bibkey: berman1972inclusion authors: Gerald Berman and K. D. Fryer year: 1972 title: ‘The Inclusion-Exclusion Principle’ doi: 10.1016/b978-0-12-092750-0.50008-9 claim: The indicator of a finite union is the alternating sum over nonempty subsets of the indicators of the corresponding intersections. strata_touched:

• D5/S0/Asymptotics/WeightedProbability/FiniteInclusionExclusion license: citation-only triage: anchor

The Inclusion-Exclusion Principle

Berman and Fryer present the classical finite inclusion-exclusion identity. For a finite family of events, the indicator of their union is the alternating sum over nonempty subfamilies of the indicator of their intersection. Applying this identity pointwise and summing against any finite weight function yields the repository’s exact weighted capture identity. No nonnegativity or normalization is needed for that linear identity.

The repository-specific complement bridge additionally assumes normalized marginals so that the frozen product sample weights sum to one. Nonnegativity enters only when the first two cardinality truncations are compared with the full sum through the frozen Bonferroni bounds.

Search log

• 2026-08-15: Queried Crossref for inclusion exclusion principle. The result for DOI 10.1016/b978-0-12-092750-0.50008-9 identified Gerald Berman and K. D. Fryer, the exact chapter title, the 1972 publication year, and the containing book Introduction to Combinatorics.
• 2026-08-15: Searched pinned Mathlib for inclusion.?exclusion|inclusion_exclusion|poincare. The exact pointwise theorem Finset.indicator_biUnion_eq_sum_powerset was found in Mathlib/Combinatorics/Enumerative/InclusionExclusion.lean and is applied directly by the Lean proof.
• 2026-08-15: Searched D5/ for capture.*inclusion, escape.*powerset, and powerset.*Captured. No exact weighted capture inclusion-exclusion declaration was present.

Verified locator

• DOI: https://doi.org/10.1016/b978-0-12-092750-0.50008-9