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
 ▶ D382: Heaviside function
Children
 ▶ D6102: Dirichlet function ▶ D382: Heaviside function ▶ D4210: Indicator function operator ▶ D109: Signum function
Results
 ▶ R4333: Binary product of indicator functions equals indicator of intersection ▶ R1868: Composition of indicator function with set endomorphism ▶ R4568: Countable indicator partition of a random complex number ▶ R4567: Countable indicator partition of a random euclidean real number ▶ R1193: Finite product of indicator functions equals indicator of intersection ▶ R2966: Indicator function under scaling of the argument ▶ R1194: Indicator function with respect to set complement ▶ R3531: 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 $I_E$.