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).
