A bipartite graph $ G(X \cup Y, E)$ has a matching that saturates X iff $|S| \leq |N(S)| \forall S \subseteq X$ , $N(S) $ is neighbour vertex set of S
I understood the meaning of this. But why are we checking all the subsets of X. Instead why can we just see on X only ie., $|X| \leq |N(X)| $as we need at least X counter vertices as neighbours in Y for all vertices in X to be included in matching. I am a beginner. Please elaborate as I am slowly understanding.