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).