Indicator vector

In mathematics, the indicator vector, characteristic vector, or incidence vector of a <a href="/facts/Subset/SJGERmbA">subset</a> T of a <a href="/facts/Set_(mathematics)/BfucfHMq">set</a> S is the vector 
 
 
 
 
 x
 
 T
 
 
 :=
 (
 
 x
 
 s
 
 
 
 )
 
 s
 ∈
 S
 
 
 
 
 {\displaystyle x_{T}:=(x_{s})_{s\in S}}
 
 such that 
 
 
 
 
 x
 
 s
 
 
 =
 1
 
 
 {\displaystyle x_{s}=1}
 
 if 
 
 
 
 s
 ∈
 T
 
 
 {\displaystyle s\in T}
 
 and 
 
 
 
 
 x
 
 s
 
 
 =
 0
 
 
 {\displaystyle x_{s}=0}
 
 if 
 
 
 
 s
 ∉
 T
 .
 
 
 {\displaystyle s\notin T.}

If S is <a href="/facts/Countable_set/Wj4DmWPv">countable</a> and its elements are numbered so that 
 
 
 
 S
 =
 {
 
 s
 
 1
 
 
 ,
 
 s
 
 2
 
 
 ,
 …
 ,
 
 s
 
 n
 
 
 }
 
 
 {\displaystyle S=\{s_{1},s_{2},\ldots ,s_{n}\}}
 
, then 
 
 
 
 
 x
 
 T
 
 
 =
 (
 
 x
 
 1
 
 
 ,
 
 x
 
 2
 
 
 ,
 …
 ,
 
 x
 
 n
 
 
 )
 
 
 {\displaystyle x_{T}=(x_{1},x_{2},\ldots ,x_{n})}
 
 where 
 
 
 
 
 x
 
 i
 
 
 =
 1
 
 
 {\displaystyle x_{i}=1}
 
 if 
 
 
 
 
 s
 
 i
 
 
 ∈
 T
 
 
 {\displaystyle s_{i}\in T}
 
 and 
 
 
 
 
 x
 
 i
 
 
 =
 0
 
 
 {\displaystyle x_{i}=0}
 
 if 
 
 
 
 
 s
 
 i
 
 
 ∉
 T
 .
 
 
 {\displaystyle s_{i}\notin T.}

To put it more simply, the indicator vector of T is a vector with one element for each element in S, with that element being one if the corresponding element of S is in T, and zero if it is not.
An indicator vector is a special (countable) case of an <a href="/facts/Indicator_function/QEuh04NM">indicator function</a>.

Indicator vector open-in-new

Indicator vector