Solution
= Solution
For every $S\subseteq V$ and $x\notin S$, including $S=\varnothing$, the <marginal increment> is
$$
\Delta_x f(S)=\max\{0,w(x)-f(S)\}.
$$
For the <empty set>, this uses $w(x)\geq f(\varnothing)=\min_{v\in V}w(v)$. The <monotone set function> property from (i) gives $f(A)\leq f(B)$ when $A\subseteq B$. Consequently
$$
\max\{0,w(x)-f(A)\}\geq\max\{0,w(x)-f(B)\}.
$$
Thus <diminishing returns> holds, so \b[$f$ is a <submodular set function>] by (b).