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In many cases, such as order theory, the inverse of the indicator function may be defined. This is commonly called the generalized Möbius function, as a generalization of the inverse of the indicator function in elementary number theory, the Möbius function. (See paragraph below about the use of the inverse in classical recursion theory.)
Example [ edit ] If S is the set of natural numbers 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 indicators have some special properties. For example, the following statements are all true for an indicator function that is trigonometrically convex at least on an interval (,): [1]: 55–57 [2]: 54–61
Indicator function – Mathematical function characterizing set membership; Linear discriminant function – Method used in statistics, pattern recognition, and other fields; Multicollinearity – Linear dependency situation in a regression model; One-hot – Bit-vector representation where only one bit can be set at a time
In the field of mathematics known as convex analysis, the characteristic function of a set is a convex function that indicates the membership (or non-membership) of a given element in that set. It is similar to the usual indicator function , and one can freely convert between the two, but the characteristic function as defined below is better ...
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Show that you might use it to multiply against an existing function to extract a piece of it, like f(x) * I[0,1] gives you the function from [0,1]. I'm personally wary of editing the article myself. Perhaps the term indicator function is a shorthand for set indicator function, because the function f specifies or indicates the set { x | f(x) = 1 }.
Although seemingly ill-defined, derivatives of the indicator function can formally be defined using the theory of distributions or generalized functions: one can obtain a well-defined prescription by postulating that the Laplacian of the indicator, for example, is defined by two integrations by parts when it