Search results
Results from the WOW.Com Content Network
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.)
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
In mathematics, the Dirichlet function [1] [2] is the indicator function of the set of rational numbers, i.e. () = if x is a rational number and () = if x is not a rational number (i.e. is an irrational number).
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 .
Dirichlet function: is an indicator function that matches 1 to rational numbers and 0 to irrationals. It is nowhere continuous. Thomae's function: is a function that is continuous at all irrational numbers and discontinuous at all rational numbers. It is also a modification of Dirichlet function and sometimes called Riemann function.
A simple function can be written in different ways as a linear combination of indicator functions, but the integral will be the same by the additivity of measures. Some care is needed when defining the integral of a real-valued simple function, to avoid the undefined expression ∞ − ∞ : one assumes that the representation
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 ...
In mathematics, a function on the real numbers is called a step function if it can be written as a finite linear combination of indicator functions of intervals. Informally speaking, a step function is a piecewise constant function having only finitely many pieces. An example of step functions (the red graph).