Abstract. Motivated by composition-closed function classes containing binary monomial polynomial functions over the field $\F_q$, we investigate subsets of residue classes modulo $q-1$ closed under the operations $(a,b,s) \mapsto as+b(1-s)$ and $a \mapsto 1-a$. By purely number theoretic means we give a necessary and sufficient condition on such a subset containing an arbitrary (but concrete) residue class modulo $q-1$.