Proper covers are defined as covers of a set 
which do not contain the entire set 
itself as a subset (Macula 1994). Of the five covers of 
, namely 
, 
, 
, 
, and 
, only 
does not contain the subset 
and so is the unique proper cover of two elements. In
general, the number of proper covers for a set of 
elements is
 |  |  |
(1)
|
 |  | ![[1/2sum_(k=0)^(N)(-1)^k(N; k)2^(2^(N-k))]-(2^(2^N))/4,](/images/equations/ProperCover/Inline17.svg) |
(2)
|
the first few of which are 0, 1, 45, 15913, 1073579193, ... (OEIS A007537).
