Let 
be the probability that 
is true, and 
be the probability that at least one of
,
,
..., 
is true. Then "the" Bonferroni inequality, also known as Boole's inequality,
states that
 |
where 
denotes the union. If 
and 
are disjoint sets for all
and 
,
then the inequality becomes an equality. A beautiful
theorem that expresses the exact relationship between the probability of unions and
probabilities of individual events is known as the inclusion-exclusion
principle.
A slightly wider class of inequalities are also known as "Bonferroni inequalities."
