Suppose $S$ is a finite set, define function $f: 2^S \rightarrow \mathbb{R}^+$, that is, a function from subsets of $S$ to positive real numbers. Also, for any disjoint $X,Y \subseteq S$ we have following three properties: $$f(X\cup Y)\le f(X)+f(Y)$$ $$f(X\cup Y) = f(\{X\})+f(Y)$$ $$f(X\cup Y) \ge f(X)+f(Y)$$
Is there any terminology to discribe $f$ if it satisfies the three properties respectively? Thanks in advance!
The second condition (assuming you meant $f(X \cup Y) = f(X) + f(Y)$) is usually called additivity (compare to $\sigma$-additivity for measures). The others might be termed "subadditivity" and "superadditivity", respectively, but I haven't seen them before.