Systems of equations of the form X_i=φ_i(X₁,...,X_n) (1≤i≤n) are considered, in which the unknowns are sets of natural numbers. Expressions φ_i may contain the operations of union, intersection and elementwise addition S + T = {m + n {pipe} m ε S, n ε T}. A system with an EXPTIME-complete least solution is constructed in the paper through a complete arithmetization of EXPTIME-completeness. At the same time, it is established that least solutions of all such systems are in EXPTIME. The general membership problem for these equations is proved to be EXPTIME-complete. Among the consequences of the result is EXPTIME-completeness of the compressed membership problem for conjunctive grammars.