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Find max of Im(f)

Source: Romanian TST 3 2007, Problem 1

May 23, 2007
functioninequalitiesalgebradomaincombinatorics proposedcombinatorics

Problem Statement

Let F\mathcal{F} be the set of all the functions f:P(S)⟶Rf : \mathcal{P}(S) \longrightarrow \mathbb{R} such that for all X,Y⊆SX, Y \subseteq S, we have f(X∩Y)=min⁡(f(X),f(Y))f(X \cap Y) = \min (f(X), f(Y)), where SS is a finite set (and P(S)\mathcal{P}(S) is the set of its subsets). Find max⁡f∈F∣Im(f)∣.\max_{f \in \mathcal{F}}| \textrm{Im}(f) |.
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