ThmDex – An index of mathematical definitions, results, and conjectures.
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Definition D41
Indicator function

Let $X$ be a D11: Set.
The indicator function on $X$ with respect to $E \subseteq X$ is the D992: Function $$X \to \{ 0, 1 \}, \quad x \mapsto \begin{cases} 1, \quad & x \in E \\ 0, \quad & x \in X \setminus E \end{cases}$$

Let $X$ be a D11: Set such that
 (i) $E \subseteq X$ is a D78: Subset of $X$
The indicator function on $X$ with respect to $E$ is the D218: Boolean function $$X \to \{ 0, 1 \}, \quad x \mapsto |E \cap \{ x \}|$$
Subdefinitions
 ▶ Heaviside function
Children
 ▶ Dirichlet function ▶ Heaviside function ▶ Indicator function operator ▶ Signum function
Results
 ▶ Binary product of indicator functions equals indicator of intersection ▶ Composition of indicator function with set endomorphism ▶ Countable indicator partition of a random complex number ▶ Countable indicator partition of a random euclidean real number ▶ Finite product of indicator functions equals indicator of intersection ▶ Indicator function under scaling of the argument ▶ Indicator function with respect to set complement ▶ Pointwise product with indicator function is lower bound for unsigned basic function
Conventions
 ▶ Convention 0 (Notation for indicator function) Let $X$ be a D11: Set. We denote the D41: Indicator function on $X$ with respect to $E \subseteq X$ by $I_E$.