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.
(In reply to solution
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....
This is a very weak argument:
If I meet the requirements for membership in Rotary that does not imply that I am a member.