Indicator vector

In mathematics, the indicator vector or 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.^[1]^[2]^[3]

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.

Notes

↑ Mirkin, Boris Grigorʹevich (1996). Mathematical Classification and Clustering. p. 112. ISBN 0-7923-4159-7. Retrieved 10 February 2014.
↑ von Luxburg, Ulrike (2007). "A Tutorial on Spectral Clustering" (PDF). Statistics and Computing. 17 (4): 2. Retrieved 10 February 2014.
↑ Taghavi, Mohammad H. (2008). Decoding Linear Codes Via Optimization and Graph-based Techniques. ProQuest. p. 21. Retrieved 10 February 2014.

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