Indicator vector

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

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

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 indicator function.

Example

If S is the set of natural numbers $\mathbb {N}$, 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.