منابع مشابه
On Representatives of Subsets
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 original...
متن کاملOn Sums of Distinct Representatives
Clearly (1) has an SDR provided that |Ai| > i for all i = 1, · · · , n, in particular an SDR of (1) exists if |A1| = · · · = |An| > n or 0 < |A1| < · · · < |An|. Let G be an additive abelian group and A1, · · · , An its subsets. We associate any SDR (2) of (1) with the sum ∑n i=1 ai and set (4) S({Ai}i=1) = S(A1, · · · , An) = {a1 + · · ·+ an : {ai}i=1 forms an SDR of {Ai}i=1} . Of course, S(A1...
متن کاملSystems of Distinct Representatives and Linear Algebra *
So me purposes of thi s paper are: (1) To take se riously the term , " term rank. " (2) To ma ke an issue of not " rea rra nging rows a nd colu mns" by not "a rranging" the m in the firs t place. (3) To promote the nu merica l use of Cra mer 's rul e. (4) To ill us tra te that the re levance of " numbe r of s teps" to "a mount of wo rk" depends on the amount of work in a step. (5) To ca ll a tt...
متن کاملDistinct volume subsets
Suppose that a and d are positive integers with a ≥ 2. Let ha,d(n) be the largest integer t such that any set of n points in Rd contains a subset of t points for which all the nonzero volumes of the (t a ) subsets of order a are distinct. Beginning with Erdős in 1957, the function h2,d(n) has been closely studied and is known to be at least a power of n. We improve the best known bound for h2,d...
متن کاملTransversals, systems of distinct representatives, mechanism design, and matching
A transversal generated by a system of distinct representatives (SDR) for a collection of sets consists of an element from each set (its representative) such that the representative uniquely identifies the set it belongs to. Theorem 1 gives a necessary and sufficient condition that an arbitrary collection, finite or infinite, of sets, finite or infinite, have an SDR. The proof is direct, short....
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ژورنال
عنوان ژورنال: Bulletin of the American Mathematical Society
سال: 1948
ISSN: 0002-9904
DOI: 10.1090/s0002-9904-1948-09098-x