Indicator function

In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, an indicator function or a characteristic function of a <a href="/facts/Subset/SJGERmbA">subset</a> of a <a href="/facts/Set_(mathematics)/BfucfHMq">set</a> is a <a href="/facts/Function_(mathematics)/CdReEKCh">function</a> that maps elements of the subset to one, and all other elements to zero. That is, if A is a subset of some set X, then the indicator function of A is the function 
 
 
 
 
 
 1
 
 
 A
 
 
 
 
 {\displaystyle \mathbf {1} _{A}}
 
 defined by 
 
 
 
 
 
 1
 
 
 A
 
 
 
 (
 x
 )
 =
 1
 
 
 {\displaystyle \mathbf {1} _{A}\!(x)=1}
 
 if 
 
 
 
 x
 ∈
 A
 ,
 
 
 {\displaystyle x\in A,}
 
 and 
 
 
 
 
 
 1
 
 
 A
 
 
 
 (
 x
 )
 =
 0
 
 
 {\displaystyle \mathbf {1} _{A}\!(x)=0}
 
 otherwise. Other common notations are 𝟙A and 
 
 
 
 
 χ
 
 A
 
 
 .
 
 
 {\displaystyle \chi _{A}.}

The indicator function of A is the <a href="/facts/Iverson_bracket/HUuSwKe8">Iverson bracket</a> of the property of belonging to A; that is, 

 
 
 
 
 
 1
 
 
 A
 
 
 (
 x
 )
 =
 
 [
 
  
 x
 ∈
 A
  
 
 ]
 
 .
 
 
 {\displaystyle \mathbf {1} _{A}(x)=\left[\ x\in A\ \right].}

For example, the <a href="/facts/Dirichlet_function/d9o708HR">Dirichlet function</a> is the indicator function of the <a href="/facts/Rational_number/VJQmyY05">rational numbers</a> as a subset of the <a href="/facts/Real_number/R02gw5Pb">real numbers</a>.

Indicator function open-in-new

Indicator function