Indicator vector

In mathematics, the indicator vector or characteristic vector or incidence vector of a subset T of a set S is the vector [math]\displaystyle{ x_T := (x_s)_{s\in S} }[/math] such that [math]\displaystyle{ x_s = 1 }[/math] if [math]\displaystyle{ s \in T }[/math] and [math]\displaystyle{ x_s = 0 }[/math] if [math]\displaystyle{ s \notin T. }[/math] If S is countable and its elements are numbered so that [math]\displaystyle{ S = \{s_1,s_2,\ldots,s_n\} }[/math], then [math]\displaystyle{ x_T = (x_1,x_2,\ldots,x_n) }[/math] where [math]\displaystyle{ x_i = 1 }[/math] if [math]\displaystyle{ s_i \in T }[/math] and [math]\displaystyle{ x_i = 0 }[/math] if [math]\displaystyle{ s_i \notin T. }[/math]

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 [math]\displaystyle{ \mathbb{N} }[/math], 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.