Definition:Characteristic Function (Set Theory)/Set

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This page is about characteristic functions in the context of set theory. For other uses, see Definition:Characteristic Function.

Definition

Let $E \subseteq S$.

The characteristic function of $E$ is the function $\chi_E: S \to \set {0, 1}$ defined as:

$\map {\chi_E} x = \begin {cases} 1 & : x \in E \\ 0 & : x \notin E \end {cases}$

That is:

$\map {\chi_E} x = \begin {cases} 1 & : x \in E \\ 0 & : x \in \relcomp S E \end {cases}$

where $\relcomp S E$ denotes the complement of $E$ relative to $S$.


Support

Let $S$ be a set

Let $E \subseteq S$ be a subset.

Let $\chi_E: S \to \left\{{0, 1}\right\}$ be the characteristic function of $E$.


The support of $\chi_E$, denoted $\operatorname{supp} \chi_E$, is the set $E$. That is:

$\operatorname{supp} \chi_E = \left\{{x \in S: \chi_E \left({x}\right) = 1}\right\}$


Also denoted as

The characteristic function of $E$ can be expressed in Iverson bracket notation as:

$\map {\chi_E} x = \sqbrk {x \in E}$


Sources