## Abstract

If*μ* is a positive measure, and*A*
_{2}, ...,*A*
_{
n
} are measurable sets, the sequences*S*
_{0}, ...,*S*
_{
n
} and*P*
_{[0]}, ...,*P*
_{[n]} are related by the inclusion-exclusion equalities. Inequalities among the*S*
_{
i
} are based on the obvious*P*
_{[k]}≧0. Letting\(M_k = S_k /\left( {\begin{array}{*{20}c} n \\ k \\ \end{array} } \right)\)=the average average measure of the intersection of*k* of the sets*A*
_{
i
}, it is shown that (−1)^{k}
*Δ*
^{k}
*M*
_{
i
}≧0 for*i*+*k*≦*n*. The case*k*=1 yields Fréchet’s inequalities, and*k*=2 yields Gumbel’s and K. L. Chung’s inequalities. Generalizations are given involving*k*-th order divided differences. Using convexity arguments, it is shown that for*S*
_{0}=1,\(\left( {\begin{array}{*{20}c} {S_1 } \\ N \\ \end{array} } \right)\) when*S*
_{1}≧*N*−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*<*N*≦*n* and*v*=0, 1, .... Asymptotic results as*n* → ∞ are obtained. In particular it is shown that for fixed*N*,\(\sum\limits_{i = 0}^N {a_i M_i } \geqq 0\) for all sequences*M*
_{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]
Kai Lai Chung, On the Probability of the Occurrence of at Least

*m*Events Among*n*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**and**942**, Paris, 1940. - [4]
E. Parzen,

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

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### Cite this article

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

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### AMS subject classification (1980)

- 05 A 20
- 05 A 05