Indexed Intersection Terms, Exact Unions and Pairwise Truncation Boundaries
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batch-20261014-advance-20261108-r05-inclusion-exclusion-principle-foundations · 2026-10-14 · 100 questions
This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.
Field order uses a source-bound presentation supplement; canonical JSONL and mathematical answers are unchanged.
Field-order and source-binding data
1. Two-Set Inclusion-Exclusion Keeps Both Sizes and Their Intersection
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).
View answer and explanation
([3, 3], 2, 4, 4, Yes)
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).
Reviewed step-by-step derivation
Inclusion-exclusion principle · qf-mod39-incexc-two-set-formula · candidate
2. Two-Set Inclusion-Exclusion Keeps Both Sizes and Their Intersection
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).
View answer and explanation
([3, 3], 2, 4, 4, Yes)
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).
Reviewed step-by-step derivation
Inclusion-exclusion principle · qf-mod39-incexc-two-set-formula · candidate
3. Two-Set Inclusion-Exclusion Keeps Both Sizes and Their Intersection
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).
View answer and explanation
([3, 3], 2, 4, 4, Yes)
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).
Reviewed step-by-step derivation
Inclusion-exclusion principle · qf-mod39-incexc-two-set-formula · candidate
4. Two-Set Inclusion-Exclusion Keeps Both Sizes and Their Intersection
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).
View answer and explanation
([3, 3], 2, 4, 4, Yes)
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).
Reviewed step-by-step derivation
Inclusion-exclusion principle · qf-mod39-incexc-two-set-formula · candidate
5. Two-Set Inclusion-Exclusion Keeps Both Sizes and Their Intersection
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).
View answer and explanation
([3, 3], 2, 4, 4, Yes)
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).
Reviewed step-by-step derivation
Inclusion-exclusion principle · qf-mod39-incexc-two-set-formula · candidate
6. Two-Set Inclusion-Exclusion Keeps Both Sizes and Their Intersection
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).
View answer and explanation
([3, 3], 2, 4, 4, Yes)
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).
Reviewed step-by-step derivation
Inclusion-exclusion principle · qf-mod39-incexc-two-set-formula · candidate
7. Two-Set Inclusion-Exclusion Keeps Both Sizes and Their Intersection
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).
View answer and explanation
([3, 3], 2, 4, 4, Yes)
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).
Reviewed step-by-step derivation
Inclusion-exclusion principle · qf-mod39-incexc-two-set-formula · candidate
8. Two-Set Inclusion-Exclusion Keeps Both Sizes and Their Intersection
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).
View answer and explanation
([3, 3], 2, 4, 4, Yes)
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).
Reviewed step-by-step derivation
Inclusion-exclusion principle · qf-mod39-incexc-two-set-formula · candidate
9. Two-Set Inclusion-Exclusion Keeps Both Sizes and Their Intersection
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).
View answer and explanation
([3, 3], 2, 4, 4, Yes)
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).
Reviewed step-by-step derivation
Inclusion-exclusion principle · qf-mod39-incexc-two-set-formula · candidate
10. Two-Set Inclusion-Exclusion Keeps Both Sizes and Their Intersection
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).
View answer and explanation
([3, 3], 2, 4, 4, Yes)
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).
Reviewed step-by-step derivation
Inclusion-exclusion principle · qf-mod39-incexc-two-set-formula · candidate
11. Two-Set Inclusion-Exclusion Keeps Both Sizes and Their Intersection
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).
View answer and explanation
([3, 3], 2, 4, 4, Yes)
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).
Reviewed step-by-step derivation
Inclusion-exclusion principle · qf-mod39-incexc-two-set-formula · candidate
12. Two-Set Inclusion-Exclusion Keeps Both Sizes and Their Intersection
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).
View answer and explanation
([3, 3], 2, 4, 4, Yes)
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).
Reviewed step-by-step derivation
Inclusion-exclusion principle · qf-mod39-incexc-two-set-formula · candidate
13. Two-Set Inclusion-Exclusion Keeps Both Sizes and Their Intersection
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).
View answer and explanation
([3, 3], 2, 4, 4, Yes)
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).
Reviewed step-by-step derivation
Inclusion-exclusion principle · qf-mod39-incexc-two-set-formula · candidate
14. Two-Set Inclusion-Exclusion Keeps Both Sizes and Their Intersection
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).
View answer and explanation
([3, 3], 2, 4, 4, Yes)
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).
Reviewed step-by-step derivation
Inclusion-exclusion principle · qf-mod39-incexc-two-set-formula · candidate
15. Two-Set Inclusion-Exclusion Keeps Both Sizes and Their Intersection
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).
View answer and explanation
([3, 3], 2, 4, 4, Yes)
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).
Reviewed step-by-step derivation
Inclusion-exclusion principle · qf-mod39-incexc-two-set-formula · candidate
16. Two-Set Inclusion-Exclusion Keeps Both Sizes and Their Intersection
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).
View answer and explanation
([3, 3], 2, 4, 4, Yes)
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).
Reviewed step-by-step derivation
Inclusion-exclusion principle · qf-mod39-incexc-two-set-formula · candidate
17. Two-Set Inclusion-Exclusion Keeps Both Sizes and Their Intersection
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).
View answer and explanation
([3, 3], 2, 4, 4, Yes)
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).
Reviewed step-by-step derivation
Inclusion-exclusion principle · qf-mod39-incexc-two-set-formula · candidate
18. Two-Set Inclusion-Exclusion Keeps Both Sizes and Their Intersection
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).
View answer and explanation
([3, 3], 2, 4, 4, Yes)
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).
Reviewed step-by-step derivation
Inclusion-exclusion principle · qf-mod39-incexc-two-set-formula · candidate
19. Two-Set Inclusion-Exclusion Keeps Both Sizes and Their Intersection
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).
View answer and explanation
([3, 3], 2, 4, 4, Yes)
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).
Reviewed step-by-step derivation
Inclusion-exclusion principle · qf-mod39-incexc-two-set-formula · candidate
20. Two-Set Inclusion-Exclusion Keeps Both Sizes and Their Intersection
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).
View answer and explanation
([3, 3], 2, 4, 4, Yes)
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).
Reviewed step-by-step derivation
Inclusion-exclusion principle · qf-mod39-incexc-two-set-formula · candidate
21. Two-Set Inclusion-Exclusion Keeps Both Sizes and Their Intersection
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).
View answer and explanation
([3, 3], 2, 4, 4, Yes)
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).
Reviewed step-by-step derivation
Inclusion-exclusion principle · qf-mod39-incexc-two-set-formula · candidate
22. Two-Set Inclusion-Exclusion Keeps Both Sizes and Their Intersection
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).
View answer and explanation
([3, 3], 2, 4, 4, Yes)
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).
Reviewed step-by-step derivation
Inclusion-exclusion principle · qf-mod39-incexc-two-set-formula · candidate
23. Two-Set Inclusion-Exclusion Keeps Both Sizes and Their Intersection
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).
View answer and explanation
([3, 3], 2, 4, 4, Yes)
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).
Reviewed step-by-step derivation
Inclusion-exclusion principle · qf-mod39-incexc-two-set-formula · candidate
24. Two-Set Inclusion-Exclusion Keeps Both Sizes and Their Intersection
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).
View answer and explanation
([3, 3], 2, 4, 4, Yes)
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).
Reviewed step-by-step derivation
Inclusion-exclusion principle · qf-mod39-incexc-two-set-formula · candidate
25. Two-Set Inclusion-Exclusion Keeps Both Sizes and Their Intersection
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).
View answer and explanation
([3, 3], 2, 4, 4, Yes)
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).
Reviewed step-by-step derivation
Inclusion-exclusion principle · qf-mod39-incexc-two-set-formula · candidate
26. Three-Set Inclusion-Exclusion Needs the Triple Intersection
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).
View answer and explanation
([[[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)
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).
Reviewed step-by-step derivation
Inclusion-exclusion principle · qf-mod39-incexc-three-set-formula · candidate
27. Three-Set Inclusion-Exclusion Needs the Triple Intersection
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).
View answer and explanation
([[[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)
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).
Reviewed step-by-step derivation
Inclusion-exclusion principle · qf-mod39-incexc-three-set-formula · candidate
28. Three-Set Inclusion-Exclusion Needs the Triple Intersection
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).
View answer and explanation
([[[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)
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).
Reviewed step-by-step derivation
Inclusion-exclusion principle · qf-mod39-incexc-three-set-formula · candidate
29. Three-Set Inclusion-Exclusion Needs the Triple Intersection
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).
View answer and explanation
([[[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)
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).
Reviewed step-by-step derivation
Inclusion-exclusion principle · qf-mod39-incexc-three-set-formula · candidate
30. Three-Set Inclusion-Exclusion Needs the Triple Intersection
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).
View answer and explanation
([[[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)
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).
Reviewed step-by-step derivation
Inclusion-exclusion principle · qf-mod39-incexc-three-set-formula · candidate
31. Three-Set Inclusion-Exclusion Needs the Triple Intersection
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).
View answer and explanation
([[[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)
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).
Reviewed step-by-step derivation
Inclusion-exclusion principle · qf-mod39-incexc-three-set-formula · candidate
32. Three-Set Inclusion-Exclusion Needs the Triple Intersection
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).
View answer and explanation
([[[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)
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).
Reviewed step-by-step derivation
Inclusion-exclusion principle · qf-mod39-incexc-three-set-formula · candidate
33. Three-Set Inclusion-Exclusion Needs the Triple Intersection
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).
View answer and explanation
([[[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)
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).
Reviewed step-by-step derivation
Inclusion-exclusion principle · qf-mod39-incexc-three-set-formula · candidate
34. Three-Set Inclusion-Exclusion Needs the Triple Intersection
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).
View answer and explanation
([[[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)
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).
Reviewed step-by-step derivation
Inclusion-exclusion principle · qf-mod39-incexc-three-set-formula · candidate
35. Three-Set Inclusion-Exclusion Needs the Triple Intersection
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).
View answer and explanation
([[[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)
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).
Reviewed step-by-step derivation
Inclusion-exclusion principle · qf-mod39-incexc-three-set-formula · candidate
36. Three-Set Inclusion-Exclusion Needs the Triple Intersection
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).
View answer and explanation
([[[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)
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).
Reviewed step-by-step derivation
Inclusion-exclusion principle · qf-mod39-incexc-three-set-formula · candidate
37. Three-Set Inclusion-Exclusion Needs the Triple Intersection
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).
View answer and explanation
([[[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)
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).
Reviewed step-by-step derivation
Inclusion-exclusion principle · qf-mod39-incexc-three-set-formula · candidate
38. Three-Set Inclusion-Exclusion Needs the Triple Intersection
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).
View answer and explanation
([[[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)
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).
Reviewed step-by-step derivation
Inclusion-exclusion principle · qf-mod39-incexc-three-set-formula · candidate
39. Three-Set Inclusion-Exclusion Needs the Triple Intersection
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).
View answer and explanation
([[[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)
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).
Reviewed step-by-step derivation
Inclusion-exclusion principle · qf-mod39-incexc-three-set-formula · candidate
40. Three-Set Inclusion-Exclusion Needs the Triple Intersection
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).
View answer and explanation
([[[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)
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).
Reviewed step-by-step derivation
Inclusion-exclusion principle · qf-mod39-incexc-three-set-formula · candidate
41. Three-Set Inclusion-Exclusion Needs the Triple Intersection
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).
View answer and explanation
([[[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)
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).
Reviewed step-by-step derivation
Inclusion-exclusion principle · qf-mod39-incexc-three-set-formula · candidate
42. Three-Set Inclusion-Exclusion Needs the Triple Intersection
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).
View answer and explanation
([[[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)
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).
Reviewed step-by-step derivation
Inclusion-exclusion principle · qf-mod39-incexc-three-set-formula · candidate
43. Three-Set Inclusion-Exclusion Needs the Triple Intersection
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).
View answer and explanation
([[[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)
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).
Reviewed step-by-step derivation
Inclusion-exclusion principle · qf-mod39-incexc-three-set-formula · candidate
44. Three-Set Inclusion-Exclusion Needs the Triple Intersection
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).
View answer and explanation
([[[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)
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).
Reviewed step-by-step derivation
Inclusion-exclusion principle · qf-mod39-incexc-three-set-formula · candidate
45. Three-Set Inclusion-Exclusion Needs the Triple Intersection
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).
View answer and explanation
([[[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)
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).
Reviewed step-by-step derivation
Inclusion-exclusion principle · qf-mod39-incexc-three-set-formula · candidate
46. Three-Set Inclusion-Exclusion Needs the Triple Intersection
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).
View answer and explanation
([[[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)
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).
Reviewed step-by-step derivation
Inclusion-exclusion principle · qf-mod39-incexc-three-set-formula · candidate
47. Three-Set Inclusion-Exclusion Needs the Triple Intersection
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).
View answer and explanation
([[[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)
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).
Reviewed step-by-step derivation
Inclusion-exclusion principle · qf-mod39-incexc-three-set-formula · candidate
48. Three-Set Inclusion-Exclusion Needs the Triple Intersection
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).
View answer and explanation
([[[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)
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).
Reviewed step-by-step derivation
Inclusion-exclusion principle · qf-mod39-incexc-three-set-formula · candidate
49. Three-Set Inclusion-Exclusion Needs the Triple Intersection
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).
View answer and explanation
([[[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)
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).
Reviewed step-by-step derivation
Inclusion-exclusion principle · qf-mod39-incexc-three-set-formula · candidate
50. Three-Set Inclusion-Exclusion Needs the Triple Intersection
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).
View answer and explanation
([[[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)
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).
Reviewed step-by-step derivation
Inclusion-exclusion principle · qf-mod39-incexc-three-set-formula · candidate
51. All Nonempty Index Subsets Are Needed for the Exact Alternating Sum
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).
View answer and explanation
(1, [[[0], 1, 3]], 3, 3, Yes, all_nonempty_index_subsets_included)
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).
Reviewed step-by-step derivation
Inclusion-exclusion principle · qf-mod39-incexc-general-alternating-sum · candidate
52. All Nonempty Index Subsets Are Needed for the Exact Alternating Sum
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).
View answer and explanation
(2, [[[0], 1, 3], [[1], 1, 3], [[0, 1], -1, 2]], 4, 4, Yes, all_nonempty_index_subsets_included)
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).
Reviewed step-by-step derivation
Inclusion-exclusion principle · qf-mod39-incexc-general-alternating-sum · candidate
53. All Nonempty Index Subsets Are Needed for the Exact Alternating Sum
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).
View answer and explanation
(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)
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).
Reviewed step-by-step derivation
Inclusion-exclusion principle · qf-mod39-incexc-general-alternating-sum · candidate
54. All Nonempty Index Subsets Are Needed for the Exact Alternating Sum
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).
View answer and explanation
(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)
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).
Reviewed step-by-step derivation
Inclusion-exclusion principle · qf-mod39-incexc-general-alternating-sum · candidate
55. All Nonempty Index Subsets Are Needed for the Exact Alternating Sum
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).
View answer and explanation
(1, [[[0], 1, 3]], 3, 3, Yes, all_nonempty_index_subsets_included)
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).
Reviewed step-by-step derivation
Inclusion-exclusion principle · qf-mod39-incexc-general-alternating-sum · candidate
56. All Nonempty Index Subsets Are Needed for the Exact Alternating Sum
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).
View answer and explanation
(2, [[[0], 1, 3], [[1], 1, 3], [[0, 1], -1, 2]], 4, 4, Yes, all_nonempty_index_subsets_included)
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).
Reviewed step-by-step derivation
Inclusion-exclusion principle · qf-mod39-incexc-general-alternating-sum · candidate
57. All Nonempty Index Subsets Are Needed for the Exact Alternating Sum
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).
View answer and explanation
(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)
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).
Reviewed step-by-step derivation
Inclusion-exclusion principle · qf-mod39-incexc-general-alternating-sum · candidate
58. All Nonempty Index Subsets Are Needed for the Exact Alternating Sum
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).
View answer and explanation
(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)
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).
Reviewed step-by-step derivation
Inclusion-exclusion principle · qf-mod39-incexc-general-alternating-sum · candidate
59. All Nonempty Index Subsets Are Needed for the Exact Alternating Sum
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).
View answer and explanation
(1, [[[0], 1, 3]], 3, 3, Yes, all_nonempty_index_subsets_included)
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).
Reviewed step-by-step derivation
Inclusion-exclusion principle · qf-mod39-incexc-general-alternating-sum · candidate
60. All Nonempty Index Subsets Are Needed for the Exact Alternating Sum
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).
View answer and explanation
(2, [[[0], 1, 3], [[1], 1, 3], [[0, 1], -1, 2]], 4, 4, Yes, all_nonempty_index_subsets_included)
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).
Reviewed step-by-step derivation
Inclusion-exclusion principle · qf-mod39-incexc-general-alternating-sum · candidate
61. All Nonempty Index Subsets Are Needed for the Exact Alternating Sum
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).
View answer and explanation
(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)
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).
Reviewed step-by-step derivation
Inclusion-exclusion principle · qf-mod39-incexc-general-alternating-sum · candidate
62. All Nonempty Index Subsets Are Needed for the Exact Alternating Sum
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).
View answer and explanation
(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)
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).
Reviewed step-by-step derivation
Inclusion-exclusion principle · qf-mod39-incexc-general-alternating-sum · candidate
63. All Nonempty Index Subsets Are Needed for the Exact Alternating Sum
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).
View answer and explanation
(1, [[[0], 1, 3]], 3, 3, Yes, all_nonempty_index_subsets_included)
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).
Reviewed step-by-step derivation
Inclusion-exclusion principle · qf-mod39-incexc-general-alternating-sum · candidate
64. All Nonempty Index Subsets Are Needed for the Exact Alternating Sum
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).
View answer and explanation
(2, [[[0], 1, 3], [[1], 1, 3], [[0, 1], -1, 2]], 4, 4, Yes, all_nonempty_index_subsets_included)
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).
Reviewed step-by-step derivation
Inclusion-exclusion principle · qf-mod39-incexc-general-alternating-sum · candidate
65. All Nonempty Index Subsets Are Needed for the Exact Alternating Sum
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).
View answer and explanation
(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)
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).
Reviewed step-by-step derivation
Inclusion-exclusion principle · qf-mod39-incexc-general-alternating-sum · candidate
66. All Nonempty Index Subsets Are Needed for the Exact Alternating Sum
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).
View answer and explanation
(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)
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).
Reviewed step-by-step derivation
Inclusion-exclusion principle · qf-mod39-incexc-general-alternating-sum · candidate
67. All Nonempty Index Subsets Are Needed for the Exact Alternating Sum
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).
View answer and explanation
(1, [[[0], 1, 3]], 3, 3, Yes, all_nonempty_index_subsets_included)
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).
Reviewed step-by-step derivation
Inclusion-exclusion principle · qf-mod39-incexc-general-alternating-sum · candidate
68. All Nonempty Index Subsets Are Needed for the Exact Alternating Sum
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).
View answer and explanation
(2, [[[0], 1, 3], [[1], 1, 3], [[0, 1], -1, 2]], 4, 4, Yes, all_nonempty_index_subsets_included)
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).
Reviewed step-by-step derivation
Inclusion-exclusion principle · qf-mod39-incexc-general-alternating-sum · candidate
69. All Nonempty Index Subsets Are Needed for the Exact Alternating Sum
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).
View answer and explanation
(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)
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).
Reviewed step-by-step derivation
Inclusion-exclusion principle · qf-mod39-incexc-general-alternating-sum · candidate
70. All Nonempty Index Subsets Are Needed for the Exact Alternating Sum
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).
View answer and explanation
(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)
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).
Reviewed step-by-step derivation
Inclusion-exclusion principle · qf-mod39-incexc-general-alternating-sum · candidate
71. All Nonempty Index Subsets Are Needed for the Exact Alternating Sum
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).
View answer and explanation
(1, [[[0], 1, 3]], 3, 3, Yes, all_nonempty_index_subsets_included)
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).
Reviewed step-by-step derivation
Inclusion-exclusion principle · qf-mod39-incexc-general-alternating-sum · candidate
72. All Nonempty Index Subsets Are Needed for the Exact Alternating Sum
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).
View answer and explanation
(2, [[[0], 1, 3], [[1], 1, 3], [[0, 1], -1, 2]], 4, 4, Yes, all_nonempty_index_subsets_included)
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).
Reviewed step-by-step derivation
Inclusion-exclusion principle · qf-mod39-incexc-general-alternating-sum · candidate
73. All Nonempty Index Subsets Are Needed for the Exact Alternating Sum
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).
View answer and explanation
(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)
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).
Reviewed step-by-step derivation
Inclusion-exclusion principle · qf-mod39-incexc-general-alternating-sum · candidate
74. All Nonempty Index Subsets Are Needed for the Exact Alternating Sum
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).
View answer and explanation
(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)
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).
Reviewed step-by-step derivation
Inclusion-exclusion principle · qf-mod39-incexc-general-alternating-sum · candidate
75. All Nonempty Index Subsets Are Needed for the Exact Alternating Sum
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).
View answer and explanation
(1, [[[0], 1, 3]], 3, 3, Yes, all_nonempty_index_subsets_included)
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).
Reviewed step-by-step derivation
Inclusion-exclusion principle · qf-mod39-incexc-general-alternating-sum · candidate
76. A Pairwise Truncation May Coincide Numerically without Becoming a General Exact Formula
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).
The final No withholds a general pairwise-only exactness claim; it does not negate the separately computed numerically_equal flag. The truncated lower bound may be negative.
View answer and explanation
(3, 5, No, No, pairwise_only_truncation_not_exact_when_higher_intersections_contribute)
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).
Reviewed step-by-step derivation
Inclusion-exclusion principle · qf-mod39-incexc-pairwise-only-boundary · candidate
77. A Pairwise Truncation May Coincide Numerically without Becoming a General Exact Formula
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).
The final No withholds a general pairwise-only exactness claim; it does not negate the separately computed numerically_equal flag. The truncated lower bound may be negative.
View answer and explanation
(3, 5, No, No, pairwise_only_truncation_not_exact_when_higher_intersections_contribute)
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).
Reviewed step-by-step derivation
Inclusion-exclusion principle · qf-mod39-incexc-pairwise-only-boundary · candidate
78. A Pairwise Truncation May Coincide Numerically without Becoming a General Exact Formula
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).
The final No withholds a general pairwise-only exactness claim; it does not negate the separately computed numerically_equal flag. The truncated lower bound may be negative.
View answer and explanation
(3, 5, No, No, pairwise_only_truncation_not_exact_when_higher_intersections_contribute)
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).
Reviewed step-by-step derivation
Inclusion-exclusion principle · qf-mod39-incexc-pairwise-only-boundary · candidate
79. A Pairwise Truncation May Coincide Numerically without Becoming a General Exact Formula
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).
The final No withholds a general pairwise-only exactness claim; it does not negate the separately computed numerically_equal flag. The truncated lower bound may be negative.
View answer and explanation
(3, 5, No, No, pairwise_only_truncation_not_exact_when_higher_intersections_contribute)
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).
Reviewed step-by-step derivation
Inclusion-exclusion principle · qf-mod39-incexc-pairwise-only-boundary · candidate
80. A Pairwise Truncation May Coincide Numerically without Becoming a General Exact Formula
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).
The final No withholds a general pairwise-only exactness claim; it does not negate the separately computed numerically_equal flag. The truncated lower bound may be negative.
View answer and explanation
(3, 5, No, No, pairwise_only_truncation_not_exact_when_higher_intersections_contribute)
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).
Reviewed step-by-step derivation
Inclusion-exclusion principle · qf-mod39-incexc-pairwise-only-boundary · candidate
81. A Pairwise Truncation May Coincide Numerically without Becoming a General Exact Formula
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).
The final No withholds a general pairwise-only exactness claim; it does not negate the separately computed numerically_equal flag. The truncated lower bound may be negative.
View answer and explanation
(3, 5, No, No, pairwise_only_truncation_not_exact_when_higher_intersections_contribute)
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).
Reviewed step-by-step derivation
Inclusion-exclusion principle · qf-mod39-incexc-pairwise-only-boundary · candidate
82. A Pairwise Truncation May Coincide Numerically without Becoming a General Exact Formula
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).
The final No withholds a general pairwise-only exactness claim; it does not negate the separately computed numerically_equal flag. The truncated lower bound may be negative.
View answer and explanation
(3, 5, No, No, pairwise_only_truncation_not_exact_when_higher_intersections_contribute)
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).
Reviewed step-by-step derivation
Inclusion-exclusion principle · qf-mod39-incexc-pairwise-only-boundary · candidate
83. A Pairwise Truncation May Coincide Numerically without Becoming a General Exact Formula
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).
The final No withholds a general pairwise-only exactness claim; it does not negate the separately computed numerically_equal flag. The truncated lower bound may be negative.
View answer and explanation
(3, 5, No, No, pairwise_only_truncation_not_exact_when_higher_intersections_contribute)
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).
Reviewed step-by-step derivation
Inclusion-exclusion principle · qf-mod39-incexc-pairwise-only-boundary · candidate
84. A Pairwise Truncation May Coincide Numerically without Becoming a General Exact Formula
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).
The final No withholds a general pairwise-only exactness claim; it does not negate the separately computed numerically_equal flag. The truncated lower bound may be negative.
View answer and explanation
(3, 5, No, No, pairwise_only_truncation_not_exact_when_higher_intersections_contribute)
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).
Reviewed step-by-step derivation
Inclusion-exclusion principle · qf-mod39-incexc-pairwise-only-boundary · candidate
85. A Pairwise Truncation May Coincide Numerically without Becoming a General Exact Formula
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).
The final No withholds a general pairwise-only exactness claim; it does not negate the separately computed numerically_equal flag. The truncated lower bound may be negative.
View answer and explanation
(3, 5, No, No, pairwise_only_truncation_not_exact_when_higher_intersections_contribute)
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).
Reviewed step-by-step derivation
Inclusion-exclusion principle · qf-mod39-incexc-pairwise-only-boundary · candidate
86. A Pairwise Truncation May Coincide Numerically without Becoming a General Exact Formula
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).
The final No withholds a general pairwise-only exactness claim; it does not negate the separately computed numerically_equal flag. The truncated lower bound may be negative.
View answer and explanation
(3, 5, No, No, pairwise_only_truncation_not_exact_when_higher_intersections_contribute)
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).
Reviewed step-by-step derivation
Inclusion-exclusion principle · qf-mod39-incexc-pairwise-only-boundary · candidate
87. A Pairwise Truncation May Coincide Numerically without Becoming a General Exact Formula
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).
The final No withholds a general pairwise-only exactness claim; it does not negate the separately computed numerically_equal flag. The truncated lower bound may be negative.
View answer and explanation
(3, 5, No, No, pairwise_only_truncation_not_exact_when_higher_intersections_contribute)
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).
Reviewed step-by-step derivation
Inclusion-exclusion principle · qf-mod39-incexc-pairwise-only-boundary · candidate
88. A Pairwise Truncation May Coincide Numerically without Becoming a General Exact Formula
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).
The final No withholds a general pairwise-only exactness claim; it does not negate the separately computed numerically_equal flag. The truncated lower bound may be negative.
View answer and explanation
(3, 5, No, No, pairwise_only_truncation_not_exact_when_higher_intersections_contribute)
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).
Reviewed step-by-step derivation
Inclusion-exclusion principle · qf-mod39-incexc-pairwise-only-boundary · candidate
89. A Pairwise Truncation May Coincide Numerically without Becoming a General Exact Formula
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).
The final No withholds a general pairwise-only exactness claim; it does not negate the separately computed numerically_equal flag. The truncated lower bound may be negative.
View answer and explanation
(3, 5, No, No, pairwise_only_truncation_not_exact_when_higher_intersections_contribute)
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).
Reviewed step-by-step derivation
Inclusion-exclusion principle · qf-mod39-incexc-pairwise-only-boundary · candidate
90. A Pairwise Truncation May Coincide Numerically without Becoming a General Exact Formula
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).
The final No withholds a general pairwise-only exactness claim; it does not negate the separately computed numerically_equal flag. The truncated lower bound may be negative.
View answer and explanation
(3, 5, No, No, pairwise_only_truncation_not_exact_when_higher_intersections_contribute)
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).
Reviewed step-by-step derivation
Inclusion-exclusion principle · qf-mod39-incexc-pairwise-only-boundary · candidate
91. A Pairwise Truncation May Coincide Numerically without Becoming a General Exact Formula
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).
The final No withholds a general pairwise-only exactness claim; it does not negate the separately computed numerically_equal flag. The truncated lower bound may be negative.
View answer and explanation
(3, 5, No, No, pairwise_only_truncation_not_exact_when_higher_intersections_contribute)
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).
Reviewed step-by-step derivation
Inclusion-exclusion principle · qf-mod39-incexc-pairwise-only-boundary · candidate
92. A Pairwise Truncation May Coincide Numerically without Becoming a General Exact Formula
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).
The final No withholds a general pairwise-only exactness claim; it does not negate the separately computed numerically_equal flag. The truncated lower bound may be negative.
View answer and explanation
(3, 5, No, No, pairwise_only_truncation_not_exact_when_higher_intersections_contribute)
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).
Reviewed step-by-step derivation
Inclusion-exclusion principle · qf-mod39-incexc-pairwise-only-boundary · candidate
93. A Pairwise Truncation May Coincide Numerically without Becoming a General Exact Formula
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).
The final No withholds a general pairwise-only exactness claim; it does not negate the separately computed numerically_equal flag. The truncated lower bound may be negative.
View answer and explanation
(3, 5, No, No, pairwise_only_truncation_not_exact_when_higher_intersections_contribute)
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).
Reviewed step-by-step derivation
Inclusion-exclusion principle · qf-mod39-incexc-pairwise-only-boundary · candidate
94. A Pairwise Truncation May Coincide Numerically without Becoming a General Exact Formula
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).
The final No withholds a general pairwise-only exactness claim; it does not negate the separately computed numerically_equal flag. The truncated lower bound may be negative.
View answer and explanation
(3, 5, No, No, pairwise_only_truncation_not_exact_when_higher_intersections_contribute)
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).
Reviewed step-by-step derivation
Inclusion-exclusion principle · qf-mod39-incexc-pairwise-only-boundary · candidate
95. A Pairwise Truncation May Coincide Numerically without Becoming a General Exact Formula
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).
The final No withholds a general pairwise-only exactness claim; it does not negate the separately computed numerically_equal flag. The truncated lower bound may be negative.
View answer and explanation
(3, 5, No, No, pairwise_only_truncation_not_exact_when_higher_intersections_contribute)
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).
Reviewed step-by-step derivation
Inclusion-exclusion principle · qf-mod39-incexc-pairwise-only-boundary · candidate
96. A Pairwise Truncation May Coincide Numerically without Becoming a General Exact Formula
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).
The final No withholds a general pairwise-only exactness claim; it does not negate the separately computed numerically_equal flag. The truncated lower bound may be negative.
View answer and explanation
(3, 5, No, No, pairwise_only_truncation_not_exact_when_higher_intersections_contribute)
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).
Reviewed step-by-step derivation
Inclusion-exclusion principle · qf-mod39-incexc-pairwise-only-boundary · candidate
97. A Pairwise Truncation May Coincide Numerically without Becoming a General Exact Formula
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).
The final No withholds a general pairwise-only exactness claim; it does not negate the separately computed numerically_equal flag. The truncated lower bound may be negative.
View answer and explanation
(3, 5, No, No, pairwise_only_truncation_not_exact_when_higher_intersections_contribute)
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).
Reviewed step-by-step derivation
Inclusion-exclusion principle · qf-mod39-incexc-pairwise-only-boundary · candidate
98. A Pairwise Truncation May Coincide Numerically without Becoming a General Exact Formula
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).
The final No withholds a general pairwise-only exactness claim; it does not negate the separately computed numerically_equal flag. The truncated lower bound may be negative.
View answer and explanation
(3, 5, No, No, pairwise_only_truncation_not_exact_when_higher_intersections_contribute)
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).
Reviewed step-by-step derivation
Inclusion-exclusion principle · qf-mod39-incexc-pairwise-only-boundary · candidate
99. A Pairwise Truncation May Coincide Numerically without Becoming a General Exact Formula
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).
The final No withholds a general pairwise-only exactness claim; it does not negate the separately computed numerically_equal flag. The truncated lower bound may be negative.
View answer and explanation
(3, 5, No, No, pairwise_only_truncation_not_exact_when_higher_intersections_contribute)
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).
Reviewed step-by-step derivation
Inclusion-exclusion principle · qf-mod39-incexc-pairwise-only-boundary · candidate
100. A Pairwise Truncation May Coincide Numerically without Becoming a General Exact Formula
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).
The final No withholds a general pairwise-only exactness claim; it does not negate the separately computed numerically_equal flag. The truncated lower bound may be negative.
View answer and explanation
(3, 5, No, No, pairwise_only_truncation_not_exact_when_higher_intersections_contribute)
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).
Reviewed step-by-step derivation
Inclusion-exclusion principle · qf-mod39-incexc-pairwise-only-boundary · candidate
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