In the set of functions $F\colon\mathcal P^n(\mathbf r)\to\mathcal P(\mathbf r)$ the subset of Boolean functions is not complete. We study the ways of partitioning the definition domain $\mathcal P^n(\mathbf r)$ of a set-valued function $F$ into equivalence classes with respect to equivalence relations generated by $F$ so that on these classes a Boolean function $f$ equal to $F$ exists.