How to construct a function from a pair of possibly empty sets? I am stuck on an elementary proof on the cardinality of sets on the following point:
Given two possibly empty sets, $A$ and $B$, I need to prove the existence of any function $f:A\rightarrow B$. Is it possible? Perhaps using AC? I'm thinking you must have at least a non-empty $B$ (all elements of $A$ mapped to the same point). Any help would be appreciated.

With the issue of possibly empty sets $A, B$, you might want to read up a bit on the existence of the Empty function.

In mathematics, an empty function is a function whose domain is the empty set. For each set A, there is exactly one such empty function

$$f_A: \varnothing \rightarrow A.$$

The graph of an empty function is a subset of the Cartesian product ∅ × A. Since the product is empty the only such subset is the empty set ∅. The empty subset is a valid graph since for every x in the domain ∅ there is a unique y in the codomain A such that (x,y) ∈ ∅ × A. This statement is an example of a vacuous truth since there is no x in the domain.

For perhaps a better explanation than that provided by Wikipedia, read this earlier post: 


*

*Why is the empty function considered a
function?.

*See also: Does there exist a bijection between empty
sets?
