There is a club called the Exclusive Club. Somebody is a member of this club if and only if he has not shaved anybody who has shaved him. In other words, X is a member of the Exclusive Club if and only if there is no Y such that X shaves Y and Y shaves X.

A barber once claimed that he had shaved every member of the Exclusive Club and nobody else. Show that the barber's claim cannot be true.

Either the barber is or is not a member of the club.

  If he is a member, then that means he shaved himself, which would mean he could not be a member of the club.

  If he is not a member of the club, then there are 4 possibilities for who shaves the barber:
1) himself: a contradiction of him only shaving club members
2) A club member: then that would contradict the requirement for that person being a member
3) A non club member: then the barber meets the requirements for membership and thus must be a member, contradicting his not being a member.
4) nobody: again, the barber would then meet the requirements for membership and thus must be a member, contradicting his not being a member

Since a contradiction is reach in every scenario, then the only conclusion is that the barber is not telling the truth.

Posted by Daniel on 2013-07-01 18:15:22