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 one way of partitioning the definition domain of a set-valued function $F\colon\mathcal P^n(\mathbf r)\to\mathcal P^n(\mathbf r)$ into equivalence classes with respect to an equivalence relation generated by $F$ so that on these classes exists a Boolean function $f$ equal to $F$, and investigate this equivalence relation for some values of $n$ and $r$.