Let A1,...,Ak be k subsets of the set V={1, ..., n}. Find a necessary and a sufficient condition for the existance of a representative susbset B, meaning that for every subset there is a marked representative that belongs to it, a representative can't represent two subset that he is an element of, but it is possible that a subset has more than one element in the representative subset.
for example: A1={1,2}, A2={2,3},A3={3,4}, A4={1,3}, B={1 (for set A1),2 (for set A2),4 (for set A3),3 (for set A4)}.
I tried to think about hall's thm but I really don't see how it satisfies hall's condition, and how to make that jump from set theory to graph theory to use the proof.
Thank you.