# Inclusion-exclusion inequalities

## Abstract

Ifμ is a positive measure, andA 2, ...,A n are measurable sets, the sequencesS 0, ...,S n andP , ...,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.

This is a preview of subscription content, access via your institution.

## References

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. 

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

3. 

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

4. 

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

Authors

## Rights and permissions

Reprints and Permissions

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