# Inclusion-exclusion inequalities

## Abstract

Ifμ is a positive measure, andA 2, ...,A n are measurable sets, the sequencesS 0, ...,S n andP [0], ...,P [n] are related by the inclusion-exclusion equalities. Inequalities among theS i are based on the obviousP [k]≧0. Letting$$M_k = S_k /\left( {\begin{array}{*{20}c} n \\ k \\ \end{array} } \right)$$=the average average measure of the intersection ofk of the setsA i , it is shown that (−1)k Δ k M i ≧0 fori+kn. The casek=1 yields Fréchet’s inequalities, andk=2 yields Gumbel’s and K. L. Chung’s inequalities. Generalizations are given involvingk-th order divided differences. Using convexity arguments, it is shown that forS 0=1,$$\left( {\begin{array}{*{20}c} {S_1 } \\ N \\ \end{array} } \right)$$ whenS 1N−1, and$$\left( {\begin{array}{*{20}c} \nu \\ {k - 1} \\ \end{array} } \right)S_N \geqq \left( {\begin{array}{*{20}c} \nu \\ {N - 1} \\ \end{array} } \right)S_k + \left( {\begin{array}{*{20}c} \nu \\ N \\ \end{array} } \right)\left( {\begin{array}{*{20}c} \nu \\ {k - 1} \\ \end{array} } \right) - \left( {\begin{array}{*{20}c} v \\ {N - 1} \\ \end{array} } \right)\left( {\begin{array}{*{20}c} \nu \\ k \\ \end{array} } \right)$$ for 1≦k<Nn andv=0, 1, .... Asymptotic results asn → ∞ are obtained. In particular it is shown that for fixedN,$$\sum\limits_{i = 0}^N {a_i M_i } \geqq 0$$ for all sequencesM 0, ...,M n of sufficiently large length if and only if$$\sum\limits_{i = 0}^N {a_i t^i } > 0$$ for 0<t<1.

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## References

1. [1]

Kai Lai Chung, On the Probability of the Occurrence of at Leastm Events Amongn Arbitrary Events,Ann. Math. Stat. 12 (1941), 328–38.

2. [2]

W. Feller,An Introduction to Probability Theory and its Applications, Vol. 1, 2nd Ed., John Wiley & Sons, Inc., N. Y., 1957.

3. [3]

M. Fréchet, Les probabilités associées à un système d’événements compatibles et dépendants,Actualités scientifiques et industrielles, 859 and942, Paris, 1940.

4. [4]

E. Parzen,Modern Probability Theory and Its Applications. John Wiley & Soms, Inc., New York, 1960.

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Hausner, M. Inclusion-exclusion inequalities. Combinatorica 5, 215–225 (1985). https://doi.org/10.1007/BF02579364