Editing Inclusion–exclusion principle (section)

==Diluted inclusion–exclusion principle==
{{See also|Bonferroni inequalities}}

In many cases where the principle could give an exact formula (in particular, counting [[prime number]]s using the [[sieve of Eratosthenes]]), the formula arising does not offer useful content because the number of terms in it is excessive. If each term individually can be estimated accurately, the accumulation of errors may imply that the inclusion–exclusion formula is not directly applicable. In [[number theory]], this difficulty was addressed by [[Viggo Brun]]. After a slow start, his ideas were taken up by others, and a large variety of [[sieve theory|sieve methods]] developed. These for example may try to find upper bounds for the "sieved" sets, rather than an exact formula.

Let ''A''<sub>1</sub>, ..., ''A''<sub>''n''</sub> be arbitrary sets and ''p''<sub>1</sub>, ..., ''p''<sub>''n''</sub> real numbers in the closed unit interval {{closed-closed|0, 1}}. Then, for every even number ''k'' in {0, ..., ''n''}, the [[indicator function]]s satisfy the inequality:<ref>{{harv|Fernández|Fröhlich|Alan D.|1992|loc=Proposition&nbsp;12.6}}</ref>

: <math>1_{A_1\cup\cdots\cup A_n} \ge \sum_{j=1}^k (-1)^{j-1}\sum_{1\le i_1<\cdots<i_j\le n} p_{i_1} \dots p_{i_j} \, 1_{A_{i_1} \cap \cdots \cap A_{i_j}}.</math>