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 $S=\{s_1,s_2,\dots ,s_n\}$, then $x_T=(x_1,x_2,\dots ,x_n)$ 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.