# Indexed Intersection Terms, Exact Unions and Pairwise Truncation Boundaries

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This question bank has corresponding bilingual step-by-step derivations. Source candidate and English ai_candidate states are retained. AI review, finite computation and Lean formal proof are labelled separately.

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](https://ocme.evemisslab.com/data/i18n/question-output-protocols.json)



## 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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0001/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0002/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0003/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0004/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0005/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0006/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0007/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0008/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0009/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0010/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0011/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0012/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0013/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0014/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0015/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0016/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0017/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0018/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0019/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0020/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0021/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0022/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0023/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0024/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0025/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0026/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0027/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0028/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0029/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0030/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0031/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0032/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0033/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0034/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0035/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0036/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0037/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0038/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0039/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0040/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0041/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0042/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0043/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0044/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0045/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0046/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0047/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0048/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0049/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0050/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0051/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0052/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0053/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0054/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0055/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0056/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0057/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0058/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0059/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0060/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0061/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0062/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0063/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0064/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0065/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0066/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0067/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0068/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0069/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0070/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0071/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0072/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0073/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0074/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0075/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0076/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0077/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0078/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0079/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0080/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0081/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0082/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0083/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0084/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0085/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0086/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0087/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0088/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0089/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0090/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0091/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0092/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0093/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0094/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0095/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0096/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0097/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0098/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0099/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r05-inclusion-exclusion-principle-0100/)

Inclusion-exclusion principle · qf-mod39-incexc-pairwise-only-boundary · candidate



[Canonical question JSONL](https://ocme.evemisslab.com/data/questions/batches/batch-20261014-advance-20261108-r05-inclusion-exclusion-principle-foundations/questions.jsonl) · [English JSONL](https://ocme.evemisslab.com/data/i18n/questions/batch-20261014-advance-20261108-r05-inclusion-exclusion-principle-foundations.en.jsonl) · [Validation report](https://ocme.evemisslab.com/data/questions/batches/batch-20261014-advance-20261108-r05-inclusion-exclusion-principle-foundations/validation-report.json)