Indicator Function

Definition

Indicator Function

Let . The indicator function, or characteristic function, of is the function defined by

It represents membership in numerically. Common notations are , and ; is the Greek letter chi.

Property or condition

For a property , write when holds and otherwise. Equivalently, indicates the set .

For a condition , notation such as or likewise equals when is true and otherwise.

Complement

The opposite condition is represented by

If is used for , this convention must be stated explicitly; it is not standard indicator notation.

Example

Membership

For , we have and .

Selecting terms in a sum

The indicator removes every term whose condition is false:

An indicator is not the selected value

Let . Then , not . To retain the value when , multiply by the indicator: