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.

"If and only if" always means necessary and sufficient.

Specifically,

"Only if" means necessary

"If" means sufficient

Or, to put it differently,

a if b means b logically implies a

a only if b means a logically implies b

If the membership requirements were necessary but not sufficient, then the sentence would say:

"Somebody is a member of this club **only if** he has not shaved anybody who has shaved him."

The misleading thing about this problem is that the club is named "the Exclusive Club", when in fact it is very Inclusive. I wouldn't be surprised to learn that 99% of the people in the world belong.