Entire Functions Mapping Arbitrary Countable Dense Sets and Their Complements Onto Each Other

In W. K. Hayman's function theory problem book [2] the following problem (attributed to P. Erdos) is stated [2; p. 17, Problem 2.31]: Let A, B be two countable dense sets in the plane. Does there exist an integral function f(z) so that f(z)eB if and only if zeA? If the answer is negative, it would be desirable to have conditions on A, B when this is so. The following theorem answers the first part of the problem affirmatively for arbitrary A and B and in addition, of course, voids the second part of the problem.