Union of Indexed Family of Sets Equal to Union of Disjoint Sets/Corollary

Corollary to Union of Indexed Family of Sets Equal to Union of Disjoint Sets
Let $\family {E_n}_{n \mathop \in \N}$ be a countable indexed family of sets where at least two $E_n$ are distinct.

Let the countable indexed family of disjoint sets $\family {F_n}_{n \mathop \in \N}$ defined by:


 * $\ds F_k = E_k \setminus \paren {\bigcup_{j \mathop = 0}^{k \mathop - 1} E_j}$

satisfying:


 * $\ds \bigsqcup_{n \mathop \in \N} F_n = \bigcup_{n \mathop \in \N} E_n$

where $\bigsqcup$ denotes disjoint union.

The countable family $\family {F_k}_{k \mathop \in \N}$ can be constructed by:


 * $\ds F_k = \bigcap_{j \mathop = 0}^{k \mathop - 1} \paren {E_k \setminus E_j}$

Proof
We have that:


 * $\ds F_k = E_k \setminus \paren {\bigcup_{j \mathop = 0}^{k \mathop - 1} E_j}$

satisfies:


 * $\ds \bigsqcup_{n \mathop \in \N} F_n = \bigcup_{n \mathop \in \N} E_n$

from Union of Indexed Family of Sets Equal to Union of Disjoint Sets.

Then by a direct application of De Morgan's laws:


 * $\ds E_k \setminus \paren {\bigcup_{j \mathop = 0}^{k \mathop - 1} E_j} = \bigcap_{j \mathop = 0}^{k \mathop - 1} \paren {E_k \setminus E_j}$