Intuitive understanding of Hall's marriage theorem

Question

Could someone help me understand Hall's marriage theorem? I don't get it either intuitively or formally, and I certainly don't understand the proof. I do not get the number of neighbours part, and I also don't get the proof in Rosen's textbook of j and j+1. And I also don't get the explanation for the example in this site when I tried to search for an explanation for dummies. I'm not sure how you can match only 2 teams to 3 days when each day has exactly 3 winners for $m=3$ in a round-robin tournament.
Notation is certainly necessary, but if you could, please match them with English explanations, thank you.

Answer 1

Imagine you have two dudes, Bill and Ted (set A), who want to marry a lady, and the goal is for each guy to marry one lady (a matching on A). If a lady is already married to one dude, she can't marry the other.

Of the dudes, there are exactly 4 possible subsets: {}, {Bill}, {Ted}, {Bill, Ted}, i.e., the empty set, only Bill, only Ted, and Bill and Ted. Let's take the only Bill set first. In order for him to marry someone, he should want to marry at least 1 Lady, i.e. he won't marry anyone he doesn't want to. In that case, N(X) = 1 = |X| = 1 (if he only desires 1 lady). He could desire two, in which case N(X) = 2 > |X| = 1, and Hall's theorem holds. The same goes for the only Ted set.

What wouldn't work, is if, say, Bill didn't want to marry any available ladies. In that case, |X| = 1, but N(X) = 0, leading us to N(X) $\ngeq$ |X|, which means you cannot find a "matching" that would make Bill and Ted each marry a lady.

Of course, if we take the Bill and Ted set, and say Ted would be ok to marry either one of 2 ladies, we would get |X| = 2 = N(X) = 2, and that's all fine and well. But because the Bill set doesn't fulfill the given condition, we know that there cannot be a matching.

Though it becomes increasingly difficult to imagine and time consuming to verify manually if |A| > 3, this holds for any number of elements in A.