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$$\sum_{A\in M} (-1)^{n-|A|}\cdot f(A)=f(S)-\max\{f(A)|A\in M, A\neq S\},$$

Source: Moldova TST 2000

August 7, 2023

function

Problem Statement

Let

S

be a finite set with

n{}

(n>1)

elements,

M{}

the set of all subsets of

S{}

and a function

f:M\rightarrow\mathbb{R}

, that verifies the relation

f(A\cap B)=\min\{f(A),f(B)\}, \forall A,B\in M

. Show that

\sum_{A\in M} (-1)^{n-|A|}\cdot f(A)=f(S)-\max\{f(A)|A\in M, A\neq S\},

where

|A|

is the number of elements of subset

A{}