Definition:Characteristic Function (Set Theory)
This page is about Characteristic Function in the context of Set Theory. For other uses, see Characteristic Function.
Definition
Set
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 $S$.
Relation
The concept of a characteristic function of a subset carries over directly to relations.
Let $\RR \subseteq S \times T$ be a relation.
The characteristic function of $\RR$ is the function $\chi_\RR: S \times T \to \set {0, 1}$ defined as:
- $\map {\chi_\RR} {x, y} = \begin {cases} 1 & : \tuple {x, y} \in \RR \\ 0 & : \tuple {x, y} \notin \RR \end{cases}$
It can be expressed in Iverson bracket notation as:
- $\map {\chi_\RR} {x, y} = \sqbrk {\tuple {x, y} \in \RR}$
More generally, let $\ds \mathbb S = \prod_{i \mathop = 1}^n S_i = S_1 \times S_2 \times \ldots \times S_n$ be the cartesian product of $n$ sets $S_1, S_2, \ldots, S_n$.
Let $\RR \subseteq \mathbb S$ be an $n$-ary relation on $\mathbb S$.
The characteristic function of $\RR$ is the function $\chi_\RR: \mathbb S \to \set {0, 1}$ defined as:
- $\map {\chi_\RR} {s_1, s_2, \ldots, s_n} = \begin {cases} 1 & : \tuple {s_1, s_2, \ldots, s_n} \in \RR \\ 0 & : \tuple {s_1, s_2, \ldots, s_n} \notin \RR \end {cases}$
It can be expressed in Iverson bracket notation as:
- $\map {\chi_\RR} {s_1, s_2, \ldots, s_n} = \sqbrk {\tuple {s_1, s_2, \ldots, s_n} \in \RR}$
Also known as
It is also known as the indicator function, and $\map {\chi_E} x$ denoted $\map {\mathbf 1_E} x$.
Some sources, in an attempt to apply consistency to the terminology, refer to this concept as a characteristic mapping, but this term appears to be rare.
Some sources use the symbol $\phi$ to denote a characteristic function.
Also see
- Results about characteristic functions of sets can be found here.