A group of Subsets of whose Union contains the given set (
) is
called a cover (or a Covering). A Minimal Cover is a cover for which removal of one member destroys
the covering property. There are various types of specialized covers, including proper covers, antichain covers, minimal
covers, -covers, and -covers. The number of possible covers for a set of elements is

the first few of which are 1, 5, 109, 32297, 2147321017, 9223372023970362989, ... (Sloane's A003465). The number of proper covers for a set of elements is

the first few of which are 0, 1, 45, 15913, 1073579193, ... (Sloane's A007537).

**References**

Eppstein, D. ``Covering and Packing.'' http://www.ics.uci.edu/~eppstein/junkyard/cover.html.

Macula, A. J. ``Covers of a Finite Set.'' *Math. Mag.* **67**, 141-144, 1994.

Sloane, N. J. A. Sequences
A003465/M4024
and A007537/M5287
in ``An On-Line Version of the Encyclopedia of Integer Sequences.''
http://www.research.att.com/~njas/sequences/eisonline.html and Sloane, N. J. A. and Plouffe, S.
*The Encyclopedia of Integer Sequences.* San Diego: Academic Press, 1995.

© 1996-9

1999-05-25