Let $G$ be a loopless graph with no isolated vertices. Let $X$ be a largest matching in $G$ and let $Y$ be a smallest set of edges of $G$ so that every vertex of $G$ is incident with at least 1 edges in $Y$. How can we prove that $|X| + |Y|=|V(G)|$?
My rigor is a little clouded here. If you could show me all the steps in the proof that would be great. I am unable to conceive a picture of this. I think that is causing most of my confusion.