In mathematics, the indicator vector, characteristic vector, or incidence vector of a

subset In mathematics, a Set (mathematics), set ''A'' is a subset of a set ''B'' if all Element (mathematics), elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they a ...

''T'' of a

set Set, The Set, SET or SETS may refer to: Science, technology, and mathematics Mathematics *Set (mathematics), a collection of elements *Category of sets, the category whose objects and morphisms are sets and total functions, respectively Electro ...

''S'' is the vector

x_T := (x_s)_

such that

x_s = 1

s \in T

and

x_s = 0

s \notin T.

If ''S'' is

countable In mathematics, a Set (mathematics), set is countable if either it is finite set, finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is ''countable'' if there exists an injective function fro ...

and its elements are numbered so that

S = \

, then

x_T = (x_1,x_2,\ldots,x_n)

where

x_i = 1

s_i \in T

and

x_i = 0

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 In mathematics, an indicator function or a characteristic function of a subset of a set is a function that maps elements of the subset to one, and all other elements to zero. That is, if is a subset of some set , then the indicator functio ...

Example

If ''S'' is the set of

natural number In mathematics, the natural numbers are the numbers 0, 1, 2, 3, and so on, possibly excluding 0. Some start counting with 0, defining the natural numbers as the non-negative integers , while others start with 1, defining them as the positive in ...

\mathbb

, and ''T'' is some subset of the natural numbers, then the indicator vector is naturally a single point in the

Cantor space In mathematics, a Cantor space, named for Georg Cantor, is a topological abstraction of the classical Cantor set: a topological space is a Cantor space if it is homeomorphic to the Cantor set. In set theory, the topological space 2ω is called "the ...

: 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 In mathematical logic, the arithmetical hierarchy, arithmetic hierarchy or Kleene–Mostowski hierarchy (after mathematicians Stephen Cole Kleene and Andrzej Mostowski) classifies certain sets based on the complexity of formulas that define th ...

Notes

{{reflist Basic concepts in set theory Vectors (mathematics and physics)