# Indicator function

In the mathematical subfield of set theory, the indicator function, or characteristic function, is a function defined on a set X which is used to indicate membership of an element in a subset A of X.

Remark. The term "characteristic function" has an unrelated meaning in probability theory; see characteristic function.

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

$\displaystyle 1_A : X \to \lbrace 0,1 \rbrace$

defined as

$\displaystyle 1_A(x) = \left\{\begin{matrix} 1 &\mbox{if}\ x \in A \\ 0 &\mbox{if}\ x \notin A \end{matrix}\right.$

The indicator function of A is sometimes denoted

$\displaystyle \ \chi_A(x)$ or $\displaystyle \ I_A(x)$ or even $\displaystyle \ A(x).$

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

The Iverson bracket allows the notation $\displaystyle [x \in A]$ .

Warning. The notation $\displaystyle 1_A$ may signify the identity function.

## Basic properties

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

If A and B are two subsets of X, then

$\displaystyle 1_{A\cap B} = \min\{1_A,1_B\} = 1_A 1_B,\,$
$\displaystyle 1_{A\cup B} = \max\{{1_A,1_B}\} = 1_A + 1_B - 1_A 1_B,$
$\displaystyle 1_{A\triangle B} = 1_A + 1_B - 2(1_{A\cap B}),$

and

$\displaystyle 1_{A^\complement} = 1-1_A.$

More generally, suppose A1, ..., An is a collection of subsets of X. For any xX,

$\displaystyle \prod_{k \in I} ( 1 - 1_{A_k}(x))$

is clearly a product of 0s and 1s. This product has the value 1 at precisely those xX which belong to none of the sets Ak and is 0 otherwise. That is

$\displaystyle \prod_{k \in I} ( 1 - 1_{A_k}) = 1_{X - \bigcup_{k} A_k} = 1 - 1_{\bigcup_{k} A_k}$

Expanding the product on the left hand side,

$\displaystyle 1_{\bigcup_{k} A_k}= 1 - \sum_{F \subseteq \{1, 2, \ldots, n\}} (-1)^{|F|} 1_{\bigcap_F A_k} = \sum_{\emptyset \neq F \subseteq \{1, 2, \ldots, n\}} (-1)^{|F|+1} 1_{\bigcap_F A_k}$

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 P and A is a measurable set, then 1A becomes a random variable whose expected value is equal to the probability of A:

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

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.