Let $G$ be a bipartite graph with vertex classes $X$ and $Y$. State Hall's necessary condition for $G$ to have a matching from $X$ to $Y$, and prove that it is sufficient.

Deduce a necessary and sufficient condition for $G$ to have $|X|-d$ independent edges, where $d$ is a natural number.

Show that the maximum size of a set of independent edges in $G$ is equal to the minimum size of a subset $S \subset V(G)$ such that every edge of $G$ has an end vertex in $S$.