# Definition:Decomposable Set

A set $S \subset \R^n$ is decomposable in $m$ sets $A_1, \ldots, A_m \subset \R^n$ if there exist isometries $\phi_1, \ldots, \phi_m: \R^n \to \R^n$ such that:
$(1):\quad \displaystyle S = \bigcup_{k \mathop = 1}^m \phi_k \left({A_k}\right)$
$(2):\quad \forall i \ne j: \phi_i \left({A_i}\right) \cap \phi_j \left({A_j}\right) = \varnothing$