Indicator vector

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

If S is countable and its elements are numbered so that ${\displaystyle S=\{s_{1},s_{2},\ldots ,s_{n}\}}${\displaystyle S=\{s_{1},s_{2},\ldots ,s_{n}\}}, then ${\displaystyle x_{T}=(x_{1},x_{2},\ldots ,x_{n})}${\displaystyle x_{T}=(x_{1},x_{2},\ldots ,x_{n})} where ${\displaystyle x_{i}=1}${\displaystyle x_{i}=1} if ${\displaystyle s_{i}\in T}${\displaystyle s_{i}\in T} and ${\displaystyle x_{i}=0}${\displaystyle x_{i}=0} if ${\displaystyle s_{i}\notin T.}${\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.^{[1] [2] [3]}

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

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

• Basic concepts in set theory
• Vectors (mathematics and physics)