ELMO SL 2011 C1

Let $S$ be a finite set, and let $F$ be a family of subsets of $S$ such that

a) If $A\subseteq S$ , then $A\in F$ if and only if $S\setminus A\notin F$ ;

b) If $A\subseteq B\subseteq S$ and $B\in F$ , then $A\in F$ .

Determine if there must exist a function $f:S\to\mathbb{R}$ such that for every $A\subseteq S$ , $A\in F$ if and only if $$\sum_{s\in A}f(s)<\sum_{s\in S\setminus A}f(s).$$

Evan O'Dorney.