Introduced here is a novel application of Satisfiability (SAT) to the set membership problem with specific focus on efficiently testing whether large sets contain a given element. Such tests can be greatly enhanced via the use of filters, probabilistic algorithms that can quickly decide whether or not a given element is in a given set. This article proposes SAT filters (i.e., filters based on SAT) and their use in the set membership problem. Both the theoretical advantages of SAT filters and experimental results show that this technique yields significant performance improvements over previous techniques. Specifically, a SAT filter is a filter construction that is simple yet efficient in terms of both query time and filter size; i.e., SAT filters asymptotically achieve the information-theoretic limit while providing fast querying. As well, this is the first application that makes use of the random k-SAT phase transition results and may drive research into efficient solvers for this and similar applications.