Characteristic Function of Disjoint Union/Corollary

Theorem

Let $X$ be a set.

Let $\set {D_1, D_2, \ldots, D_N}$ be a set of pairwise disjoint subsets of $X$.

Let:

$\ds D = \bigcup_{n \mathop = 1}^N D_n$

Then:

$\ds \chi_D = \sum_{n \mathop = 1}^N \chi_{D_n}$

where:

$\chi_D$ is the characteristic function of $D$

$\chi_{D_n}$ is the characteristic function of $D_n$.

Proof

We can extend $\set {D_1, D_2, \ldots, D_N}$ to a sequence $\sequence {D_n}_{n \mathop \in \N}$ of subsets of $X$ by setting:

$D_i = \O$ for $i \ge N + 1$

Clearly, from Intersection with Empty Set, we have:

$D_i \cap D_j = \O$ for $i \ge N + 1$ and all $j$.

So $\sequence {D_n}_{n \mathop \in \N}$ is a sequence of pairwise disjoint subsets of $X$ with:

$\ds D = \bigcup_{n \mathop = 1}^\infty D_n$

We therefore have, by Characteristic Function of Disjoint Union:

$\ds \chi_D = \sum_{n \mathop = 1}^\infty \chi_{D_n}$

We can write:

\(\ds \sum_{n \mathop = 1}^\infty \chi_{D_n}\)

\(=\)

\(\ds \sum_{n \mathop = 1}^N \chi_{D_n} + \sum_{n \mathop = N + 1}^\infty \chi_{D_n}\)

\(\ds \)

\(=\)

\(\ds \sum_{n \mathop = 1}^N \chi_{D_n} + \sum_{n \mathop = N + 1}^\infty \chi_\O\)

\(\ds \)

\(=\)

\(\ds \sum_{n \mathop = 1}^N \chi_{D_n} + \sum_{n \mathop = N + 1}^\infty 0\)

Characteristic Function of Empty Set

\(\ds \)

\(=\)

\(\ds \sum_{n \mathop = 1}^N \chi_{D_n}\)

$\blacksquare$