The inclusion-exclusion formula expresses the probability of a union of $n$ events as an alternating sum involving all possible intersections — a sum with $2^n - 1$ terms that quickly becomes impractical to compute. The Bonferroni inequalities say that truncating this alternating sum always produces a valid bound: stopping after an odd number of terms gives an upper bound, and stopping after an even number gives a lower bound. The simplest case, truncating at the first term, recovers the union bound ([Countable Subadditivity](/theorems/1108)). These inequalities are a powerful tool whenever exact inclusion-exclusion is too expensive. In combinatorics, they provide the rigorous backbone of sieve methods — for instance, the Bonferroni inequalities are what make the "at most $k$ intersections" truncations of the sieve of Eratosthenes valid. In probability, the second-order lower bound $\mathbb{P}(\bigcup A_i) \ge \sum \mathbb{P}(A_i) - \sum_{i<j} \mathbb{P}(A_i \cap A_j)$ is frequently used to show that a union of events has positive probability when the pairwise overlaps are small — a technique that appears throughout the probabilistic method in combinatorics and in the second-moment method.