Indicator vector

In mathematics, the indicator vector, characteristic vector, or incidence vector of a subset T of a set S is the vector ${\displaystyle x_{T}:=(x_{s})_{s\in S}}$ such that ${\displaystyle x_{s}=1}$ if ${\displaystyle s\in T}$ and ${\displaystyle x_{s}=0}$ if ${\displaystyle s\notin T.}$

If S is countable and its elements are numbered so that ${\displaystyle S=\{s_{1},s_{2},\ldots ,s_{n}\}}$, then ${\displaystyle x_{T}=(x_{1},x_{2},\ldots ,x_{n})}$ where ${\displaystyle x_{i}=1}$ if ${\displaystyle s_{i}\in T}$ and ${\displaystyle x_{i}=0}$ if ${\displaystyle s_{i}\notin T.}$

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.^[1][2][3]

An indicator vector is a special (countable) case of an indicator function.

Example

If S is the set of natural numbers ${\displaystyle \mathbb {N} }$, and T is some subset of the natural numbers, then the indicator vector is naturally a single point in the Cantor space: that is, an infinite sequence of 1's and 0's, indicating membership, or lack thereof, in T. Such vectors commonly occur in the study of arithmetical hierarchy.