On Representatives of Subsets
نویسنده
چکیده
1. Let a set S of raw things be divided into m classes of n things each in two distinct ways, (a) and (6); so that there are m (a)-classes and m (6)-classes. Then it is always possible to find a set B of m things of S which is at one and the same time a C.S.R. ( = complete system of representatives) for the (a)-classes, and also a C.S.R. for the (ft)-classes. This remarkable result was originally obtained (in the form of a theorem about graphs) by D. KonigJ. In the present note we are concerned with a slightly different problem, viz. with the problem of the existence of a C.D.R. ( = complete system of distinct representatives) for a finite collection of (arbitrarily overlapping) subsets of any given set of things. The solution, Theorem 1, is very simple. From it may be deduced a general criterion, viz. Theorem 3, for the existence of a common C.S.R. for two distinct classifications of a given set; where it is not assumed, as in Konig's theorem, that all the classes have the same number of terms. Konig's theorem follows as an immediate corollary.
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تاریخ انتشار 2006