Systems of finitely many equations of the form phi(X-1, ..., X-n) = psi(X-1, ..., X-n) are considered, in which the unknowns X; are sets of natural numbers, while the expressions phi, psi may contain singleton constants and the operations of union and pairwise addition S + T = {m + n vertical bar m is an element of S, n is an element of T}. It is shown that the family of sets representable by unique (least, greatest) solutions of such systems is exactly the family of recursive (r.e., co-r.e., respectively) sets of numbers. Basic decision problems for these systems are located in the arithmetical hierarchy. The same results are established for equations with addition and intersection. (C) 2014 Elsevier Inc. All rights reserved.