Indicator function

An indicator function assigns 1 to elements in a specified set and 0 to elements outside it. It encodes set membership as a numerical function.

Characteristic function: A function that assigns 1 to members of a set and 0 to nonmembers. This is a common alternative name for the same set-membership function.
Intersection: The set of elements shared by two or more sets. Multiplying two indicator functions gives the indicator of their intersection.
Expected value: The probability-weighted average of a random variable’s possible values. An event’s indicator has expectation equal to the event’s probability.
Heaviside step function: A function that jumps from zero to one at a threshold. It is an indicator of a half-line, while general indicators can represent arbitrary sets.
Set: A collection of distinct objects treated as a mathematical object. The specified set determines exactly where the function takes value one.
Union: The set of elements belonging to at least one of two or more sets. The maximum of two indicators represents membership in the union.
Law of large numbers: A theorem stating that sample averages converge to expected values under suitable conditions. Averages of independent event indicators estimate the event’s probability.
Sign function: A function giving the sign of a real number, usually −1, 0, or 1. It distinguishes positive and negative inputs rather than membership in an arbitrary set.
Set complement: The set of elements in a universe that are not in a given set. The indicator is zero precisely on the specified set’s complement.
Disjoint union: A union of pairwise nonoverlapping sets. Indicators add exactly when the represented sets are disjoint.

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