Definition:Characteristic Function (Set Theory)/Relation

From ProofWiki
Jump to navigation Jump to search

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

Definition

The concept of a characteristic function of a subset carries over directly to relations.


Let $\mathcal R \subseteq S \times T$ be a relation.

The characteristic function of $\mathcal R$ is the function $\chi_{\mathcal R}: S \times T \to \set {0, 1}$ defined as:

$\map {\chi_{\mathcal R} } {x, y} = \begin {cases} 1 & : \tuple {x, y} \in \mathcal R \\ 0 & : \tuple {x, y} \notin \mathcal R \end{cases}$


It can be expressed in Iverson bracket notation as:

$\map {\chi_{\mathcal R} } {x, y} = \sqbrk {\tuple {x, y} \in \mathcal R}$


More generally, let $\displaystyle \mathbb S = \prod_{i \mathop = 1}^n S_i = S_1 \times S_2 \times \ldots \times S_n$ be the cartesian product of $n$ sets $S_1, S_2, \ldots, S_n$.

Let $\mathcal R \subseteq \mathbb S$ be an $n$-ary relation on $\mathbb S$.

The characteristic function of $\mathcal R$ is the function $\chi_{\mathcal R}: \mathbb S \to \set {0, 1}$ defined as:

$\map {\chi_{\mathcal R} } {s_1, s_2, \ldots, s_n} = \begin {cases} 1 & : \tuple {s_1, s_2, \ldots, s_n} \in \mathcal R \\ 0 & : \tuple {s_1, s_2, \ldots, s_n} \notin \mathcal R \end {cases}$


It can be expressed in Iverson bracket notation as:

$\map {\chi_{\mathcal R} } {s_1, s_2, \ldots, s_n} = \sqbrk {\tuple {s_1, s_2, \ldots, s_n} \in \mathcal R}$