Witnessing differences without redundancies

Abstract

We show that n 1 elements suffice to witness the differences of n pairwise distinct sets, and provide sufficient conditions for an infinite family of pairwise distinct sets to have a minimal collection of elements witnessing the differences between any two of its members. By the Extensionality Axiom, the difference between two distinct sets a and b is witnessed by at least one element d such that d E a b or d E b a; in fact any element in the symmetric difference aAb = (a b) U (b a) witnesses such a difference. For that reason we say that aAb is a differentiating set for {a, b}. Since all the elements in aAb but one are redundant for that purpose, unless a'Ab is a singleton, we say that aAb is a redundant or non-minimal differentiating set for {a, b}, while for any d E aAb, {d} is an irredundant or minimal differentiating set for {a, b}. Suppose now that n pairwise distinct sets ai,. .. , an are given; how many elements do we need to witness their being different from each other? Equivalently, given a differentiating set D for {ai,... , an} , how many redundant elements are to be found in D ? Two extreme cases immediately come under attention. If a,,... ,an can be arranged into an increasing chain with respect to inclusion, or else if a,,... ,an are pairwise disjoint, then obviously we need exactly n 1 elements to witness their differences and any differentiating set for {ai, ... , an} of cardinality m has at least m n + 1 redundant elements. In general it is obvious that we need at most (n) elements to witness the differences of n pairwise distinct sets a1,... ,an . However (n) is by far an excessively large bound; in this note we offer an extremely simple proof that n 1 elements always suffice to witness the differences among n distinct sets (see Proposition 1). For an earlier proof of this result in the special case in which the n sets are subsets of an n-elements domain see [Bon72, Bol86]. Even from the first rough estimate, it is clear that in the case of finitely many pairwise distinct sets a,, ... , an , an irredundant differentiating set can be obtained from any finite differentiating set by suppressing one after the other the elements which are redundant and remain so as the procedure goes on. It is quite natural to enquire whether that holds also for infinite families of pairwise distinct sets. Any sequence of sets densely ordered with respect to inclusion readily provides an example of a family of pairwise distinct sets for which no minimal differentiating set can exist (see Proposition 2 below). However, by making an essential use of the Received by the editors February 7, 1994 and, in revised form, August 28, 1995. 1991 Mathematics Subject Classification. Primary 03E05; Secondary 03E25. This work has been supported by funds MURST 40% and 60% of Italy. (?)1997 American Mathematical Society