Indicator function

In mathematics, an indicator function or a characteristic function is a function defined on a set $X$ that indicates membership of an element in a subset $A$ of $X$ .

Remark. The term "characteristic function" has an unrelated meaning in probability theory. For this reason, probabilists use the term indicator function for the function defined here almost exclusively, while mathematicians in other fields are more likely to use the term characteristic function to describe the function which indicates membership in a set.

The indicator function of a subset $A$ of a set $X$ is a function

\mathbf{1}_A : X \to \lbrace 0,1 \rbrace \,

defined as

\mathbf{1}_A(x) =

\left\{\begin{matrix} 1 &\mbox{if}\ x \in A, \\ 0 &\mbox{if}\ x \notin A. \end{matrix}\right.

The indicator function of $A$ is sometimes denoted

\chi_A(x)

\mathbf{I}_A(x)

or even

A(x).

(The Greek letter χ because it is the initial letter of the Greek etymon of the word characteristic.)

The Iverson bracket allows the notation $\in A$ .

Warning. The notation $\mathbf{1}_A$ may signify the identity function.

Basic properties

The mapping which associates a subset $A$ of $X$ to its indicator function $\mathbf{1}_A$ is injective; its range is the set of functions $f : X \to \{0,1\}$ .

If $A$ and $B$ are two subsets of $X$ , then

\mathbf{1}_{A\cap B} = \min\{\mathbf{1}_A,\mathbf{1}_B\} = \mathbf{1}_A \mathbf{1}_B,\,

\mathbf{1}_{A\cup B} = \max\ (-1)^{|F|} \mathbf{1}_{\bigcap_F A_k} = \sum_{\emptyset \neq F \subseteq \{1, 2, \ldots, n\}} (-1)^{|F|+1} \mathbf{1}_{\bigcap_F A_k}

where $|F|$ is the cardinality of $F$ . This is one form of the principle of inclusion-exclusion.

As suggested by the previous example, the indicator function is a useful notational device in combinatorics. The notation is used in other places as well, for instance in probability theory: if $X$ is a probability space with probability measure $\mathbb{P}$ and $A$ is a measurable set, then $\mathbf{1}_A$ becomes a random variable whose expected value is equal to the probability of $A:$

E(\mathbf{1}_A)= \int_{X} \mathbf{1}_A(x)\,dP = \int_{A} dP = P(A).\quad

This identity is used in a simple proof of Markov's inequality.

References

Folland, G.B.; Real Analysis: Modern Techniques and Their Applications, 2nd ed, John Wiley & Sons, Inc., 1999.
Thomas H. Cormen, Charles E. Leiserson, Ronald L. Rivest, and Clifford Stein. Introduction to Algorithms, Second Edition. MIT Press and McGraw-Hill, 2001. ISBN 0262032937. Section 5.2: Indicator random variables, pp.94–99.

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Basic properties

References

See also