1899ee9da81f6883982e9cb34ca01daddd03cb","family_id":"qf-mod39-incexc-general-alternating-sum","seed":202611085063,"semantic_template":"qf-mod39-incexc-general-alternating-sum/{\"sets\":[[12,13,14]]}"},"fields":{"title":"All Nonempty Index Subsets Are Needed for the Exact Alternating Sum","stem":"Use the complete indexed finite integer-set family sets=[[12, 13, 14]] in supplied index order. Each set has distinct elements; equal sets at different indices remain separate indexed entries. Every nonempty index subset contributes its intersection cardinality, with positive odd-order and negative even-order sign. Keep zero-size intersections and source increasing-bitmask term order. Give (family_size, terms, union_size, alternating_sum, equal, certificate).","answer":{"kind":"expression","value":"(1, [[[0], 1, 3]], 3, 3, Yes, all_nonempty_index_subsets_included)","explanation":"Include all2^k−1 nonempty index subsets for a family of size k. Equal indexed sets are not deduplicated. The empty indexed family has empty union, no terms and alternating sum0. The exact result, retaining the requested order, is (1, [[[0], 1, 3]], 3, 3, Yes, all_nonempty_index_subsets_included)."},"skills":["All Nonempty Index Subsets Are Needed for the Exact Alternating Sum"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0064","batch_id":"batch-20261014-advance-20261108-r05-inclusion-exclusion-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"146a96dadd03f1e6b7d336509245f301171a54f8c77f6d2895e5ae1ba25bf578","family_id":"qf-mod39-incexc-general-alternating-sum","seed":202611085064,"semantic_template":"qf-mod39-incexc-general-alternating-sum/{\"sets\":[[13,14,15],[13,15,16]]}"},"fields":{"title":"All Nonempty Index Subsets Are Needed for the Exact Alternating Sum","stem":"Use the complete indexed finite integer-set family sets=[[13, 14, 15], [13, 15, 16]] in supplied index order. Each set has distinct elements; equal sets at different indices remain separate indexed entries. Every nonempty index subset contributes its intersection cardinality, with positive odd-order and negative even-order sign. Keep zero-size intersections and source increasing-bitmask term order. Give (family_size, terms, union_size, alternating_sum, equal, certificate).","answer":{"kind":"expression","value":"(2, [[[0], 1, 3], [[1], 1, 3], [[0, 1], -1, 2]], 4, 4, Yes, all_nonempty_index_subsets_included)","explanation":"Include all2^k−1 nonempty index subsets for a family of size k. Equal indexed sets are not deduplicated. The empty indexed family has empty union, no terms and alternating sum0. The exact result, retaining the requested order, is (2, [[[0], 1, 3], [[1], 1, 3], [[0, 1], -1, 2]], 4, 4, Yes, all_nonempty_index_subsets_included)."},"skills":["All Nonempty Index Subsets Are Needed for the Exact Alternating Sum"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0065","batch_id":"batch-20261014-advance-20261108-r05-inclusion-exclusion-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3c17d5346f13607cff746db117e15396b326aa76e30a1ff3a487fc5b309f1da8","family_id":"qf-mod39-incexc-general-alternating-sum","seed":202611085065,"semantic_template":"qf-mod39-incexc-general-alternating-sum/{\"sets\":[[14,15,16],[14,16,17],[14,16,18]]}"},"fields":{"title":"All Nonempty Index Subsets Are Needed for the Exact Alternating Sum","stem":"Use the complete indexed finite integer-set family sets=[[14, 15, 16], [14, 16, 17], [14, 16, 18]] in supplied index order. Each set has distinct elements; equal sets at different indices remain separate indexed entries. Every nonempty index subset contributes its intersection cardinality, with positive odd-order and negative even-order sign. Keep zero-size intersections and source increasing-bitmask term order. Give (family_size, terms, union_size, alternating_sum, equal, certificate).","answer":{"kind":"expression","value":"(3, [[[0], 1, 3], [[1], 1, 3], [[0, 1], -1, 2], [[2], 1, 3], [[0, 2], -1, 2], [[1, 2], -1, 2], [[0, 1, 2], 1, 2]], 5, 5, Yes, all_nonempty_index_subsets_included)","explanation":"Include all2^k−1 nonempty index subsets for a family of size k. Equal indexed sets are not deduplicated. The empty indexed family has empty union, no terms and alternating sum0. The exact result, retaining the requested order, is (3, [[[0], 1, 3], [[1], 1, 3], [[0, 1], -1, 2], [[2], 1, 3], [[0, 2], -1, 2], [[1, 2], -1, 2], [[0, 1, 2], 1, 2]], 5, 5, Yes, all_nonempty_index_subsets_included)."},"skills":["All Nonempty Index Subsets Are Needed for the Exact Alternating Sum"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0066","batch_id":"batch-20261014-advance-20261108-r05-inclusion-exclusion-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"40fe0b0e87face036da12609b0a5c847f9e447d77e29e72e299b9d41baec597d","family_id":"qf-mod39-incexc-general-alternating-sum","seed":202611085066,"semantic_template":"qf-mod39-incexc-general-alternating-sum/{\"sets\":[[15,16,17],[15,17,18],[15,17,19],[15,20]]}"},"fields":{"title":"All Nonempty Index Subsets Are Needed for the Exact Alternating Sum","stem":"Use the complete indexed finite integer-set family sets=[[15, 16, 17], [15, 17, 18], [15, 17, 19], [15, 20]] in supplied index order. Each set has distinct elements; equal sets at different indices remain separate indexed entries. Every nonempty index subset contributes its intersection cardinality, with positive odd-order and negative even-order sign. Keep zero-size intersections and source increasing-bitmask term order. Give (family_size, terms, union_size, alternating_sum, equal, certificate).","answer":{"kind":"expression","value":"(4, [[[0], 1, 3], [[1], 1, 3], [[0, 1], -1, 2], [[2], 1, 3], [[0, 2], -1, 2], [[1, 2], -1, 2], [[0, 1, 2], 1, 2], [[3], 1, 2], [[0, 3], -1, 1], [[1, 3], -1, 1], [[0, 1, 3], 1, 1], [[2, 3], -1, 1], [[0, 2, 3], 1, 1], [[1, 2, 3], 1, 1], [[0, 1, 2, 3], -1, 1]], 6, 6, Yes, all_nonempty_index_subsets_included)","explanation":"Include all2^k−1 nonempty index subsets for a family of size k. Equal indexed sets are not deduplicated. The empty indexed family has empty union, no terms and alternating sum0. The exact result, retaining the requested order, is (4, [[[0], 1, 3], [[1], 1, 3], [[0, 1], -1, 2], [[2], 1, 3], [[0, 2], -1, 2], [[1, 2], -1, 2], [[0, 1, 2], 1, 2], [[3], 1, 2], [[0, 3], -1, 1], [[1, 3], -1, 1], [[0, 1, 3], 1, 1], [[2, 3], -1, 1], [[0, 2, 3], 1, 1], [[1, 2, 3], 1, 1], [[0, 1, 2, 3], -1, 1]], 6, 6, Yes, all_nonempty_index_subsets_included)."},"skills":["All Nonempty Index Subsets Are Needed for the Exact Alternating Sum"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0067","batch_id":"batch-20261014-advance-20261108-r05-inclusion-exclusion-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"301c8fe76f5f1336fdd2f6033b88b97f79d37241258de98b424e3679c1fa7acc","family_id":"qf-mod39-incexc-general-alternating-sum","seed":202611085067,"semantic_template":"qf-mod39-incexc-general-alternating-sum/{\"sets\":[[16,17,18]]}"},"fields":{"title":"All Nonempty Index Subsets Are Needed for the Exact Alternating Sum","stem":"Use the complete indexed finite integer-set family sets=[[16, 17, 18]] in supplied index order. Each set has distinct elements; equal sets at different indices remain separate indexed entries. Every nonempty index subset contributes its intersection cardinality, with positive odd-order and negative even-order sign. Keep zero-size intersections and source increasing-bitmask term order. Give (family_size, terms, union_size, alternating_sum, equal, certificate).","answer":{"kind":"expression","value":"(1, [[[0], 1, 3]], 3, 3, Yes, all_nonempty_index_subsets_included)","explanation":"Include all2^k−1 nonempty index subsets for a family of size k. Equal indexed sets are not deduplicated. The empty indexed family has empty union, no terms and alternating sum0. The exact result, retaining the requested order, is (1, [[[0], 1, 3]], 3, 3, Yes, all_nonempty_index_subsets_included)."},"skills":["All Nonempty Index Subsets Are Needed for the Exact Alternating Sum"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0068","batch_id":"batch-20261014-advance-20261108-r05-inclusion-exclusion-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"615fd35c849c9d7f9bf303c921a6b7151c398da127a16bdaae1ba7a024fca88d","family_id":"qf-mod39-incexc-general-alternating-sum","seed":202611085068,"semantic_template":"qf-mod39-incexc-general-alternating-sum/{\"sets\":[[17,18,19],[17,19,20]]}"},"fields":{"title":"All Nonempty Index Subsets Are Needed for the Exact Alternating Sum","stem":"Use the complete indexed finite integer-set family sets=[[17, 18, 19], [17, 19, 20]] in supplied index order. Each set has distinct elements; equal sets at different indices remain separate indexed entries. Every nonempty index subset contributes its intersection cardinality, with positive odd-order and negative even-order sign. Keep zero-size intersections and source increasing-bitmask term order. Give (family_size, terms, union_size, alternating_sum, equal, certificate).","answer":{"kind":"expression","value":"(2, [[[0], 1, 3], [[1], 1, 3], [[0, 1], -1, 2]], 4, 4, Yes, all_nonempty_index_subsets_included)","explanation":"Include all2^k−1 nonempty index subsets for a family of size k. Equal indexed sets are not deduplicated. The empty indexed family has empty union, no terms and alternating sum0. The exact result, retaining the requested order, is (2, [[[0], 1, 3], [[1], 1, 3], [[0, 1], -1, 2]], 4, 4, Yes, all_nonempty_index_subsets_included)."},"skills":["All Nonempty Index Subsets Are Needed for the Exact Alternating Sum"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0069","batch_id":"batch-20261014-advance-20261108-r05-inclusion-exclusion-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"63f2018ac97c4f5a2e7f5238b29bd1c285237e802f6726052b4c65576a040a53","family_id":"qf-mod39-incexc-general-alternating-sum","seed":202611085069,"semantic_template":"qf-mod39-incexc-general-alternating-sum/{\"sets\":[[18,19,20],[18,20,21],[18,20,22]]}"},"fields":{"title":"All Nonempty Index Subsets Are Needed for the Exact Alternating Sum","stem":"Use the complete indexed finite integer-set family sets=[[18, 19, 20], [18, 20, 21], [18, 20, 22]] in supplied index order. Each set has distinct elements; equal sets at different indices remain separate indexed entries. Every nonempty index subset contributes its intersection cardinality, with positive odd-order and negative even-order sign. Keep zero-size intersections and source increasing-bitmask term order. Give (family_size, terms, union_size, alternating_sum, equal, certificate).","answer":{"kind":"expression","value":"(3, [[[0], 1, 3], [[1], 1, 3], [[0, 1], -1, 2], [[2], 1, 3], [[0, 2], -1, 2], [[1, 2], -1, 2], [[0, 1, 2], 1, 2]], 5, 5, Yes, all_nonempty_index_subsets_included)","explanation":"Include all2^k−1 nonempty index subsets for a family of size k. Equal indexed sets are not deduplicated. The empty indexed family has empty union, no terms and alternating sum0. The exact result, retaining the requested order, is (3, [[[0], 1, 3], [[1], 1, 3], [[0, 1], -1, 2], [[2], 1, 3], [[0, 2], -1, 2], [[1, 2], -1, 2], [[0, 1, 2], 1, 2]], 5, 5, Yes, all_nonempty_index_subsets_included)."},"skills":["All Nonempty Index Subsets Are Needed for the Exact Alternating Sum"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0070","batch_id":"batch-20261014-advance-20261108-r05-inclusion-exclusion-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8058929c086442457ecc0218afa7213139808d7943f6b1e712fe8d4914881912","family_id":"qf-mod39-incexc-general-alternating-sum","seed":202611085070,"semantic_template":"qf-mod39-incexc-general-alternating-sum/{\"sets\":[[19,20,21],[19,21,22],[19,21,23],[19,24]]}"},"fields":{"title":"All Nonempty Index Subsets Are Needed for the Exact Alternating Sum","stem":"Use the complete indexed finite integer-set family sets=[[19, 20, 21], [19, 21, 22], [19, 21, 23], [19, 24]] in supplied index order. Each set has distinct elements; equal sets at different indices remain separate indexed entries. Every nonempty index subset contributes its intersection cardinality, with positive odd-order and negative even-order sign. Keep zero-size intersections and source increasing-bitmask term order. Give (family_size, terms, union_size, alternating_sum, equal, certificate).","answer":{"kind":"expression","value":"(4, [[[0], 1, 3], [[1], 1, 3], [[0, 1], -1, 2], [[2], 1, 3], [[0, 2], -1, 2], [[1, 2], -1, 2], [[0, 1, 2], 1, 2], [[3], 1, 2], [[0, 3], -1, 1], [[1, 3], -1, 1], [[0, 1, 3], 1, 1], [[2, 3], -1, 1], [[0, 2, 3], 1, 1], [[1, 2, 3], 1, 1], [[0, 1, 2, 3], -1, 1]], 6, 6, Yes, all_nonempty_index_subsets_included)","explanation":"Include all2^k−1 nonempty index subsets for a family of size k. Equal indexed sets are not deduplicated. The empty indexed family has empty union, no terms and alternating sum0. The exact result, retaining the requested order, is (4, [[[0], 1, 3], [[1], 1, 3], [[0, 1], -1, 2], [[2], 1, 3], [[0, 2], -1, 2], [[1, 2], -1, 2], [[0, 1, 2], 1, 2], [[3], 1, 2], [[0, 3], -1, 1], [[1, 3], -1, 1], [[0, 1, 3], 1, 1], [[2, 3], -1, 1], [[0, 2, 3], 1, 1], [[1, 2, 3], 1, 1], [[0, 1, 2, 3], -1, 1]], 6, 6, Yes, all_nonempty_index_subsets_included)."},"skills":["All Nonempty Index Subsets Are Needed for the Exact Alternating Sum"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0071","batch_id":"batch-20261014-advance-20261108-r05-inclusion-exclusion-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"50690e89c3a61720133872bf184562e139e14623cd05e4c7453d8ce58751f441","family_id":"qf-mod39-incexc-general-alternating-sum","seed":202611085071,"semantic_template":"qf-mod39-incexc-general-alternating-sum/{\"sets\":[[20,21,22]]}"},"fields":{"title":"All Nonempty Index Subsets Are Needed for the Exact Alternating Sum","stem":"Use the complete indexed finite integer-set family sets=[[20, 21, 22]] in supplied index order. Each set has distinct elements; equal sets at different indices remain separate indexed entries. Every nonempty index subset contributes its intersection cardinality, with positive odd-order and negative even-order sign. Keep zero-size intersections and source increasing-bitmask term order. Give (family_size, terms, union_size, alternating_sum, equal, certificate).","answer":{"kind":"expression","value":"(1, [[[0], 1, 3]], 3, 3, Yes, all_nonempty_index_subsets_included)","explanation":"Include all2^k−1 nonempty index subsets for a family of size k. Equal indexed sets are not deduplicated. The empty indexed family has empty union, no terms and alternating sum0. The exact result, retaining the requested order, is (1, [[[0], 1, 3]], 3, 3, Yes, all_nonempty_index_subsets_included)."},"skills":["All Nonempty Index Subsets Are Needed for the Exact Alternating Sum"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0072","batch_id":"batch-20261014-advance-20261108-r05-inclusion-exclusion-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e036c5c088bc882864bf3c744ff824a4b67194f2045c201d0231d4db358a0441","family_id":"qf-mod39-incexc-general-alternating-sum","seed":202611085072,"semantic_template":"qf-mod39-incexc-general-alternating-sum/{\"sets\":[[21,22,23],[21,23,24]]}"},"fields":{"title":"All Nonempty Index Subsets Are Needed for the Exact Alternating Sum","stem":"Use the complete indexed finite integer-set family sets=[[21, 22, 23], [21, 23, 24]] in supplied index order. Each set has distinct elements; equal sets at different indices remain separate indexed entries. Every nonempty index subset contributes its intersection cardinality, with positive odd-order and negative even-order sign. Keep zero-size intersections and source increasing-bitmask term order. Give (family_size, terms, union_size, alternating_sum, equal, certificate).","answer":{"kind":"expression","value":"(2, [[[0], 1, 3], [[1], 1, 3], [[0, 1], -1, 2]], 4, 4, Yes, all_nonempty_index_subsets_included)","explanation":"Include all2^k−1 nonempty index subsets for a family of size k. Equal indexed sets are not deduplicated. The empty indexed family has empty union, no terms and alternating sum0. The exact result, retaining the requested order, is (2, [[[0], 1, 3], [[1], 1, 3], [[0, 1], -1, 2]], 4, 4, Yes, all_nonempty_index_subsets_included)."},"skills":["All Nonempty Index Subsets Are Needed for the Exact Alternating Sum"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0073","batch_id":"batch-20261014-advance-20261108-r05-inclusion-exclusion-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7996389352d662f4ac23fba5123c17152a8cfe4d5067a57bfa9d1243a94e2dd5","family_id":"qf-mod39-incexc-general-alternating-sum","seed":202611085073,"semantic_template":"qf-mod39-incexc-general-alternating-sum/{\"sets\":[[22,23,24],[22,24,25],[22,24,26]]}"},"fields":{"title":"All Nonempty Index Subsets Are Needed for the Exact Alternating Sum","stem":"Use the complete indexed finite integer-set family sets=[[22, 23, 24], [22, 24, 25], [22, 24, 26]] in supplied index order. Each set has distinct elements; equal sets at different indices remain separate indexed entries. Every nonempty index subset contributes its intersection cardinality, with positive odd-order and negative even-order sign. Keep zero-size intersections and source increasing-bitmask term order. Give (family_size, terms, union_size, alternating_sum, equal, certificate).","answer":{"kind":"expression","value":"(3, [[[0], 1, 3], [[1], 1, 3], [[0, 1], -1, 2], [[2], 1, 3], [[0, 2], -1, 2], [[1, 2], -1, 2], [[0, 1, 2], 1, 2]], 5, 5, Yes, all_nonempty_index_subsets_included)","explanation":"Include all2^k−1 nonempty index subsets for a family of size k. Equal indexed sets are not deduplicated. The empty indexed family has empty union, no terms and alternating sum0. The exact result, retaining the requested order, is (3, [[[0], 1, 3], [[1], 1, 3], [[0, 1], -1, 2], [[2], 1, 3], [[0, 2], -1, 2], [[1, 2], -1, 2], [[0, 1, 2], 1, 2]], 5, 5, Yes, all_nonempty_index_subsets_included)."},"skills":["All Nonempty Index Subsets Are Needed for the Exact Alternating Sum"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0074","batch_id":"batch-20261014-advance-20261108-r05-inclusion-exclusion-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"095ee51ef12ace26ef177965c4bb186769856469bc0b75242ba2175f8a184f7b","family_id":"qf-mod39-incexc-general-alternating-sum","seed":202611085074,"semantic_template":"qf-mod39-incexc-general-alternating-sum/{\"sets\":[[23,24,25],[23,25,26],[23,25,27],[23,28]]}"},"fields":{"title":"All Nonempty Index Subsets Are Needed for the Exact Alternating Sum","stem":"Use the complete indexed finite integer-set family sets=[[23, 24, 25], [23, 25, 26], [23, 25, 27], [23, 28]] in supplied index order. Each set has distinct elements; equal sets at different indices remain separate indexed entries. Every nonempty index subset contributes its intersection cardinality, with positive odd-order and negative even-order sign. Keep zero-size intersections and source increasing-bitmask term order. Give (family_size, terms, union_size, alternating_sum, equal, certificate).","answer":{"kind":"expression","value":"(4, [[[0], 1, 3], [[1], 1, 3], [[0, 1], -1, 2], [[2], 1, 3], [[0, 2], -1, 2], [[1, 2], -1, 2], [[0, 1, 2], 1, 2], [[3], 1, 2], [[0, 3], -1, 1], [[1, 3], -1, 1], [[0, 1, 3], 1, 1], [[2, 3], -1, 1], [[0, 2, 3], 1, 1], [[1, 2, 3], 1, 1], [[0, 1, 2, 3], -1, 1]], 6, 6, Yes, all_nonempty_index_subsets_included)","explanation":"Include all2^k−1 nonempty index subsets for a family of size k. Equal indexed sets are not deduplicated. The empty indexed family has empty union, no terms and alternating sum0. The exact result, retaining the requested order, is (4, [[[0], 1, 3], [[1], 1, 3], [[0, 1], -1, 2], [[2], 1, 3], [[0, 2], -1, 2], [[1, 2], -1, 2], [[0, 1, 2], 1, 2], [[3], 1, 2], [[0, 3], -1, 1], [[1, 3], -1, 1], [[0, 1, 3], 1, 1], [[2, 3], -1, 1], [[0, 2, 3], 1, 1], [[1, 2, 3], 1, 1], [[0, 1, 2, 3], -1, 1]], 6, 6, Yes, all_nonempty_index_subsets_included)."},"skills":["All Nonempty Index Subsets Are Needed for the Exact Alternating Sum"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0075","batch_id":"batch-20261014-advance-20261108-r05-inclusion-exclusion-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7e7462ae49c4b71c7a433e3511e6b9c6286d403c5b2a7524493c4ae50652408f","family_id":"qf-mod39-incexc-general-alternating-sum","seed":202611085075,"semantic_template":"qf-mod39-incexc-general-alternating-sum/{\"sets\":[[24,25,26]]}"},"fields":{"title":"All Nonempty Index Subsets Are Needed for the Exact Alternating Sum","stem":"Use the complete indexed finite integer-set family sets=[[24, 25, 26]] in supplied index order. Each set has distinct elements; equal sets at different indices remain separate indexed entries. Every nonempty index subset contributes its intersection cardinality, with positive odd-order and negative even-order sign. Keep zero-size intersections and source increasing-bitmask term order. Give (family_size, terms, union_size, alternating_sum, equal, certificate).","answer":{"kind":"expression","value":"(1, [[[0], 1, 3]], 3, 3, Yes, all_nonempty_index_subsets_included)","explanation":"Include all2^k−1 nonempty index subsets for a family of size k. Equal indexed sets are not deduplicated. The empty indexed family has empty union, no terms and alternating sum0. The exact result, retaining the requested order, is (1, [[[0], 1, 3]], 3, 3, Yes, all_nonempty_index_subsets_included)."},"skills":["All Nonempty Index Subsets Are Needed for the Exact Alternating Sum"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0076","batch_id":"batch-20261014-advance-20261108-r05-inclusion-exclusion-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"25dc9b0aa22333dda4e4b85a9b6a6b46d5846b882d8b6ce1fd3129b8256e0e1d","family_id":"qf-mod39-incexc-pairwise-only-boundary","seed":202611085076,"semantic_template":"qf-mod39-incexc-pairwise-only-boundary/{\"sets\":[[0,1,2],[0,2,3],[0,2,4]]}"},"fields":{"title":"A Pairwise Truncation May Coincide Numerically without Becoming a General Exact Formula","stem":"Use the complete indexed finite integer-set family sets=[[0, 1, 2], [0, 2, 3], [0, 2, 4]] in supplied index order. Each set has distinct elements; equal sets at different indices remain separate indexed entries. Every nonempty index subset contributes its intersection cardinality, with positive odd-order and negative even-order sign. Keep zero-size intersections and source increasing-bitmask term order. Give (pairwise_truncation, union_size, numerically_equal, general_exactness_claim, boundary).","answer":{"kind":"expression","value":"(3, 5, No, No, pairwise_only_truncation_not_exact_when_higher_intersections_contribute)","explanation":"Require at least three indexed sets. Compute singles minus pairs, the actual union size and their numerical equality separately. The final No withholds a general pairwise-only exactness claim; it does not negate that equality flag. This Bonferroni lower bound may even be negative, so it is not itself an asserted cardinality. The exact result, retaining the requested order, is (3, 5, No, No, pairwise_only_truncation_not_exact_when_higher_intersections_contribute)."},"skills":["A Pairwise Truncation May Coincide Numerically without Becoming a General Exact Formula"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0077","batch_id":"batch-20261014-advance-20261108-r05-inclusion-exclusion-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a2a5d41803af9f50cc6b66a14a84620e0e5296f18b85842a11a5b818433ac93b","family_id":"qf-mod39-incexc-pairwise-only-boundary","seed":202611085077,"semantic_template":"qf-mod39-incexc-pairwise-only-boundary/{\"sets\":[[1,2,3],[1,3,4],[1,3,5]]}"},"fields":{"title":"A Pairwise Truncation May Coincide Numerically without Becoming a General Exact Formula","stem":"Use the complete indexed finite integer-set family sets=[[1, 2, 3], [1, 3, 4], [1, 3, 5]] in supplied index order. Each set has distinct elements; equal sets at different indices remain separate indexed entries. Every nonempty index subset contributes its intersection cardinality, with positive odd-order and negative even-order sign. Keep zero-size intersections and source increasing-bitmask term order. Give (pairwise_truncation, union_size, numerically_equal, general_exactness_claim, boundary).","answer":{"kind":"expression","value":"(3, 5, No, No, pairwise_only_truncation_not_exact_when_higher_intersections_contribute)","explanation":"Require at least three indexed sets. Compute singles minus pairs, the actual union size and their numerical equality separately. The final No withholds a general pairwise-only exactness claim; it does not negate that equality flag. This Bonferroni lower bound may even be negative, so it is not itself an asserted cardinality. The exact result, retaining the requested order, is (3, 5, No, No, pairwise_only_truncation_not_exact_when_higher_intersections_contribute)."},"skills":["A Pairwise Truncation May Coincide Numerically without Becoming a General Exact Formula"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0078","batch_id":"batch-20261014-advance-20261108-r05-inclusion-exclusion-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c37e391d047f37da70facf7bfa559203dc7938efdc221509c1d42bba55356739","family_id":"qf-mod39-incexc-pairwise-only-boundary","seed":202611085078,"semantic_template":"qf-mod39-incexc-pairwise-only-boundary/{\"sets\":[[2,3,4],[2,4,5],[2,4,6]]}"},"fields":{"title":"A Pairwise Truncation May Coincide Numerically without Becoming a General Exact Formula","stem":"Use the complete indexed finite integer-set family sets=[[2, 3, 4], [2, 4, 5], [2, 4, 6]] in supplied index order. Each set has distinct elements; equal sets at different indices remain separate indexed entries. Every nonempty index subset contributes its intersection cardinality, with positive odd-order and negative even-order sign. Keep zero-size intersections and source increasing-bitmask term order. Give (pairwise_truncation, union_size, numerically_equal, general_exactness_claim, boundary).","answer":{"kind":"expression","value":"(3, 5, No, No, pairwise_only_truncation_not_exact_when_higher_intersections_contribute)","explanation":"Require at least three indexed sets. Compute singles minus pairs, the actual union size and their numerical equality separately. The final No withholds a general pairwise-only exactness claim; it does not negate that equality flag. This Bonferroni lower bound may even be negative, so it is not itself an asserted cardinality. The exact result, retaining the requested order, is (3, 5, No, No, pairwise_only_truncation_not_exact_when_higher_intersections_contribute)."},"skills":["A Pairwise Truncation May Coincide Numerically without Becoming a General Exact Formula"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0079","batch_id":"batch-20261014-advance-20261108-r05-inclusion-exclusion-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"168f83739e54baffab841d95b7752e50f1b87da1bfcfcac97747c92d232cc6fe","family_id":"qf-mod39-incexc-pairwise-only-boundary","seed":202611085079,"semantic_template":"qf-mod39-incexc-pairwise-only-boundary/{\"sets\":[[3,4,5],[3,5,6],[3,5,7]]}"},"fields":{"title":"A Pairwise Truncation May Coincide Numerically without Becoming a General Exact Formula","stem":"Use the complete indexed finite integer-set family sets=[[3, 4, 5], [3, 5, 6], [3, 5, 7]] in supplied index order. Each set has distinct elements; equal sets at different indices remain separate indexed entries. Every nonempty index subset contributes its intersection cardinality, with positive odd-order and negative even-order sign. Keep zero-size intersections and source increasing-bitmask term order. Give (pairwise_truncation, union_size, numerically_equal, general_exactness_claim, boundary).","answer":{"kind":"expression","value":"(3, 5, No, No, pairwise_only_truncation_not_exact_when_higher_intersections_contribute)","explanation":"Require at least three indexed sets. Compute singles minus pairs, the actual union size and their numerical equality separately. The final No withholds a general pairwise-only exactness claim; it does not negate that equality flag. This Bonferroni lower bound may even be negative, so it is not itself an asserted cardinality. The exact result, retaining the requested order, is (3, 5, No, No, pairwise_only_truncation_not_exact_when_higher_intersections_contribute)."},"skills":["A Pairwise Truncation May Coincide Numerically without Becoming a General Exact Formula"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0080","batch_id":"batch-20261014-advance-20261108-r05-inclusion-exclusion-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"104796d257849b3da5f68ab81f1052567174ad754cb8e5489290ec7a31f7314c","family_id":"qf-mod39-incexc-pairwise-only-boundary","seed":202611085080,"semantic_template":"qf-mod39-incexc-pairwise-only-boundary/{\"sets\":[[4,5,6],[4,6,7],[4,6,8]]}"},"fields":{"title":"A Pairwise Truncation May Coincide Numerically without Becoming a General Exact Formula","stem":"Use the complete indexed finite integer-set family sets=[[4, 5, 6], [4, 6, 7], [4, 6, 8]] in supplied index order. Each set has distinct elements; equal sets at different indices remain separate indexed entries. Every nonempty index subset contributes its intersection cardinality, with positive odd-order and negative even-order sign. Keep zero-size intersections and source increasing-bitmask term order. Give (pairwise_truncation, union_size, numerically_equal, general_exactness_claim, boundary).","answer":{"kind":"expression","value":"(3, 5, No, No, pairwise_only_truncation_not_exact_when_higher_intersections_contribute)","explanation":"Require at least three indexed sets. Compute singles minus pairs, the actual union size and their numerical equality separately. The final No withholds a general pairwise-only exactness claim; it does not negate that equality flag. This Bonferroni lower bound may even be negative, so it is not itself an asserted cardinality. The exact result, retaining the requested order, is (3, 5, No, No, pairwise_only_truncation_not_exact_when_higher_intersections_contribute)."},"skills":["A Pairwise Truncation May Coincide Numerically without Becoming a General Exact Formula"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0081","batch_id":"batch-20261014-advance-20261108-r05-inclusion-exclusion-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ddfe1af36db8ec82f8e43046e21c2f28bcede09c5da4d182318cd8887be2fc40","family_id":"qf-mod39-incexc-pairwise-only-boundary","seed":202611085081,"semantic_template":"qf-mod39-incexc-pairwise-only-boundary/{\"sets\":[[5,6,7],[5,7,8],[5,7,9]]}"},"fields":{"title":"A Pairwise Truncation May Coincide Numerically without Becoming a General Exact Formula","stem":"Use the complete indexed finite integer-set family sets=[[5, 6, 7], [5, 7, 8], [5, 7, 9]] in supplied index order. Each set has distinct elements; equal sets at different indices remain separate indexed entries. Every nonempty index subset contributes its intersection cardinality, with positive odd-order and negative even-order sign. Keep zero-size intersections and source increasing-bitmask term order. Give (pairwise_truncation, union_size, numerically_equal, general_exactness_claim, boundary).","answer":{"kind":"expression","value":"(3, 5, No, No, pairwise_only_truncation_not_exact_when_higher_intersections_contribute)","explanation":"Require at least three indexed sets. Compute singles minus pairs, the actual union size and their numerical equality separately. The final No withholds a general pairwise-only exactness claim; it does not negate that equality flag. This Bonferroni lower bound may even be negative, so it is not itself an asserted cardinality. The exact result, retaining the requested order, is (3, 5, No, No, pairwise_only_truncation_not_exact_when_higher_intersections_contribute)."},"skills":["A Pairwise Truncation May Coincide Numerically without Becoming a General Exact Formula"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0082","batch_id":"batch-20261014-advance-20261108-r05-inclusion-exclusion-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8c0640a690796aafa655f81c982fb3f47579e7e41d49231731791115bfdf971b","family_id":"qf-mod39-incexc-pairwise-only-boundary","seed":202611085082,"semantic_template":"qf-mod39-incexc-pairwise-only-boundary/{\"sets\":[[6,7,8],[6,8,10],[6,8,9]]}"},"fields":{"title":"A Pairwise Truncation May Coincide Numerically without Becoming a General Exact Formula","stem":"Use the complete indexed finite integer-set family sets=[[6, 7, 8], [6, 8, 9], [6, 8, 10]] in supplied index order. Each set has distinct elements; equal sets at different indices remain separate indexed entries. Every nonempty index subset contributes its intersection cardinality, with positive odd-order and negative even-order sign. Keep zero-size intersections and source increasing-bitmask term order. Give (pairwise_truncation, union_size, numerically_equal, general_exactness_claim, boundary).","answer":{"kind":"expression","value":"(3, 5, No, No, pairwise_only_truncation_not_exact_when_higher_intersections_contribute)","explanation":"Require at least three indexed sets. Compute singles minus pairs, the actual union size and their numerical equality separately. The final No withholds a general pairwise-only exactness claim; it does not negate that equality flag. This Bonferroni lower bound may even be negative, so it is not itself an asserted cardinality. The exact result, retaining the requested order, is (3, 5, No, No, pairwise_only_truncation_not_exact_when_higher_intersections_contribute)."},"skills":["A Pairwise Truncation May Coincide Numerically without Becoming a General Exact Formula"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0083","batch_id":"batch-20261014-advance-20261108-r05-inclusion-exclusion-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"eab2e1dfd68a421047cbf6d7dd87b64ca0fa869fbe63e98a3e6e6e60678d7e06","family_id":"qf-mod39-incexc-pairwise-only-boundary","seed":202611085083,"semantic_template":"qf-mod39-incexc-pairwise-only-boundary/{\"sets\":[[7,8,9],[7,9,10],[7,9,11]]}"},"fields":{"title":"A Pairwise Truncation May Coincide Numerically without Becoming a General Exact Formula","stem":"Use the complete indexed finite integer-set family sets=[[7, 8, 9], [7, 9, 10], [7, 9, 11]] in supplied index order. Each set has distinct elements; equal sets at different indices remain separate indexed entries. Every nonempty index subset contributes its intersection cardinality, with positive odd-order and negative even-order sign. Keep zero-size intersections and source increasing-bitmask term order. Give (pairwise_truncation, union_size, numerically_equal, general_exactness_claim, boundary).","answer":{"kind":"expression","value":"(3, 5, No, No, pairwise_only_truncation_not_exact_when_higher_intersections_contribute)","explanation":"Require at least three indexed sets. Compute singles minus pairs, the actual union size and their numerical equality separately. The final No withholds a general pairwise-only exactness claim; it does not negate that equality flag. This Bonferroni lower bound may even be negative, so it is not itself an asserted cardinality. The exact result, retaining the requested order, is (3, 5, No, No, pairwise_only_truncation_not_exact_when_higher_intersections_contribute)."},"skills":["A Pairwise Truncation May Coincide Numerically without Becoming a General Exact Formula"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0084","batch_id":"batch-20261014-advance-20261108-r05-inclusion-exclusion-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d497fc0b7d53630edb6d027ccb5ce3c6676a68411104bac2a5804042a43e3f92","family_id":"qf-mod39-incexc-pairwise-only-boundary","seed":202611085084,"semantic_template":"qf-mod39-incexc-pairwise-only-boundary/{\"sets\":[[8,10,11],[8,10,12],[8,9,10]]}"},"fields":{"title":"A Pairwise Truncation May Coincide Numerically without Becoming a General Exact Formula","stem":"Use the complete indexed finite integer-set family sets=[[8, 9, 10], [8, 10, 11], [8, 10, 12]] in supplied index order. Each set has distinct elements; equal sets at different indices remain separate indexed entries. Every nonempty index subset contributes its intersection cardinality, with positive odd-order and negative even-order sign. Keep zero-size intersections and source increasing-bitmask term order. Give (pairwise_truncation, union_size, numerically_equal, general_exactness_claim, boundary).","answer":{"kind":"expression","value":"(3, 5, No, No, pairwise_only_truncation_not_exact_when_higher_intersections_contribute)","explanation":"Require at least three indexed sets. Compute singles minus pairs, the actual union size and their numerical equality separately. The final No withholds a general pairwise-only exactness claim; it does not negate that equality flag. This Bonferroni lower bound may even be negative, so it is not itself an asserted cardinality. The exact result, retaining the requested order, is (3, 5, No, No, pairwise_only_truncation_not_exact_when_higher_intersections_contribute)."},"skills":["A Pairwise Truncation May Coincide Numerically without Becoming a General Exact Formula"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0085","batch_id":"batch-20261014-advance-20261108-r05-inclusion-exclusion-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e93fb0d2eb047b4278f3f0f65d189e202a346acec8b85f896af4685879ffd2b5","family_id":"qf-mod39-incexc-pairwise-only-boundary","seed":202611085085,"semantic_template":"qf-mod39-incexc-pairwise-only-boundary/{\"sets\":[[9,10,11],[9,11,12],[9,11,13]]}"},"fields":{"title":"A Pairwise Truncation May Coincide Numerically without Becoming a General Exact Formula","stem":"Use the complete indexed finite integer-set family sets=[[9, 10, 11], [9, 11, 12], [9, 11, 13]] in supplied index order. Each set has distinct elements; equal sets at different indices remain separate indexed entries. Every nonempty index subset contributes its intersection cardinality, with positive odd-order and negative even-order sign. Keep zero-size intersections and source increasing-bitmask term order. Give (pairwise_truncation, union_size, numerically_equal, general_exactness_claim, boundary).","answer":{"kind":"expression","value":"(3, 5, No, No, pairwise_only_truncation_not_exact_when_higher_intersections_contribute)","explanation":"Require at least three indexed sets. Compute singles minus pairs, the actual union size and their numerical equality separately. The final No withholds a general pairwise-only exactness claim; it does not negate that equality flag. This Bonferroni lower bound may even be negative, so it is not itself an asserted cardinality. The exact result, retaining the requested order, is (3, 5, No, No, pairwise_only_truncation_not_exact_when_higher_intersections_contribute)."},"skills":["A Pairwise Truncation May Coincide Numerically without Becoming a General Exact Formula"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0086","batch_id":"batch-20261014-advance-20261108-r05-inclusion-exclusion-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"99c82dcca1d3bc33a7e62a726b4cdf6e101bf827505dbbcd7831bfc08919cd24","family_id":"qf-mod39-incexc-pairwise-only-boundary","seed":202611085086,"semantic_template":"qf-mod39-incexc-pairwise-only-boundary/{\"sets\":[[10,11,12],[10,12,13],[10,12,14]]}"},"fields":{"title":"A Pairwise Truncation May Coincide Numerically without Becoming a General Exact Formula","stem":"Use the complete indexed finite integer-set family sets=[[10, 11, 12], [10, 12, 13], [10, 12, 14]] in supplied index order. Each set has distinct elements; equal sets at different indices remain separate indexed entries. Every nonempty index subset contributes its intersection cardinality, with positive odd-order and negative even-order sign. Keep zero-size intersections and source increasing-bitmask term order. Give (pairwise_truncation, union_size, numerically_equal, general_exactness_claim, boundary).","answer":{"kind":"expression","value":"(3, 5, No, No, pairwise_only_truncation_not_exact_when_higher_intersections_contribute)","explanation":"Require at least three indexed sets. Compute singles minus pairs, the actual union size and their numerical equality separately. The final No withholds a general pairwise-only exactness claim; it does not negate that equality flag. This Bonferroni lower bound may even be negative, so it is not itself an asserted cardinality. The exact result, retaining the requested order, is (3, 5, No, No, pairwise_only_truncation_not_exact_when_higher_intersections_contribute)."},"skills":["A Pairwise Truncation May Coincide Numerically without Becoming a General Exact Formula"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0087","batch_id":"batch-20261014-advance-20261108-r05-inclusion-exclusion-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a8b6c83281e592a6a7a255d365fedd16e465f33cf685f7a6bb1ac1e218827542","family_id":"qf-mod39-incexc-pairwise-only-boundary","seed":202611085087,"semantic_template":"qf-mod39-incexc-pairwise-only-boundary/{\"sets\":[[11,12,13],[11,13,14],[11,13,15]]}"},"fields":{"title":"A Pairwise Truncation May Coincide Numerically without Becoming a General Exact Formula","stem":"Use the complete indexed finite integer-set family sets=[[11, 12, 13], [11, 13, 14], [11, 13, 15]] in supplied index order. Each set has distinct elements; equal sets at different indices remain separate indexed entries. Every nonempty index subset contributes its intersection cardinality, with positive odd-order and negative even-order sign. Keep zero-size intersections and source increasing-bitmask term order. Give (pairwise_truncation, union_size, numerically_equal, general_exactness_claim, boundary).","answer":{"kind":"expression","value":"(3, 5, No, No, pairwise_only_truncation_not_exact_when_higher_intersections_contribute)","explanation":"Require at least three indexed sets. Compute singles minus pairs, the actual union size and their numerical equality separately. The final No withholds a general pairwise-only exactness claim; it does not negate that equality flag. This Bonferroni lower bound may even be negative, so it is not itself an asserted cardinality. The exact result, retaining the requested order, is (3, 5, No, No, pairwise_only_truncation_not_exact_when_higher_intersections_contribute)."},"skills":["A Pairwise Truncation May Coincide Numerically without Becoming a General Exact Formula"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0088","batch_id":"batch-20261014-advance-20261108-r05-inclusion-exclusion-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"34110881c2709057e83edaa57183115251d89aaa75f8f9434ca2852ee33cea63","family_id":"qf-mod39-incexc-pairwise-only-boundary","seed":202611085088,"semantic_template":"qf-mod39-incexc-pairwise-only-boundary/{\"sets\":[[12,13,14],[12,14,15],[12,14,16]]}"},"fields":{"title":"A Pairwise Truncation May Coincide Numerically without Becoming a General Exact Formula","stem":"Use the complete indexed finite integer-set family sets=[[12, 13, 14], [12, 14, 15], [12, 14, 16]] in supplied index order. Each set has distinct elements; equal sets at different indices remain separate indexed entries. Every nonempty index subset contributes its intersection cardinality, with positive odd-order and negative even-order sign. Keep zero-size intersections and source increasing-bitmask term order. Give (pairwise_truncation, union_size, numerically_equal, general_exactness_claim, boundary).","answer":{"kind":"expression","value":"(3, 5, No, No, pairwise_only_truncation_not_exact_when_higher_intersections_contribute)","explanation":"Require at least three indexed sets. Compute singles minus pairs, the actual union size and their numerical equality separately. The final No withholds a general pairwise-only exactness claim; it does not negate that equality flag. This Bonferroni lower bound may even be negative, so it is not itself an asserted cardinality. The exact result, retaining the requested order, is (3, 5, No, No, pairwise_only_truncation_not_exact_when_higher_intersections_contribute)."},"skills":["A Pairwise Truncation May Coincide Numerically without Becoming a General Exact Formula"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0089","batch_id":"batch-20261014-advance-20261108-r05-inclusion-exclusion-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7df1282c3c6f48cb85be1677c73c2bad507a05afc1bb24e3a901e1870b6008a9","family_id":"qf-mod39-incexc-pairwise-only-boundary","seed":202611085089,"semantic_template":"qf-mod39-incexc-pairwise-only-boundary/{\"sets\":[[13,14,15],[13,15,16],[13,15,17]]}"},"fields":{"title":"A Pairwise Truncation May Coincide Numerically without Becoming a General Exact Formula","stem":"Use the complete indexed finite integer-set family sets=[[13, 14, 15], [13, 15, 16], [13, 15, 17]] in supplied index order. Each set has distinct elements; equal sets at different indices remain separate indexed entries. Every nonempty index subset contributes its intersection cardinality, with positive odd-order and negative even-order sign. Keep zero-size intersections and source increasing-bitmask term order. Give (pairwise_truncation, union_size, numerically_equal, general_exactness_claim, boundary).","answer":{"kind":"expression","value":"(3, 5, No, No, pairwise_only_truncation_not_exact_when_higher_intersections_contribute)","explanation":"Require at least three indexed sets. Compute singles minus pairs, the actual union size and their numerical equality separately. The final No withholds a general pairwise-only exactness claim; it does not negate that equality flag. This Bonferroni lower bound may even be negative, so it is not itself an asserted cardinality. The exact result, retaining the requested order, is (3, 5, No, No, pairwise_only_truncation_not_exact_when_higher_intersections_contribute)."},"skills":["A Pairwise Truncation May Coincide Numerically without Becoming a General Exact Formula"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0090","batch_id":"batch-20261014-advance-20261108-r05-inclusion-exclusion-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"462033bb43c0d398b3d6c65b8c5b4e6f920e982fa126b54911a3f598fb87c7a3","family_id":"qf-mod39-incexc-pairwise-only-boundary","seed":202611085090,"semantic_template":"qf-mod39-incexc-pairwise-only-boundary/{\"sets\":[[14,15,16],[14,16,17],[14,16,18]]}"},"fields":{"title":"A Pairwise Truncation May Coincide Numerically without Becoming a General Exact Formula","stem":"Use the complete indexed finite integer-set family sets=[[14, 15, 16], [14, 16, 17], [14, 16, 18]] in supplied index order. Each set has distinct elements; equal sets at different indices remain separate indexed entries. Every nonempty index subset contributes its intersection cardinality, with positive odd-order and negative even-order sign. Keep zero-size intersections and source increasing-bitmask term order. Give (pairwise_truncation, union_size, numerically_equal, general_exactness_claim, boundary).","answer":{"kind":"expression","value":"(3, 5, No, No, pairwise_only_truncation_not_exact_when_higher_intersections_contribute)","explanation":"Require at least three indexed sets. Compute singles minus pairs, the actual union size and their numerical equality separately. The final No withholds a general pairwise-only exactness claim; it does not negate that equality flag. This Bonferroni lower bound may even be negative, so it is not itself an asserted cardinality. The exact result, retaining the requested order, is (3, 5, No, No, pairwise_only_truncation_not_exact_when_higher_intersections_contribute)."},"skills":["A Pairwise Truncation May Coincide Numerically without Becoming a General Exact Formula"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0091","batch_id":"batch-20261014-advance-20261108-r05-inclusion-exclusion-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e24e696ba1ccaafdc486e7da1b124f39206a433e55838cc19ea46837c5c6707f","family_id":"qf-mod39-incexc-pairwise-only-boundary","seed":202611085091,"semantic_template":"qf-mod39-incexc-pairwise-only-boundary/{\"sets\":[[15,16,17],[15,17,18],[15,17,19]]}"},"fields":{"title":"A Pairwise Truncation May Coincide Numerically without Becoming a General Exact Formula","stem":"Use the complete indexed finite integer-set family sets=[[15, 16, 17], [15, 17, 18], [15, 17, 19]] in supplied index order. Each set has distinct elements; equal sets at different indices remain separate indexed entries. Every nonempty index subset contributes its intersection cardinality, with positive odd-order and negative even-order sign. Keep zero-size intersections and source increasing-bitmask term order. Give (pairwise_truncation, union_size, numerically_equal, general_exactness_claim, boundary).","answer":{"kind":"expression","value":"(3, 5, No, No, pairwise_only_truncation_not_exact_when_higher_intersections_contribute)","explanation":"Require at least three indexed sets. Compute singles minus pairs, the actual union size and their numerical equality separately. The final No withholds a general pairwise-only exactness claim; it does not negate that equality flag. This Bonferroni lower bound may even be negative, so it is not itself an asserted cardinality. The exact result, retaining the requested order, is (3, 5, No, No, pairwise_only_truncation_not_exact_when_higher_intersections_contribute)."},"skills":["A Pairwise Truncation May Coincide Numerically without Becoming a General Exact Formula"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0092","batch_id":"batch-20261014-advance-20261108-r05-inclusion-exclusion-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d3a0afaa73fa3827e8b7bf15aa461c37f2dfbef70eb61e0740c81d4437f08c36","family_id":"qf-mod39-incexc-pairwise-only-boundary","seed":202611085092,"semantic_template":"qf-mod39-incexc-pairwise-only-boundary/{\"sets\":[[16,17,18],[16,18,19],[16,18,20]]}"},"fields":{"title":"A Pairwise Truncation May Coincide Numerically without Becoming a General Exact Formula","stem":"Use the complete indexed finite integer-set family sets=[[16, 17, 18], [16, 18, 19], [16, 18, 20]] in supplied index order. Each set has distinct elements; equal sets at different indices remain separate indexed entries. Every nonempty index subset contributes its intersection cardinality, with positive odd-order and negative even-order sign. Keep zero-size intersections and source increasing-bitmask term order. Give (pairwise_truncation, union_size, numerically_equal, general_exactness_claim, boundary).","answer":{"kind":"expression","value":"(3, 5, No, No, pairwise_only_truncation_not_exact_when_higher_intersections_contribute)","explanation":"Require at least three indexed sets. Compute singles minus pairs, the actual union size and their numerical equality separately. The final No withholds a general pairwise-only exactness claim; it does not negate that equality flag. This Bonferroni lower bound may even be negative, so it is not itself an asserted cardinality. The exact result, retaining the requested order, is (3, 5, No, No, pairwise_only_truncation_not_exact_when_higher_intersections_contribute)."},"skills":["A Pairwise Truncation May Coincide Numerically without Becoming a General Exact Formula"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0093","batch_id":"batch-20261014-advance-20261108-r05-inclusion-exclusion-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f55cf5908001cd1c1a5fadddce3b7a984f3fe04095dc95f5bb7dfffaedd18489","family_id":"qf-mod39-incexc-pairwise-only-boundary","seed":202611085093,"semantic_template":"qf-mod39-incexc-pairwise-only-boundary/{\"sets\":[[17,18,19],[17,19,20],[17,19,21]]}"},"fields":{"title":"A Pairwise Truncation May Coincide Numerically without Becoming a General Exact Formula","stem":"Use the complete indexed finite integer-set family sets=[[17, 18, 19], [17, 19, 20], [17, 19, 21]] in supplied index order. Each set has distinct elements; equal sets at different indices remain separate indexed entries. Every nonempty index subset contributes its intersection cardinality, with positive odd-order and negative even-order sign. Keep zero-size intersections and source increasing-bitmask term order. Give (pairwise_truncation, union_size, numerically_equal, general_exactness_claim, boundary).","answer":{"kind":"expression","value":"(3, 5, No, No, pairwise_only_truncation_not_exact_when_higher_intersections_contribute)","explanation":"Require at least three indexed sets. Compute singles minus pairs, the actual union size and their numerical equality separately. The final No withholds a general pairwise-only exactness claim; it does not negate that equality flag. This Bonferroni lower bound may even be negative, so it is not itself an asserted cardinality. The exact result, retaining the requested order, is (3, 5, No, No, pairwise_only_truncation_not_exact_when_higher_intersections_contribute)."},"skills":["A Pairwise Truncation May Coincide Numerically without Becoming a General Exact Formula"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0094","batch_id":"batch-20261014-advance-20261108-r05-inclusion-exclusion-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"029e72f425e41367b6384e4300ba39b0d6c7fc4feec8bb9e255d92f0c3c4e2f0","family_id":"qf-mod39-incexc-pairwise-only-boundary","seed":202611085094,"semantic_template":"qf-mod39-incexc-pairwise-only-boundary/{\"sets\":[[18,19,20],[18,20,21],[18,20,22]]}"},"fields":{"title":"A Pairwise Truncation May Coincide Numerically without Becoming a General Exact Formula","stem":"Use the complete indexed finite integer-set family sets=[[18, 19, 20], [18, 20, 21], [18, 20, 22]] in supplied index order. Each set has distinct elements; equal sets at different indices remain separate indexed entries. Every nonempty index subset contributes its intersection cardinality, with positive odd-order and negative even-order sign. Keep zero-size intersections and source increasing-bitmask term order. Give (pairwise_truncation, union_size, numerically_equal, general_exactness_claim, boundary).","answer":{"kind":"expression","value":"(3, 5, No, No, pairwise_only_truncation_not_exact_when_higher_intersections_contribute)","explanation":"Require at least three indexed sets. Compute singles minus pairs, the actual union size and their numerical equality separately. The final No withholds a general pairwise-only exactness claim; it does not negate that equality flag. This Bonferroni lower bound may even be negative, so it is not itself an asserted cardinality. The exact result, retaining the requested order, is (3, 5, No, No, pairwise_only_truncation_not_exact_when_higher_intersections_contribute)."},"skills":["A Pairwise Truncation May Coincide Numerically without Becoming a General Exact Formula"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0095","batch_id":"batch-20261014-advance-20261108-r05-inclusion-exclusion-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"77e5dab154169b0fadb30a950f819f2a7358006190fc933d31414ea8a53f1254","family_id":"qf-mod39-incexc-pairwise-only-boundary","seed":202611085095,"semantic_template":"qf-mod39-incexc-pairwise-only-boundary/{\"sets\":[[19,20,21],[19,21,22],[19,21,23]]}"},"fields":{"title":"A Pairwise Truncation May Coincide Numerically without Becoming a General Exact Formula","stem":"Use the complete indexed finite integer-set family sets=[[19, 20, 21], [19, 21, 22], [19, 21, 23]] in supplied index order. Each set has distinct elements; equal sets at different indices remain separate indexed entries. Every nonempty index subset contributes its intersection cardinality, with positive odd-order and negative even-order sign. Keep zero-size intersections and source increasing-bitmask term order. Give (pairwise_truncation, union_size, numerically_equal, general_exactness_claim, boundary).","answer":{"kind":"expression","value":"(3, 5, No, No, pairwise_only_truncation_not_exact_when_higher_intersections_contribute)","explanation":"Require at least three indexed sets. Compute singles minus pairs, the actual union size and their numerical equality separately. The final No withholds a general pairwise-only exactness claim; it does not negate that equality flag. This Bonferroni lower bound may even be negative, so it is not itself an asserted cardinality. The exact result, retaining the requested order, is (3, 5, No, No, pairwise_only_truncation_not_exact_when_higher_intersections_contribute)."},"skills":["A Pairwise Truncation May Coincide Numerically without Becoming a General Exact Formula"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0096","batch_id":"batch-20261014-advance-20261108-r05-inclusion-exclusion-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cd532808b2959e4d0c7248b15c629136966e39f5a624362b4c938b64d0b1c58f","family_id":"qf-mod39-incexc-pairwise-only-boundary","seed":202611085096,"semantic_template":"qf-mod39-incexc-pairwise-only-boundary/{\"sets\":[[20,21,22],[20,22,23],[20,22,24]]}"},"fields":{"title":"A Pairwise Truncation May Coincide Numerically without Becoming a General Exact Formula","stem":"Use the complete indexed finite integer-set family sets=[[20, 21, 22], [20, 22, 23], [20, 22, 24]] in supplied index order. Each set has distinct elements; equal sets at different indices remain separate indexed entries. Every nonempty index subset contributes its intersection cardinality, with positive odd-order and negative even-order sign. Keep zero-size intersections and source increasing-bitmask term order. Give (pairwise_truncation, union_size, numerically_equal, general_exactness_claim, boundary).","answer":{"kind":"expression","value":"(3, 5, No, No, pairwise_only_truncation_not_exact_when_higher_intersections_contribute)","explanation":"Require at least three indexed sets. Compute singles minus pairs, the actual union size and their numerical equality separately. The final No withholds a general pairwise-only exactness claim; it does not negate that equality flag. This Bonferroni lower bound may even be negative, so it is not itself an asserted cardinality. The exact result, retaining the requested order, is (3, 5, No, No, pairwise_only_truncation_not_exact_when_higher_intersections_contribute)."},"skills":["A Pairwise Truncation May Coincide Numerically without Becoming a General Exact Formula"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0097","batch_id":"batch-20261014-advance-20261108-r05-inclusion-exclusion-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ea8318822423514afe5c10e8bd7957eb0cb287808357c4a6582be1b94a38e285","family_id":"qf-mod39-incexc-pairwise-only-boundary","seed":202611085097,"semantic_template":"qf-mod39-incexc-pairwise-only-boundary/{\"sets\":[[21,22,23],[21,23,24],[21,23,25]]}"},"fields":{"title":"A Pairwise Truncation May Coincide Numerically without Becoming a General Exact Formula","stem":"Use the complete indexed finite integer-set family sets=[[21, 22, 23], [21, 23, 24], [21, 23, 25]] in supplied index order. Each set has distinct elements; equal sets at different indices remain separate indexed entries. Every nonempty index subset contributes its intersection cardinality, with positive odd-order and negative even-order sign. Keep zero-size intersections and source increasing-bitmask term order. Give (pairwise_truncation, union_size, numerically_equal, general_exactness_claim, boundary).","answer":{"kind":"expression","value":"(3, 5, No, No, pairwise_only_truncation_not_exact_when_higher_intersections_contribute)","explanation":"Require at least three indexed sets. Compute singles minus pairs, the actual union size and their numerical equality separately. The final No withholds a general pairwise-only exactness claim; it does not negate that equality flag. This Bonferroni lower bound may even be negative, so it is not itself an asserted cardinality. The exact result, retaining the requested order, is (3, 5, No, No, pairwise_only_truncation_not_exact_when_higher_intersections_contribute)."},"skills":["A Pairwise Truncation May Coincide Numerically without Becoming a General Exact Formula"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0098","batch_id":"batch-20261014-advance-20261108-r05-inclusion-exclusion-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"aee9d0f0431bc313f71a3478687fbafa382b50a81fcd9c07143ff623bccfce79","family_id":"qf-mod39-incexc-pairwise-only-boundary","seed":202611085098,"semantic_template":"qf-mod39-incexc-pairwise-only-boundary/{\"sets\":[[22,23,24],[22,24,25],[22,24,26]]}"},"fields":{"title":"A Pairwise Truncation May Coincide Numerically without Becoming a General Exact Formula","stem":"Use the complete indexed finite integer-set family sets=[[22, 23, 24], [22, 24, 25], [22, 24, 26]] in supplied index order. Each set has distinct elements; equal sets at different indices remain separate indexed entries. Every nonempty index subset contributes its intersection cardinality, with positive odd-order and negative even-order sign. Keep zero-size intersections and source increasing-bitmask term order. Give (pairwise_truncation, union_size, numerically_equal, general_exactness_claim, boundary).","answer":{"kind":"expression","value":"(3, 5, No, No, pairwise_only_truncation_not_exact_when_higher_intersections_contribute)","explanation":"Require at least three indexed sets. Compute singles minus pairs, the actual union size and their numerical equality separately. The final No withholds a general pairwise-only exactness claim; it does not negate that equality flag. This Bonferroni lower bound may even be negative, so it is not itself an asserted cardinality. The exact result, retaining the requested order, is (3, 5, No, No, pairwise_only_truncation_not_exact_when_higher_intersections_contribute)."},"skills":["A Pairwise Truncation May Coincide Numerically without Becoming a General Exact Formula"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0099","batch_id":"batch-20261014-advance-20261108-r05-inclusion-exclusion-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a34fd16d96e04c815af3db1e7ce1f456c874086462872a43946c8b2223ed5df2","family_id":"qf-mod39-incexc-pairwise-only-boundary","seed":202611085099,"semantic_template":"qf-mod39-incexc-pairwise-only-boundary/{\"sets\":[[23,24,25],[23,25,26],[23,25,27]]}"},"fields":{"title":"A Pairwise Truncation May Coincide Numerically without Becoming a General Exact Formula","stem":"Use the complete indexed finite integer-set family sets=[[23, 24, 25], [23, 25, 26], [23, 25, 27]] in supplied index order. Each set has distinct elements; equal sets at different indices remain separate indexed entries. Every nonempty index subset contributes its intersection cardinality, with positive odd-order and negative even-order sign. Keep zero-size intersections and source increasing-bitmask term order. Give (pairwise_truncation, union_size, numerically_equal, general_exactness_claim, boundary).","answer":{"kind":"expression","value":"(3, 5, No, No, pairwise_only_truncation_not_exact_when_higher_intersections_contribute)","explanation":"Require at least three indexed sets. Compute singles minus pairs, the actual union size and their numerical equality separately. The final No withholds a general pairwise-only exactness claim; it does not negate that equality flag. This Bonferroni lower bound may even be negative, so it is not itself an asserted cardinality. The exact result, retaining the requested order, is (3, 5, No, No, pairwise_only_truncation_not_exact_when_higher_intersections_contribute)."},"skills":["A Pairwise Truncation May Coincide Numerically without Becoming a General Exact Formula"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0100","batch_id":"batch-20261014-advance-20261108-r05-inclusion-exclusion-principle-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e06dd52981d47c44e94f3267d77e241c404e8e0c6348e2fe4de1cc79163ec59b","family_id":"qf-mod39-incexc-pairwise-only-boundary","seed":202611085100,"semantic_template":"qf-mod39-incexc-pairwise-only-boundary/{\"sets\":[[24,25,26],[24,26,27],[24,26,28]]}"},"fields":{"title":"A Pairwise Truncation May Coincide Numerically without Becoming a General Exact Formula","stem":"Use the complete indexed finite integer-set family sets=[[24, 25, 26], [24, 26, 27], [24, 26, 28]] in supplied index order. Each set has distinct elements; equal sets at different indices remain separate indexed entries. Every nonempty index subset contributes its intersection cardinality, with positive odd-order and negative even-order sign. Keep zero-size intersections and source increasing-bitmask term order. Give (pairwise_truncation, union_size, numerically_equal, general_exactness_claim, boundary).","answer":{"kind":"expression","value":"(3, 5, No, No, pairwise_only_truncation_not_exact_when_higher_intersections_contribute)","explanation":"Require at least three indexed sets. Compute singles minus pairs, the actual union size and their numerical equality separately. The final No withholds a general pairwise-only exactness claim; it does not negate that equality flag. This Bonferroni lower bound may even be negative, so it is not itself an asserted cardinality. The exact result, retaining the requested order, is (3, 5, No, No, pairwise_only_truncation_not_exact_when_higher_intersections_contribute)."},"skills":["A Pairwise Truncation May Coincide Numerically without Becoming a General Exact Formula"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}