Having trouble applying Hall's Theorem

[junho]

1. Let k>0 and let A1,A2,...,An be a collection of k-subsets of finite set X such that every element of X appears in exactly k sets in the collection. Prove that |X| = n



2. Teacher provided us with the following hints:
-- Use a variation of Hall's Theorem.



3. I have no clue how to approach this problem. I don't even understand how proving |X|=n implies anything?

 

[HallsofIvy]

?? |X|= n is the conclusion you want to prove! Why should it imply anything?

"k- subsets" here means subsets of X containing exactly k members? The "overlapping" is the problem here so imaging "painting" the members of each A_i a different color! In other words, imagine that these are all distinct items. Since there are n sets, each containing k items, how many items are there altogether.

But we are told that each item in X appears in exactly k sets. In other words each item in X is counted k times in the formula above.

 

[junho]

?? |X|= n is the conclusion you want to prove! Why should it imply anything?

By that, I meant that I don't know understand what solving for |X| = n proves and therefore I can't figure out the reasoning and logic to bring me to that solution.

But thanks for the tips! I'll see if I can use it to help me figure it all out.

 

[HallsofIvy]

By that, I meant that I don't know understand what solving for |X| = n proves and therefore I can't figure out the reasoning and logic to bring me to that solution.

But thanks for the tips! I'll see if I can use it to help me figure it all out.

? It proves just what it says: that X contains n elements!

I can't think what else you could mean.

 

[junho]

? It proves just what it says: that X contains n elements!

I can't think what else you could mean.

Hahaha, well I guess that's the meaning that I missed. Sorry for my horrible math skills. The last time I took any university level math was 5 years ago. I'm just going back to school now for some upper level math courses to complete a minor in Math.

I've basically forgotten everything.