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Finite Set, Mapping, Subsets

Source: ILL 1970 - Problem 23.

May 24, 2011
combinatorics unsolvedcombinatorics

Problem Statement

Let EE be a finite set, PEP_E the family of its subsets, and ff a mapping from PEP_E to the set of non-negative reals, such that for any two disjoint subsets A,BA,B of EE, f(A∪B)=f(A)+f(B)f(A\cup B)=f(A)+f(B). Prove that there exists a subset FF of EE such that if with each A⊂EA \subset E, we associate a subset A′A' consisting of elements of AA that are not in FF, then f(A)=f(A′)f(A)=f(A') and f(A)f(A) is zero if and only if AA is a subset of FF.
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