A family of nonempty subsets of whose union contains the given set (and which contains no duplicated subsets) is called a cover (or covering) of . For example, there is only a single cover of , namely . However, there are five covers of , namely , , , , and .
A minimal cover is a cover for which removal of one member destroys the covering property. For example, of the five covers of , only and are minimal covers. There are various other types of specialized covers, including proper covers, antichain covers, -covers, and -covers (Macula 1994).
The number of possible covers for a set of elements are
the first few of which are 1, 5, 109, 32297, 2147321017, 9223372023970362989, ... (OEIS A003465).