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Proper Cover

Proper covers are defined as covers of a set X which do not contain the entire set X itself as a subset (Macula 1994). Of the five covers of {1,2}, namely {{1},{2}}, {{1,2}}, {{1},{1,2}}, {{2},{1,2}}, and {{1},{2},{1,2}}, only {{1},{2}} does not contain the subset {1,2} and so is the unique proper cover of two elements. In general, the number of proper covers for a set of N elements is

|C^'(N)|=|C(N)|-1/42^(2^N)
(1)
=[1/2sum_(k=0)^(N)(-1)^k(N; k)2^(2^(N-k))]-(2^(2^N))/4,
(2)

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

See also

Cover, Minimal Cover

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References

Macula, A. J. "Covers of a Finite Set." Math. Mag. 67, 141-144, 1994.Sloane, N. J. A. Sequence A007537/M5287 in "The On-Line Encyclopedia of Integer Sequences."

Referenced on Wolfram|Alpha

Proper Cover

Cite this as:

Weisstein, Eric W. "Proper Cover." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/ProperCover.html

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