Definition:Decomposable
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Definition
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:
- $(1):\quad \displaystyle S = \bigcup_{k=1}^m \phi_k \left({A_k}\right)$
- $(2):\quad \forall i \ne j: \phi_i \left({A_i}\right) \cap \phi_j \left({A_j}\right) = \varnothing$
Such a union is known as a decomposition.
Point out that this is a partition. I'd do it myself but I'm working on something else at the moment.
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Also see
- Irreducible: describes a set which can not be decomposed.
