Indicator function: Difference between revisions

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[[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 to zero. That is, if {{mvar|A}} is a subset of thesome set to{{mvar|X}}, zero.one Thehas <math>\mathbf{1}_{A}(x)=1</math> if <math>x\in A,</math> and <math>\mathbf{1}_{A}(x)=0</math> otherwise, where <math>\mathbf{1}_A</math> is a common notation for the indicator function. ofOther acommon subsetnotations are <math>I_A,</math> and <math>\chi_A.</math>

{{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 may be denoted as <math>\mathbf{1}_A,</math> by <math>I_A,</math> or by <math>\chi_A.</math>

The indicator function of {{mvar|A}} is the [[Iverson bracket]] of the property of belonging to {{mvar|A}}; that is,