Let $A$ and $B$ be non empty, define $B^A = \{F|F : A \rightarrow B\}$, and $A^B = \{F|F : B \rightarrow A\}$. (Basically $A^B$ and $B^A$ are the number of functions for $F:A \rightarrow B$ and $F:B \rightarrow A$ respectively.)

Then is the following:

$B^A\,\&\, A^B$ are both countable $\implies$ $A\, \& \,B$ are both countable.

true or false.

I know that if $A\, \& \,B$ are countable then $A^B$ and $B^A$ need not be countable. But the given statement is the converse. How to determine if it is true or false?