Let P(E_i) be the probability that E_i is true, and P( union _(i = 1)^n E_i) be the probability that at least one of E_1, E_2, ..., E_n is true. Then "the" Bonferroni inequality, also known as Boole's inequality, states that P( union _(i = 1)^n E_i)<= sum_(i = 1)^n P(E_i), where union denotes the union. If E_i and E_j are disjoint sets for all i and j, 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."
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