Indicator function: Difference between revisions

Revision as of 01:06, 27 December 2021

Revision as of 01:09, 27 December 2021

Line 6:

[[Image:Indicator function illustration.png|right|thumb|A three-dimensional plot of an indicator function, shown over a square two-dimensional ___domain (set {{mvar|X}}): the "raised" portion overlays those two-dimensional points which are members of the "indicated" subset ({{mvar|A}}).]]

In [[mathematics]], an '''indicator function''' or a '''characteristic function''' of a [[subset]] of a [[Set (mathematics)|set]] is a [[Function (mathematics)|function]] that maps elements of the subset to one, and all other elements of the set to zero. The indicator function of a subset

{{mvar|A}} of a set {{mvar|X}} maps {{mvar|X}} to the two-element set <math>\{ 0, 1 \}</math>; <math>\mathbf{1}_{A}(x)=1</math> if an element <math>x</math> in {{mvar|X}} belongs to {{mvar|A}}, and <math>\mathbf{1}_{A}(x)=0</math> if <math>x</math> does not belong to {{mvar|A}}. It ismay typicallybe denoteddenoteed as <math>\mathbf{1}_A \colon X \to \{ 0</math>, 1 \}by <math>I_A</math>, or by <math>I_A\chi_A</math> to emphasize the fact that this function identifies the [[subset]] {{mvar|A}} of {{mvar|X}}.

In other contexts, such as [[computer science]], this would more often be described as a '''boolean [[Predicate (mathematical logic)|predicate]] function''' (to test set inclusion).

The [[Dirichlet function]], the indicator function of the [[rational number]]s as a subset of the [[real number]]s, is an example of an indicator function.