MathDB

Denote by

\mathbb{S}

the set of all proper subsets of

\mathbb{Z}_{>0}

. Find all functions

f : \mathbb{S} \mapsto \mathbb{Z}_{>0}

that satisfy the following:\\
[color=#FFFFFF]___1. For all sets

A, B \in \mathbb{S}

we have

f(A \cap B) = \text{min}(f(A), f(B)).

[color=#FFFFFF]___2. For all positive integers

n

we have

\sum \limits_{X \subseteq [1, n]} f(X) = 2^{n+1}-1.

(Here, by a proper subset

X

\mathbb{Z}_{>0}

we mean

X \subset \mathbb{Z}_{>0}

with

X \ne \mathbb{Z}_{>0}

. It is allowed for

X

to have infinite size.) \\
Proposed by MV Adhitya, Kanav Talwar, Siddharth Choppara, Archit Manas

Problem Statement