principle of inclusion and exclusion | sieve formula

Let left bracketing bar A right bracketing bar denote the cardinal number of set A, then it follows immediately that left bracketing bar A union B right bracketing bar = left bracketing bar A right bracketing bar + left bracketing bar B right bracketing bar - left bracketing bar A intersection B right bracketing bar , where union denotes union, and intersection denotes intersection. The more general statement left bracketing bar union _(i = 1)^N E_i right bracketing bar <= sum_(i = 1)^N left bracketing bar E_i right bracketing bar , also holds, and is known as Boole's inequality or one of the Bonferroni inequalities.

Bayes' theorem | Bonferroni inequalities