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 re(3): solution by Ady TZIDON)

taking the second sentence at face value I would have to disagree with you.  Let's break the sentence into two separate sentences and analyze it that way:
first, let p(x) be the statement that x satisfies the requirements for membership and m(x) be the statement that x is a member.
1) Somebody is a member of this club if he has not shaved anybody who has shaved him.  This is the same as p(x) implies m(x).
2) Somebody is a member of this club only if he has not shaved anybody who has shaved him.  This is the same as m(x) implies p(x).
thus we have necessary and sufficient conditions.

When you add the qualification that you are simply free to join would contradict #1 above.

Perhaps another example would better illustrate my point.  Consider the statement:
If today is Monday, then you are going to work.  There is no wiggle room here, if today is Monday, you are not simply free to go to work, by this statement you must go to work.  Otherwise this statement is not entirely truthful.  If you were simply free to go to work, then it would have had to be something like "If today is Monday, then you are able/free to go work."

Posted by Daniel on 2013-07-02 05:15:43