Let $X$ a finite nonempty set. A function $f: \powerset{X} \to \R $ is submodular if \[ f(S \cup T) + f(S \cap T) \leq f(S) + f(T) \] for all $S, T \subset X$.
\[ f(S \cup T) + f(S \cap T) \leq f(S) + f(T) \]