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Research Report CS-RR-253
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H. Koizumi, A. Maruoka and M.S. Paterson,
Consistency of Natural Relations on Sets
(October 1, 1993).
Abstract
Five natural relations for sets, such as inclusion, disjointness,
intersection, etc., are introduced in terms of the emptiness of the
subsets defined by Boolean combinations of the sets. Let N denote
{1,2,...,n} and (N 2) denote {(i,j) | i,j in N and i < j}. A function mu
on (N 2) specifies one of these relations for each pair of indices. Then
mu is said to be "consistent on" M, a subset of N, if and only if there
exists a collection of sets corresponding to indices in M such that
the relations specified by mu hold between each associated pair of the
sets. In this paper it is proved that if mu is consistent on all subsets
of N of size three then mu is consistent on N. Furthermore, conditions
that make mu consistent on a subset of size three are given explicitly.
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