MathDB

Let

S_n = \{1, \cdots, n\}

and let

f

be a function that maps every subset of

S_n

into a positive real number and satisfies the following condition: For all

A \subseteq S_n

and

x, y \in S_n, x \neq y, f(A \cup \{x\})f(A \cup \{y\}) \le f(A \cup \{x, y\})f(A)

. Prove that for all

A,B \subseteq S_n

the following inequality holds:

f(A) \cdot f(B) \le f(A \cup B) \cdot f(A \cap B)

Problem Statement