# Axiomatic Systems Problem solving, touch question!! will give lifesaver review!

###### Question:

The town of Greendale has an interesting club system. Each club is just a list of townspeople and no two clubs have exactly the same membership list. Here are theimportant rules regarding the clubs that are followed by every person in town and every club: a) Every two people have a club that they both belong to, but it isunique, meaning that for each pair of people, there is only one such club. b)every club has at least two members. c)No club contains all the townspeople. d)If you namea club and then a person who is not a member of that club, there will be one and only one club that person belongs to that has no members in common with the firstclub.
Now, suppose that we have a valid club system in Greendale(where there are 4 or more people). Let X be a club with memebers {x1,...xn} and let Y1 be any other clubthat has x1 as a member. We know from Axiom (a) that none of x2,x3....,xn can be in Y1. So using axiom (d), we know that for each xi(i=2,....,n), there is a club Yisuch that xi is in Yi and Yi and Y1 have no members in common.
a)Draw a picture that depicts this situation.
b)Carefully prove that every person in Greendale belongs to exactly one of the clubs Y1, Y2,...,Yn.

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