# of functions from A to B

The question of how to compute the number of functions from any given set A to any given set B came up in class, with the suggestion that it’s the cardinality of the second set to the power of the cardinality of the first set.

That indeed is correct, at least for non-deviant cases (see some links for interesting discussions on the internet, but beware of some questionable reasoning when it comes to the number of ‘onto’ functions and other more complex questions).

The deviant cases are when one of the sets is empty. In that case the only relation between A and B is the empty set. If one counts that as a function (because it vacuously satisfies the requirements for being a function), that would mean the result in the non-deviant cases is incorrect. Also 0 to the 0th power is undefined. So things don’t seem to work out for these cases in the same way…